Ultimate Guide: Mathematics: Analysis and Approaches (IB)

Unit 1: Numbers and Algebra

Toolkit

Standard form:

Some numbers are too big or too small to write easily, so we put them into standard form. In standard form, numbers are always written in the form ax10^k where

• 1≤a≤10 (there must be one non-zero digit before the decimal point)

• k ∈ ℤ (k must be an integer)

• k > 0 for large numbers (how many times a is multiplied by 10)

• k < 0 for small numbers (how many times a is divided by 10)

Laws of indices:

These allow us to simplify and manipulate expressions involving exponents. The key index laws you need to know are:

• (xy)^m = x^m x y^m

• (x/y)^m = x^m/y^m

• x^mxx^n = x^m^+^n

• x^m/x^n = x^m^-^n

• (x^m)^n = x^m^n

• x^1 = x

• x^0 = 1

• 1/x^m = x^-^n

• x^1^/^n = n√x

• x^m^/^n = n√x^m

These laws are NOT in the formula booklet.

Partial fractions:

These allow us to simplify rational expressions into the sum of two or more fractions with constant numerators, allowing for integration of rational functions.

Step 1: factorise the denominator into the product of two linear factors

Step 2: split the fraction into a sum of two fractions (use A, B and C to represent the unknown factors)

Step 3: multiply through by the denominator to eliminate fractions

Step 4: substitute values into the identity and solve for the unknown constants

Step 5: Write the original as partial fractions

Exponentials and Logarithms

Logarithms:

A logarithm is the inverse of an exponent. If a^x=b, loga b = x. Two important cases to know are:

• ln x = loge x

  • where e is the mathematical constant 2.718…

• log x = log10 x

The laws of logarithms (which are in the formula booklet) are equivalent to the laws of indices.

• loga xy = loga x + loga y

  • this relates to a^x x a^y = a^x^+^y

• loga x/y = loga x - loga y

  • this relates to a^x/a^y = a^x^-^y

• loga x^m = mloga x

  • this relates to (a^x)^y = a^x^y

Sequences and Series

Arithmetic Sequences & Series:

In an arithmetic sequence, the difference between consecutive terms is constant. This constant difference is known as the common difference, d. An arithmetic constant can be increasing (positive common difference) or decreasing (negative common difference). The nth term formula for an arithmetic sequence is given as:

An arithmetic series is the sum of all of the terms in an arithmetic sequence given by the following formulas:

Geometric Sequences & Series:

In a geometric sequence there is a common ratio, r, between consecutive terms in the sequence. the sequence can be increasing (r>1) or decreasing (0<r<1). If the common ratio is negative, the terms will alternate between being negative and being positive. The nth term formula for a geometric sequence is given as:

A geometric series is the sum of a certain number of terms in a geometric sequence given by the following formulas:

Sum to Infinity:

A geometric sequence will either increase or decrease away from zero or the terms will get progressively closer to 0 (in this case it is said to converge). If |r| doesn’t converge (i.e. is larger than or equal to 1), we cannot find the sum to infinity. This is found using the formula:

Proof

Language of Proofs:

You need to be familiar with the different sets of numbers:

• ℕ - the set of natural numbers

• ℤ - the set of integers

• ℚ - the set of quotients (rational numbers)

• ℝ - the set of real numbers

Proof by Deduction:

A proof by deduction question will often require showing that a result is true for all integers, consecutive integers or even or odd numbers.

Proof by Induction:

This is a way of proving a result is true for a set of integers by showing that if it is true for one integer than it is true for the next integer. It is important that you set this up correctly, following the steps below, to gain full marks on these questions.

Proof by Contradiction:

Proof by contradiction is a way of proving that a result is true by showing that the negation (opposite) of the result is not true. Examples of negation can be found below:

An example of a question (and the working out) are below:

Binomial Theorem

Binomial Theorem:

The binomial theorem gives a method for expanding a two-term expression raised to a power. This can be done using the formula:

Where:

We can also use the binomial theorem for fractional and negative indices by using the formula below:

Permutations and Combinations

Permutations:

A permutation is the number of possible arrangements of a set of objects when the order of the arrangement matters. You can find the value of a permutation easily using your GDC.

Combinations:

This is the number of possible arrangements of a set of objects when the order of the arrangement does not matter.

Complex Numbers

Argand Diagram:

This is also sometimes known as the complex plane and we use it to illustrate complex numbers. It has two axes, the real axis (x-axis) and imaginary axis (y-axis). Complex numbers are often represented as vectors, with an arrow away from the origin. This allows for geometrical representations of complex numbers.

Cartesian Form:

This is one of the forms that we can write complex numbers in. Cartesian form is when we write it as z = a + bi, where a, b ∈ ℝ. In general for z = a + bi, a = Re(z) and b = Im(z).

To add or subtract numbers in Cartesian Form, simplify the real and imaginary parts separately. For example:

ADDITION: (a + bi) + (c + di) = (a + c) + (b + d)i

SUBTRACTION: (a + bi) - (c + di) = (a - c) + (b - d)i

We can also multiply numbers in Cartesian form. They can be multiplied by a constant just like algebraic expressions or by each other like two linear expressions. REMEMBER: i² = -1.

MULTIPLICATION BY A CONSTANT: k(a + bi) = ka + kbi

MULTIPLICATION: (a + bi) x (c + di) = ac (ad + bc)i + bdi² = ac + (ad + bc)i - bd

It is also important that you know the value of i to higher powers

• i² = -1

• i^3 = i² x i = i x (-1) = -i

• i^4 = i² x i² = (-1) x (-1) = 1

• i^5 = i² x i³ = (-1) x (-i) = i

• i^6 = i³ x i³ = (-i) x (-i) = i² = -1

To be able to divide complex numbers, you must use the complex conjugate to change the denominator to a real number. This complex conjugate is z* = a - bi

• if z = a - bi, z* = a + bi

• z + z* is always real

  • (a + bi) + (a - bi) = 2a

• z - z* is always imaginary

  • (a + bi) - (a - bi) = 2bi

• z x z* is always real

  • (a + bi) x (a - bi) = a² + b²

Then to divide complex numbers, follow these steps:

Step 1: express the calculation in the form of a fraction

Step 2: multiply the numerator and the denominator by the conjugate of the denominator

Step 3: multiply out and simplify your answer

Step 4: write your answer in Cartesian form as two terms simplifying them if needed

Modulus and Argument:

The modulus of a complex number is its distance from the origin when plotted on an Argand diagram. The modulus of z is written as |z|. If z = x + iy, we can use Pythagoras’ Theorem to show that |z| = √x² + y². A modulus is never negative.

The modulus is related to the conjugate, z*, by:

• zz* = z*z = |z|²

  • because zz* = (x + iy)(x - iy) = x² + y²

The argument of a complex number is the angle that it makes on an Argand diagram. The angle must be taken from the positive real axis and must be measured in a counter-clockwise direction. Arguments are measured in radians and can be given in terms of π. The argument of z is written as arg z. They can be calculated using right-angled trigonometry and are usually in the range -π < arg z ≤ π.

• Negative arguments are for complex numbers in the third and fourth quadrants.

• The argument of 0, arg 0, is undefined (as no angle can be drawn).

When two complex numbers are multiplied their moduli are also multiplied and their arguments are added. When they are divided, their moduli are also divided and their arguments are subtracted.

Geometry of Complex Addition and Subtraction:

This is what addition looks like on an Argand diagram:

And this is what subtraction looks like:

The rules of the geometric representations of addition and subtraction of complex numbers (assuming w=a + bi) are:

• adding w to z results in z being translated by:

• subtracting w from z, results in z being translated by:

When we multiply by a complex number by a real number, a, there will be an enlargement of the vector by scale factor a. For positive values of a the direction of the vector will not change, only the distance of the point from the origin.

Unit 2: IB Functions

Linear Functions

Forms

• y = mx + c

           “Gradient-intercept form”/”Slope-intercept form”

             m is the gradient/slope

             c is the y-intercept

Example:

y = 2x - 1

Gradient = 2

y - intercept at y = -1

• ax + by = c

           “Standard Form”

Example:

y = (2/3)x - 1

-(2/3)x + y + 1 = 0

Answer: -2x + 3y + 3 = 0

• y - y₁ = m(x - x₁)

           “Point - slope form”

            useful for finding an equation from slope (m) and point (x₁,y₁)

Example:

Find line with slope 2 that passes through (1,3)

y - 3 = 2(x - 1)

y - 3 = 2x - 2

Answer: y = 2x + 1

Relationships

• Parallel lines: two or more lines that lie in the same plane and do not intersect. Have same slope

• Perpendicular lines: lines that intersect at a 90-degree angle

• Intersection: found by “setting equal to each other”; substitution

Example:

Intersection of y = 2x -1 & y =-x + 8

2x - 1= -x + 8

3x = 9

x = 3

Equation 1: y = 2(3) - 1

y = 5

Answer: (3,5)

Function Concepts

• Function: series of operations that will output one specific value (y) for any input (x)

Notation: y = 4x - 3 or f(x) = 4x - 3

• Domain: set of x values that can be plugged in. can be limited by division by zero, log of 0 or negative, sqrt of negative

Example:

Domain of f(x) = ln(x + 1)

x + 1 > 0

x > -1 or {xR|x > -1} or (-1,∞)

• A restricted domain can be used to make a one-to-many “function” into a one-to-one function

Example:

y = sqrt(x) is one-to-many

f(x) = sqrt(x); x0 is a function

• Range: set of y values that can be outputted

Example:

Range of g(x) = 2x² - 3

x² > = 0

g(x) > = -3 or [-3,∞)

• Inverse: reverses the effect of the original function; x and y are switched only a function if original function is one-on-one, if output of original function is plugged in, the input will be returned

1. graph is reflected over y = x

2. Notation: f^-1(x)

Example:

f(x) = 2x - 4

y = 2x - 4

x = 2y - 4

x + 4 = 2y

y = (1/2)x + 2

f^-1(x) = (1/2)x + 2

• Self-inverse: A function that is an inverse of itself. If f(x) = f^-1(x) or f(f(x)) = x

Example:

f(x) = 0.5/x is a self - inverse function

f(f(x)) = 0.5/f(x) = 0.5/(0.5/x) = 0.5*x/0.5 = x

• Even function: f(-x) = f(x)

• Odd function: f(-x) = -f(x)

Graphs

Features

• Absolute Maximum/Minimum: The highest/lowest point on a graph

• Relative Maximum/Minimum: “turning points” of the graph
  

• Intercepts: where the graph crosses the x - axis (y = 0) or the y - axis (x = 0)

• Line of symmetry: mainly quadratics, splits the graph in half
  

• Vertex: Max or min points for quadratic function

• Zero/Roots: Where y = 0, in other words, the x - intercept

• Asymptotes: A line that a graph will get infinitely close to but will never touch, as x or y tends to infinity

1. Vertical Asymptotes: x value of a restricted domain E.X. divide by 0

2. Horizontal Asymptotes: Find what the graph get closer to by plugging in very large/very small values for x
   

Composite Functions

• Composite Function: plug in a whole other function as x into a function, creating a new function of x

            Notation: (fg) (x) = f(g(x))

Example:

f(x) = 6x - 5 & g(x) = x² - x ,find (gf)(1)

f(1) = 6(1) - 5 = 1

g(f(1)) = g(1) = (1)² - 1 = 0

Composite functions can be used to check inverse functions. Inverse functions must follow:

f(f^-1(x)) = x

Quadratics

• Quadratics: polynomial with degree of 2

Form

• f(x) = ax² + bx + c

           “Standard Form”

                       y-intercept: (0, c)

                       c is known as the constant

                       Axis of symmetry: x = -b/2a

• f(x) = a(x-p)(x-q)

            “Factored Form”

                       Obtained through factoring

                        Roots: x = p & x = q

• f(x)=a(x-h)²+k

            “Vertex Form”

                       Obtained through completing square

                       Vertex: (h, k)

                        a: leading coefficient

Factorization

• Factorization: converting from standard form to factored form

            When a = 1:

1. (x+p)(x+q) = x² + (p+q)x + pq

Find numbers p and q that add up to b and multiply to c

Example:

x² + 3x + 2

p + q = 3

pq = 2

p = 1, q = 2

(x+p)(x+q) = (x+1)(x+2)

            When a1: 

2. (mx+p)(nx+q) = ax² + bx + c 

General idea: We must find m, n, p, c such that mn=a, pq=c, mq+np=b. We can do this with various methods (e.g. Star Method, Funnel Method) or trial-and-error.

Factoring by grouping: One method of factoring involving separating quadratic into groups and finding GCF

Example:

2x²+11x+12

= (2x² + 3x) + (8x + 12)   (Grouping) 

= x(2x + 3) + 4(2x + 3)    (Factor out GCF)

= (x + 4)(2x + 3)   (Combine like terms)

Solving

• Zero product property: solve either bracket equal to zero to find 2 roots

Example:

(x + 4)(2x + 3)

x + 4 = 0

x = -4

2x + 3 = 0

2x = -3

x = -3/2

Answer: x = -4 or x = -3/2

• Completing the square: converting from standard form to vertex form

ax²+bx+c=a(x-h)²+k

Example:

4x² + 20x - 24 = 0

x² + 5x - 6 = 0   (divide by a)

[x² + 5x + (5/2)²] - 6 = (5/2)²   (add [b/(2a)])

[x + (5/2)]² = (5/2)² + (24/4)   (rewrite as squared term)

[x + (5/2)]² = 49/4

x + (5/2) = (7/2)   (take the square root, positive and negative roots!)

x = -(5/2) (7/2)

x = 1 or x = -6

Generalized to derive the quadratic formula!

• Discriminant: from the quadratic formula. determines how many roots a quadratic has

∆ = b² - 4ac

If ∆ > 0, there are two distinct roots

If ∆ = 0, there is one repeated root

If ∆ < 0, there are no real roots

Hidden Quadratics

• u - substitution: replacing u with a function to make a solvable quadratic

Example:

2e^2x + 5e^x = 3

2(e^x)² + 5(e^x) - 3 = 0

u = e^x

2u² + 5u - 3 = 0

(2u - 1)(u + 3) = 0

2u - 1 = 0

2u = 1

u = 0.5

e^x = 0.5

x = ln(0.5)

u + 3 = 0

u = -3

e^x = -3 (undefined)

x = ln(0.5)

Rational functions

Rational functions: a fraction where both the numerator and the denominator are polynomials

defined by the location of its asymptotes:

f(x) = (ax+b) / (cx+d)

1. Vertical Asymptotes: finding an x value that cannot be used, i.e. that makes the denominator 0

cx + d = 0; x = -d/c

2. Horizontal Asymptotes: finding a value that the fraction never reaches, but gets infinitely close to. Plugging in a very large x value shows that the b and d become negligible. f(x) tends to ax/cx or just y = a/c

f(x) = (ax² + bx + c) / (dx + e): Degree of numerator > degree of denominator, can result in oblique asymptote

Oblique (diagonal) asymptote: Divide using long division

f(x) = (ax + b) / (cx² + dx + e): Denominator rises much faster, so horizontal asymptote is 0

Reciprocal functions: f(x) = 1/x

horizontal and vertical asymptote at 0

self-inverse, symmetrical about y = x, f^-1(x) = f(x)

Exponential and Logarithmic Functions

• Exponential Function: variable is the exponent

f(x) = a(b)^x + c

b - base

• Logarithmic Function: Inverse of exponential function

f(x) = a*log_bx + c

  b - base

Example:

f(x) = e^x

f^-1(x) = ln(x)

Transformations

Translations

y = f(x) + b: adding b to every y-coordinate, shifting up by “b” units

y = f(x) - b: subtracting b from every y-coordinate, shifting down by “b” units

y = f(x - a): taking y-coordinate from the original x that is a to the right, shift right by “a” units

y = f(x + a): taking y-coordinate from the original x that is a to the left, shift left by “a” units

Stretches

y = p * f(x): multiply every y-coordinate by p, stretching vertically by scale factor “p”

y = f(qx): taking y-coordinate from the value qx, stretching horizontally by scale factor “1/q”

Reflections

y = -f(x): change the sign of every y-coordinate, reflect over the x-axis

y = f(-x): take the y-coordinate from the -x value, reflect over the y-axis

Horizontal translations and stretches are counterintuitive

Absolute Value

y = |f(x)|: Replace any section of the graph that’s below the x-axis with a reflection across the x-axis

y = f(|x|): The portion of the graph to the right of the y-axis is reflected over the y-axis, replacing what was on the left

Polynomials

• Polynomials: general term for functions in the form f(x) = a_nxⁿ + a_n-1x^n-1 +... + a₀

• Degree: the largest exponent of a polynomial; n

Examples: cubics - 3, quadratics - 2, quartics - 4

• Remainder Theorem: If a polynomial, P(x), is divided by (x-c), the remainder is P(c)

• Factor Theorem: (x - c) is a factor of polynomial P(x) if and only if P(c) = 0

Division Methods

Synthetic division: Use this method when dividing by (x - c)

Long division: More general method, a bit more time-consuming

Roots

Sum of roots: -a_n-1/a_n

Product of roots: (-1)ⁿa₀/a_n

Example:

Quadratic

f(x) = ax² + bx + c 

Sum of roots: -b/a

Product of roots: c/a

A polynomial of degree n has exactly n roots (imaginary and real)

Conjugate roots: If z is a root, then z^* is a root (z = a+bi & z^* = a - bi)

Unit 3: Algebra

Sequences and Series

Arithmetic sequences

• In an arithmetic sequence, the difference between consecutive terms in the sequence is constant

• This constant difference is known as the common difference, d, of the sequence

The n^th term formula:

u_n ​= u₁ ​+ (n−1) × d

Where:

• u_n is the n^th term of the sequence

• u₁​ is the first term

• d is the common difference

• n is the term number

Sum of an arithmetic sequence:

S_n = (n / 2) × (u₁ + u_n)

Where:

• S_n is the sum of the first nnn terms

• u₁ is the first term

• u_n is the n^th term

• d is the common difference

• n is the number of terms

Geometric sequences

• In a geometric sequence, the ratio between consecutive terms in the sequence is constant.

• This constant ratio is known as the common ratio, r, of the sequence.

The n^th term formula:

u_n = u₁ × r^{(n - 1)}

Where:

• u_n is the nth term of the sequence

• u₁ is the first term

• r is the common ratio

• n is the term number

Sum of a geometric sequence:

S_n = u₁ × (1 - rⁿ) / (1 - r) (for r ≠ 1)

Where:

• Sn is the sum of the first n terms

• u₁ is the first term

• r is the common ratio

• n is the number of terms

For an infinite geometric sequence (when |r| < 1), the sum is:

S_∞ = u₁ / (1 - r)

Where:

• S_∞ is the sum of the infinite terms

• u₁ is the first term

• r is the common ratio

Sigma notation and recurrence relations

Sigma notation

• shorthand way of writing the sum of terms in a sequence. It uses the symbol ∑ to represent summation.

The general form is:

∑(from i = a to b) of f(i)

Where:

• i is the index, starting at a and ending at b

• f(i) is the expression to sum

• a and b are the limits of summation

For example, ∑(from i = 1 to 4) of i means 1 + 2 + 3 + 4 = 10.

Recurrence relations

• define each term in a sequence using previous terms.

A general form is:

u_n = f(u_(n-1), u_(n-2), ...)

For example, in the Fibonacci sequence, u_n = u_(n-1) + u_(n-2). Each term is the sum of the two previous terms.

Exponents and Logarithms

Laws of exponents

• The laws of exponents are rules that simplify expressions with powers.

The main laws are:

1. Product of powers: a^m × aⁿ = a^(m+n)

   • When multiplying powers with the same base, add the exponents.

2. Quotient of powers: a^m / aⁿ = a^(m-n)

   • When dividing powers with the same base, subtract the exponents.

3. Power of a power: (a^m)ⁿ = a^(m×n)

   • When raising a power to another power, multiply the exponents.

4. Power of a product: (a × b)ⁿ = aⁿ × bⁿ

   • When raising a product to a power, apply the power to each factor.

5. Power of a quotient: (a / b)ⁿ = aⁿ / bⁿ

   • When raising a quotient to a power, apply the power to both the numerator and the denominator.

6. Zero exponent: a⁰ = 1 (where a ≠ 0)

   • Any nonzero number raised to the power of 0 equals 1.

7. Negative exponent: a^(-n) = 1 / aⁿ

   • A negative exponent means the reciprocal of the base raised to the positive exponent.

Logarithm laws

The laws of logarithms are rules that simplify expressions with logarithms.

The main laws are:

1. Product rule: log_a(xy) = log_a(x) + log_a(y)

   • The logarithm of a product is the sum of the logarithms.

2. Quotient rule: log_a(x / y) = log_a(x) - log_a(y)

   • The logarithm of a quotient is the difference of the logarithms.

3. Power rule: log_a(xⁿ) = n × log_a(x)

   • The logarithm of a number raised to a power is the exponent times the logarithm of the base.

4. Change of base formula: log_a(x) = log_b(x) / log_b(a)

   • To change the base of a logarithm, divide the logarithms of the new base and the original base.

5. Logarithm of 1: log_a(1) = 0

   • The logarithm of 1 with any base is always 0.

6. Logarithm of the base: log_a(a) = 1

   • The logarithm of the base with itself is always 1.

Change of base formula

• The change of base formula allows you to convert a logarithm from one base to another.

The formula is:

log_a(x) = log_b(x) / log_b(a)

Where:

• log_a(x) is the logarithm of x with base a

• log_b(x) is the logarithm of x with base b

• log_b(a) is the logarithm of a with base b

Binomial Theorem

Expansion of (a+b)ⁿ using binomial coefficients

• The expansion of (a + b)^n using binomial coefficients is given by the Binomial Theorem.

The general form is:

(a + b)ⁿ = ∑ (from k = 0 to n) [C(n, k) × a^(n-k) × b^k]

Where:

• n is the exponent

• C(n, k) is the binomial coefficient, calculated as C(n, k) = n! / (k!(n - k)!)

• k is the index of summation, ranging from 0 to n

• a^(n-k) is the power of a

• b^k is the power of b

Pascal’s Triangle

• Triangular array of numbers that represents the binomial coefficients. Each number in the triangle is the sum of the two numbers directly above it. The nth row of Pascal's Triangle corresponds to the coefficients of the expansion of (a + b)ⁿ.

The first few rows of Pascal's Triangle are as follows:

Row 0: 1
Row 1: 1 1
Row 2: 1 2 1
Row 3: 1 3 3 1
Row 4: 1 4 6 4 1
Row 5: 1 5 10 10 5 1

How to Use Pascal's Triangle for Binomial Expansions:

• The numbers in each row are the coefficients for the terms in the binomial expansion.

• The 0th row corresponds to (a + b)⁰, the 1st row corresponds to (a + b)¹, the 2nd row corresponds to (a + b)², and so on.

For example, using the 4th row (1, 4, 6, 4, 1), we get the binomial expansion of (a + b)⁴:

(a + b)⁴ = 1 a⁴ + 4 a³ b + 6 a² b² + 4 a b³ + 1 b⁴

Functions

Definition and Types of Functions

• One-to-One Function (Injective Function)
  A function is one-to-one (injective) if each element in the domain has a distinct image in the co-domain.

  • Example: f(x) = 3x + 5 is a one-to-one function. Each input x maps to a unique output, so no two different values of x give the same result.

  • Graph: The graph of a one-to-one function will pass the horizontal line test, meaning no horizontal line intersects the graph at more than one point.

• Many-to-One Function
  A function is many-to-one if there are at least two distinct elements in the domain that map to the same element in the co-domain.

  • Example: f(x) = x² + 1 is a many-to-one function. Multiple values of x, such as 2 and -2, give the same output, 5.

  • Graph: The graph of this function will not pass the horizontal line test because horizontal lines can intersect the graph at multiple points.

• Onto Function (Surjective Function)
  A function is onto if every element in the co-domain has at least one pre-image in the domain.

  • Example: f(x) = x² is onto when the co-domain is restricted to non-negative real numbers, because every non-negative number has a corresponding x-value that maps to it.

  • Graph: The graph of an onto function will cover the entire co-domain, meaning the function’s range matches the co-domain.

• Into Function
  A function is into if there exists at least one element in the co-domain that is not an image of any element in the domain.

  • Example: f(x) = |x|, where A = B = {x: 1 ≤ x ≤ 1} and the range is [0, 1], is into because there are elements in the co-domain (negative numbers) that do not correspond to any input.

  • Graph: The graph of an into function does not cover the entire co-domain, leaving some elements without a corresponding input.

• Polynomial Function
  A polynomial function is of the form f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀, where the exponents of x are non-negative integers.

  • Example: f(x) = x² + 2x + 1 is a polynomial function of degree 2 (quadratic).

  • Graph: The graph of a polynomial function depends on the degree. For example, a quadratic function (degree 2) has a parabolic graph.

• Linear Function
  A linear function is a first-degree polynomial of the form f(x) = mx + b.

  • Example: f(x) = 2x + 1 is a linear function.

  • Graph: The graph of a linear function is always a straight line.

• Identical Function
  Two functions f and g are identical if their domains, ranges, and outputs are the same for all inputs.

  • Example: f(x) = x is an identical function because it maps every value in the domain to itself.

  • Graph: The graph of an identical function is a straight line passing through the origin (y = x).

• Quadratic Function
  A quadratic function is a second-degree polynomial of the form f(x) = ax² + bx + c, where a ≠ 0.

  • Example: f(x) = 2x² + x – 1 is a quadratic function.

  • Graph: The graph of a quadratic function is a parabola. If a > 0, it opens upwards, and if a < 0, it opens downwards.

• Algebraic Functions
  Algebraic functions involve operations like addition, subtraction, multiplication, division, and taking powers or roots of variables.

  • Example: f(x) = √(x² + 1) is an algebraic function.

  • Graph: The graph of an algebraic function depends on its specific form, but it typically involves curves, roots, and other common mathematical shapes.

• Cubic Function
  A cubic function is a third-degree polynomial of the form f(x) = ax³ + bx² + cx + d, where a ≠ 0.

  • Example: f(x) = x³ – 3x² + 2x is a cubic function.

  • Graph: The graph of a cubic function can have one or two turning points, and it can cross the x-axis up to three times.

• Modulus Function
  The modulus function is defined as f(x) = |x|, which returns the absolute value of x.

  • Example: f(x) = |x| is a modulus function.

  • Graph: The graph of a modulus function is a V-shaped curve that touches the x-axis at x = 0 and increases as x moves away from 0.

• Signum Function
  The signum function gives the sign of a real number: f(x) = 1 if x > 0, f(x) = 0 if x = 0, and f(x) = -1 if x < 0.

  • Example: f(x) = sign(x) is the signum function.

  • Graph: The graph of the signum function consists of three horizontal lines, one at 1 for positive values, one at -1 for negative values, and one at 0 for x = 0.

• Greatest Integer Function
  The greatest integer function returns the greatest integer less than or equal to a given number.

  • Example: f(x) = [x] is the greatest integer function.

  • Graph: The graph is a step function, consisting of horizontal line segments at each integer value.

• Fractional Part Function
  The fractional part function gives the fractional part of a number, defined as f(x) = x - [x].

  • Example: f(x) = {x} is the fractional part function.

  • Graph: The graph of the fractional part function consists of a series of horizontal line segments that repeat every integer.

• Even and Odd Functions
  A function is even if f(x) = f(-x), and odd if f(x) = -f(-x).

  • Example (Even): f(x) = x² is even because f(x) = f(-x).

  • Example (Odd): f(x) = x³ is odd because f(x) = -f(-x).

  • Graph: Even functions have symmetrical graphs about the y-axis, while odd functions have rotational symmetry around the origin.

• Periodic Function
  A periodic function repeats its values at regular intervals.

  • Example: f(x) = sin(x) is periodic with a period of 2π.

  • Graph: The graph of a periodic function repeats at regular intervals, like the sine or cosine wave.

• Composite Function
  A composite function is formed by combining two functions, where f(g(x)) is the composition of f and g.

  • Example: If f(x) = x² and g(x) = 2x, then f(g(x)) = (2x)² = 4x².

  • Graph: The graph of a composite function depends on the functions involved.

• Constant Function
  A constant function has the form f(x) = c, where c is a constant.

  • Example: f(x) = 5 is a constant function.

  • Graph: The graph is a horizontal line at y = c.

• Identity Function
  The identity function maps every element to itself, f(x) = x.

  • Example: f(x) = x is an identity function.

  • Graph: The graph is a straight line passing through the origin with a slope of 1.

Domain, range, and inverse functions

• Domain: The set of all possible input values (x-values) that a function can accept.

  • Example: For f(x) = √(x - 1), the domain is x ≥ 1.

• Range: The set of all possible output values (y-values) the function can produce.

  • Example: For f(x) = x², the range is y ≥ 0.

• Inverse Function: A function that "undoes" the operation of the original function. If f(x) = y, then f⁻¹(y) = x.

  • Example: If f(x) = 2x + 3, then f⁻¹(x) = (x - 3) / 2.

• Finding the Inverse:

  • Replace f(x) with y.

  • Swap x and y.

  • Solve for y.

  • Replace y with f⁻¹(x).

  • Example: For f(x) = 3x - 5, the inverse is f⁻¹(x) = (x + 5) / 3.

• Inverse Function Domain and Range:

  • The domain of the inverse is the range of the original function.

  • The range of the inverse is the domain of the original function.

  • Example: For f(x) = x² (x ≥ 0), the domain and range are both [0, ∞), and for the inverse f⁻¹(x) = √x, the domain and range are also [0, ∞).

• One-to-One and Inverse Functions: For a function to have an inverse, it must be one-to-one (injective), meaning each input maps to a unique output.

  • Example: f(x) = 2x + 3 is one-to-one and has an inverse.

  • Horizontal line test can determine if a function is one-to-one.

Composite functions

• Composite Functions: A composite function is created when one function is applied to the result of another function. It's denoted as (g ∘ f)(x), meaning g(f(x)).

• How to Find a Composite Function:

  • Take the output of the first function, f(x), and plug it into the second function, g(x).

  • The result is the composite function, (g ∘ f)(x).

  • Example: If f(x) = x + 2 and g(x) = 3x, then (g ∘ f)(x) = g(f(x)) = 3(x + 2) = 3x + 6.

• Domain of Composite Functions:

  • The domain of (g ∘ f)(x) is determined by the domain of f and the domain of g applied to the range of f.

  • Example: If f(x) = √x (domain: x ≥ 0) and g(x) = x + 1 (domain: all real numbers), the domain of the composite function is x ≥ 0.

• Notation:

  • Composite functions can also be written as f(g(x)) or g(f(x)).

  • The order matters; (g ∘ f)(x) is not necessarily the same as (f ∘ g)(x).

Transformations of Functions

Translations

• Vertical Shift: If f(x) is shifted up by k units, the new function is f(x) + k. If shifted down, it's f(x) - k.

• Horizontal Shift: If f(x) is shifted right by h units, the new function is f(x - h). If shifted left, it's f(x + h).

• Example: If f(x) = x², then f(x) + 3 shifts the graph 3 units up.

Reflections

• Reflection in the x-axis: The graph of -f(x) reflects the graph of f(x) over the x-axis.

• Reflection in the y-axis: The graph of f(-x) reflects the graph of f(x) over the y-axis.

• Example: If f(x) = x², then -f(x) = -x² reflects the graph of f(x) over the x-axis.

Stretches an compressions

• Vertical Stretch/Compression: If the function is multiplied by a factor of a (where |a| > 1), the graph is stretched vertically; if |a| < 1, it is compressed vertically.

• Horizontal Stretch/Compression: If the input is multiplied by a factor of b (where |b| > 1), the graph is compressed horizontally; if |b| < 1, it is stretched horizontally.

• Example: If f(x) = x², then 2f(x) = 2x² is a vertical stretch, and f(2x) = (2x)² is a horizontal compression.

Polynomial Functions

• Definition: A polynomial function is a mathematical expression involving a sum of powers of a variable, each multiplied by a coefficient. Its general form is:

  • f(x) = aₙxⁿ + aₙ₋₁xⁿ⁻¹ + ... + a₁x + a₀ where aₙ, aₙ₋₁, ..., a₁, a₀ are constants, and n is a non-negative integer.

• Graph Behavior:

  • The graph of a polynomial function depends on its degree (the highest power of x) and its leading coefficient (the coefficient of the highest power).

  • Odd-degree polynomials (like cubic functions) have opposite end behaviors (one side goes up, the other goes down).

  • Even-degree polynomials (like quadratic functions) have the same end behavior (both sides go up or both go down).

  • The shape of the graph is also influenced by the number of turning points (degree - 1).

Factor and remainder theorems

• Factor Theorem:

  • If x - c is a factor of a polynomial f(x), then f(c) = 0.

  • Usage: To check if x - c is a factor, substitute c into the polynomial. If the result is 0, x - c is a factor.

• Remainder Theorem:

  • When dividing a polynomial f(x) by x - c, the remainder is f(c).

  • Usage: To find the remainder of the division of f(x) by x - c, substitute c into the polynomial.

Finding roots and graphing behavior

• Finding Roots:

  • Roots (or zeros) of a polynomial function are the values of x that make the function equal to zero, i.e., when f(x) = 0.

  • To find the roots, you can factor the polynomial (if possible) or use methods like synthetic division or the quadratic formula (for second-degree polynomials).

• Graphing Behavior:

  • The end behavior of a polynomial is determined by the degree and the sign of the leading coefficient.

  • Odd-degree polynomials have opposite end behaviors (one side goes up, the other goes down).

  • Even-degree polynomials have the same end behavior (both sides go up or both go down).

  • The turning points of the graph correspond to where the function changes direction. The number of turning points is one less than the degree of the polynomial.

Rational Functions

• Definition: A rational function is the ratio of two polynomials, expressed as:

  • f(x) = P(x) / Q(x), where P(x) and Q(x) are polynomials, and Q(x) ≠ 0.

Asymptotes (vertical, horizontal, oblique)

• Vertical Asymptotes:

  • Occur where the denominator (Q(x)) is zero but the numerator (P(x)) is not zero.

  • These represent values of x where the function approaches infinity or negative infinity.

  • To find vertical asymptotes, solve Q(x) = 0 and check if P(x) ≠ 0 at those values.

• Horizontal Asymptotes:

  • Represent the end behavior of the function as x approaches ±∞.

  • If the degree of P(x) is less than the degree of Q(x), the horizontal asymptote is y = 0.

  • If the degrees of P(x) and Q(x) are equal, the horizontal asymptote is y = the ratio of the leading coefficients of P(x) and Q(x).

  • If the degree of P(x) is greater than the degree of Q(x), there is no horizontal asymptote.

• Oblique (Slant) Asymptotes:

  • Occur when the degree of P(x) is exactly one more than the degree of Q(x).

  • To find the oblique asymptote, divide P(x) by Q(x) using polynomial long division. The quotient (ignoring the remainder) gives the equation of the oblique asymptote.

Intercepts and end behavior

• Intercepts:

  • x-intercepts: Set the numerator (P(x)) equal to 0 and solve for x. These are the points where the graph crosses the x-axis.

  • y-intercepts: Set x = 0 and solve for f(0) = P(0) / Q(0). This is the point where the graph crosses the y-axis.

• End Behavior:

  • As x approaches ±∞, the function behaves according to the degrees and leading coefficients of the numerator and denominator (determining horizontal or oblique asymptotes).

  • If the degree of the numerator is less than the denominator, the function approaches 0 as x → ±∞.

  • If the degree of the numerator is greater than the denominator, the function increases or decreases without bound, depending on the sign of the leading terms.

Exponential and Logarithmic Functions

Exponential Functions

• Definition: An exponential function is of the form f(x) = a * bˣ, where:

  • a is a constant (vertical stretch/shrink).

  • b is the base (b > 0, b ≠ 1).

  • The function represents exponential growth or decay.

• Properties:

  • Domain: All real numbers (-∞, ∞).

  • Range: For b > 1, the range is (0, ∞); for 0 < b < 1, the range is (0, ∞).

  • Asymptote: Horizontal asymptote at y = 0.

  • Growth: If b > 1, the function shows exponential growth (increases rapidly as x increases).

  • Decay: If 0 < b < 1, the function shows exponential decay (decreases rapidly as x increases).

• Graph:

  • For b > 1: The graph increases and has a horizontal asymptote at y = 0.

  • For 0 < b < 1: The graph decreases and has a horizontal asymptote at y = 0.

Logarithmic Functions

• Definition: A logarithmic function is the inverse of an exponential function, written as f(x) = logₓ(x), where:

  • b is the base (b > 0, b ≠ 1).

  • The function answers the question: "To what power must b be raised to get x?"

• Properties:

  • Domain: (0, ∞), since logarithms are only defined for positive real numbers.

  • Range: All real numbers (-∞, ∞).

  • Asymptote: Vertical asymptote at x = 0.

  • Increasing Function: For b > 1, the function increases as x increases.

  • Decreasing Function: For 0 < b < 1, the function decreases as x increases.

• Graph:

  • The graph increases to the right of x = 0 and has a vertical asymptote at x = 0.

  • For b > 1: The graph rises gradually.

  • For 0 < b < 1: The graph falls gradually.

Applications to growth and decay

• Exponential Growth:

  • Formula: P(t) = P₀ * eᵏᵗ, where:

    • P(t) is the population or quantity at time t.

    • P₀ is the initial quantity.

    • k is the growth rate (k > 0).

    • e is Euler’s number (~2.718).

  • Examples: Population growth, compound interest, viral spread.

• Exponential Decay:

  • Formula: A(t) = A₀ * e⁻ᵏᵗ, where:

    • A(t) is the amount remaining at time t.

    • A₀ is the initial amount.

    • k is the decay rate (k > 0).

  • Examples: Radioactive decay, depreciation of assets, cooling of an object.

• Logarithmic Applications:

  • Decibels: Used to measure sound intensity.

  • pH scale: Measures the acidity or alkalinity of a solution, based on the logarithm of hydrogen ion concentration.

  • Richter scale: Measures the magnitude of earthquakes using a logarithmic scale.

Trigonometric Functions

Sine, Cosine, and Tangent Functions

• Sine Function:

  • Definition: f(x) = sin(x), where x is the angle in radians.

  • Graph: A smooth wave oscillating between -1 and 1, with a period of 2π.

  • Key Features:

    • Periodicity: Repeats every 2π units.

    • Amplitude: 1 (oscillates between -1 and 1).

    • X-intercepts: Occur at multiples of π.

    • Maximum value: 1; Minimum value: -1.

• Cosine Function:

  • Definition: f(x) = cos(x), where x is the angle in radians.

  • Graph: A wave similar to sine, but starts at a maximum value (1) at x = 0.

  • Key Features:

    • Periodicity: Repeats every 2π units.

    • Amplitude: 1.

    • Maximum value: 1; Minimum value: -1.

    • X-intercepts: Occur at odd multiples of π/2.

• Tangent Function:

  • Definition: f(x) = tan(x) = sin(x) / cos(x).

  • Graph: A wave with vertical asymptotes where cos(x) = 0 (at x = ±π/2, ±3π/2, etc.).

  • Key Features:

    • Periodicity: Repeats every π units.

    • No amplitude (extends to ±∞).

    • Asymptotes: Occur at odd multiples of π/2.

Transformations of sine, cosine, and tangent

Transformation of Functions:

• Vertical Shift: f(x) + c shifts the graph vertically by c units.

• Horizontal Shift: f(x - h) shifts the graph horizontally by h units.

• Amplitude Change: A * f(x) stretches or compresses the graph vertically.

• Period Change: f(bx) compresses/stretch the graph horizontally by a factor of 1/b.

Periodicity and amplitude

• Periodicity:

  • Sine, cosine, and tangent functions repeat their values in regular intervals:

    • Sine and Cosine: Period of 2π.

    • Tangent: Period of π.

• Amplitude:

  • The amplitude is the maximum height from the midline of the graph:

    • Sine and Cosine: Amplitude is |A| in A sin(x) or A cos(x).

    • Tangent: No fixed amplitude; the function has no bounds.

Reciprocal functions (cosecant, secant, cotangent)

• Cosecant (csc(x)):

  • Definition: csc(x) = 1 / sin(x).

  • Graph: Reciprocal of the sine function, with vertical asymptotes where sin(x) = 0.

• Secant (sec(x)):

  • Definition: sec(x) = 1 / cos(x).

  • Graph: Reciprocal of the cosine function, with vertical asymptotes where cos(x) = 0.

• Cotangent (cot(x)):

  • Definition: cot(x) = 1 / tan(x) = cos(x) / sin(x).

  • Graph: Reciprocal of the tangent function, with vertical asymptotes where sin(x) = 0.

Inverse Trigonometric Functions

sin⁻¹(x), cos⁻¹(x), tan⁻¹(x)

• sin⁻¹(x) (Arcsine):

  • Definition: sin⁻¹(x) is the inverse of sin(x). It returns the angle whose sine is x.

  • Domain: -1 ≤ x ≤ 1.

  • Range: -π/2 ≤ y ≤ π/2 (angles between -90° and 90°).

  • Graph: A curve starting at (-1, -π/2), passing through (0, 0), and ending at (1, π/2).

• cos⁻¹(x) (Arccosine):

  • Definition: cos⁻¹(x) is the inverse of cos(x). It returns the angle whose cosine is x.

  • Domain: -1 ≤ x ≤ 1.

  • Range: 0 ≤ y ≤ π (angles between 0° and 180°).

  • Graph: A curve starting at (-1, π), passing through (0, π/2), and ending at (1, 0).

• tan⁻¹(x) (Arctangent):

  • Definition: tan⁻¹(x) is the inverse of tan(x). It returns the angle whose tangent is x.

  • Domain: All real numbers.

  • Range: -π/2 < y < π/2 (angles between -90° and 90°).

  • Graph: A curve with horizontal asymptotes at y = -π/2 and y = π/2.

Graphs and domain/range

• Graphs:

  • The graphs of inverse trigonometric functions are reflections of the graphs of the original trigonometric functions across the line y = x.

  • They are restricted to specific domains and ranges to ensure that each x value corresponds to only one y value (this is important because functions need to be one-to-one to have an inverse).

• Domain:

  • The domain of the inverse functions is the range of the corresponding original function.

  • For sin⁻¹(x) and cos⁻¹(x), the domain is -1 ≤ x ≤ 1, as those are the possible values of sine and cosine.

  • For tan⁻¹(x), the domain is all real numbers, as tangent can take any real value.

• Range:

  • The range of the inverse functions is the domain of the corresponding original function, ensuring the angle (output) is within a specific interval:

    • sin⁻¹(x) has a range of -π/2 ≤ y ≤ π/2.

    • cos⁻¹(x) has a range of 0 ≤ y ≤ π.

    • tan⁻¹(x) has a range of -π/2 < y < π/2.

Piecewise and Absolute Value Functions

• Piecewise Functions:

  • These are functions that are defined by different expressions for different parts of their domain.

  • Example: The absolute value function can be written as:

    • f(x) = |x| =

      • x, for x ≥ 0

      • -x, for x < 0

  • The graph of a piecewise function consists of different "pieces" that join together at certain points.

• Absolute Value Functions:

  • Definition: The absolute value function outputs the distance of a number from zero, regardless of its sign.

  • Graph: A V-shaped curve, where the function is non-negative for all x. It reflects positive and negative values as positive.

  • Key Features:

    • Vertex at (0, 0).

    • Symmetric about the y-axis.

    • Domain: All real numbers.

    • Range: y ≥ 0.

Trigonometry

Trigonometric Identities

• Pythagorean Identities:

  • These identities are based on the Pythagorean theorem and relate the sine, cosine, and tangent functions:

    • sin²(x) + cos²(x) = 1

    • 1 + tan²(x) = sec²(x)

    • 1 + cot²(x) = csc²(x)

• Double-Angle and Half-Angle Identities:

  • Double-Angle Identities: These are used to express trigonometric functions of 2θ (double the angle):

    • sin(2θ) = 2sin(θ)cos(θ)

    • cos(2θ) = cos²(θ) - sin²(θ)

    • tan(2θ) = (2tan(θ)) / (1 - tan²(θ))

  • Half-Angle Identities: These are used to find trigonometric values of half of an angle:

    • sin(θ/2) = ±√[(1 - cos(θ)) / 2]

    • cos(θ/2) = ±√[(1 + cos(θ)) / 2]

    • tan(θ/2) = ±√[(1 - cos(θ)) / (1 + cos(θ))]

• Sum and Difference Formulas:

  • These formulas help simplify the trigonometric functions of the sum or difference of two angles:

    • sin(A + B) = sin(A)cos(B) + cos(A)sin(B)

    • cos(A + B) = cos(A)cos(B) - sin(A)sin(B)

    • tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))

    • sin(A - B) = sin(A)cos(B) - cos(A)sin(B)

    • cos(A - B) = cos(A)cos(B) + sin(A)sin(B)

    • tan(A - B) = (tan(A) - tan(B)) / (1 + tan(A)tan(B))

Solving Trigonometric Equations

• General Approach:

  1. To solve trigonometric equations, one often uses trigonometric identities to simplify the equation and then find the values of the angle that satisfy the equation.

  2. Example:
     Solve sin(x) = 1/2 for x in the interval [0, 2π].

     • The solutions are x = π/6 and x = 5π/6.

• Basic Steps:

  1. Simplify the equation using identities.

  2. Solve for the angle.

  3. Consider the general solution by adding 2πn to account for periodicity.

The Unit Circle and Radian Measure

• Unit Circle:

  • A unit circle is a circle with a radius of 1 centered at the origin of the coordinate plane.

  • It is used to define the trigonometric functions for all angles.

  • The angle θ corresponds to the arc length on the unit circle, and the coordinates (x, y) of a point on the circle give the values of cos(θ) and sin(θ), respectively.

• Radian Measure:

  • Radians measure angles based on the radius of the circle, with 2π radians equal to 360°.

  • Conversion between degrees and radians:

    • θ (radians) = θ (degrees) × π / 180

    • θ (degrees) = θ (radians) × 180 / π

Sine and Cosine Rule

• Sine Rule:

  • The sine rule relates the sides and angles of a triangle:

    • a / sin(A) = b / sin(B) = c / sin(C)

  • This rule is used to solve non-right triangles.

• Cosine Rule:

  • The cosine rule relates the sides and angles of a triangle:

    • c² = a² + b² - 2ab cos(C)

  • It is used to find an unknown side or angle in non-right triangles.

Applications in Geometry

• Right-Angle Triangles:

  • Trigonometry is often used to solve problems involving right-angled triangles. The sine, cosine, and tangent ratios help find unknown sides or angles.

• Non-Right Triangles:

  • The sine and cosine rules are used to solve problems involving non-right triangles (oblique triangles). These rules help find unknown sides or angles.

• Area of a Triangle:

  • The area of a triangle can be found using trigonometry:

    • Area = 1/2 × a × b × sin(C)

    • Where a and b are two sides, and C is the included angle between them.

Calculus

Limits and Continuity

Limits:

• A limit describes the behavior of a function as it approaches a specific point. It is foundational in calculus for defining derivatives and integrals.
  Notation:

  • lim (x → a) f(x) = L means as x approaches a, f(x) approaches L.

• L'Hopital's Rule

L'Hopital's Rule is a method used to evaluate limits of indeterminate forms. It applies when the limit of a function results in an indeterminate form such as 0/0 or ∞/∞. The rule states that:

If lim(x→a) f(x)/g(x) = 0/0 or ∞/∞, and the derivatives of f(x) and g(x) exist near x = a, then:

lim(x→a) [f(x)/g(x)] = lim(x→a) [f'(x)/g'(x)],

provided the limit on the right-hand side exists.

• Steps to Apply L'Hopital's Rule

1. Identify the Indeterminate Form
   Check if the limit results in 0/0 or ∞/∞.

2. Differentiate the Numerator and Denominator
   Take the derivatives of both the numerator and the denominator separately.

3. Evaluate the New Limit
   Compute the limit of the ratio of the derivatives. If the result is still an indeterminate form, you can apply L'Hopital’s Rule again.

Continuity:

• A function is continuous at a point if:

  1. The function is defined at that point.

  2. The limit exists as x approaches that point.

  3. The limit equals the function’s value at that point.

Differentiation

• Definition:
  The derivative of a function measures the rate of change of the function at any given point. It is the slope of the tangent line to the curve at that point.
  Notation:

  • f'(x) or dy/dx is the derivative of f(x).

• Rules:

  • Product Rule:

    • (d/dx)[u(x) * v(x)] = u'(x)v(x) + u(x)v'(x)

  • Quotient Rule:

    • (d/dx)[u(x) / v(x)] = (v(x)u'(x) - u(x)v'(x)) / (v(x))²

  • Chain Rule:

    • (d/dx)[f(g(x))] = f'(g(x)) * g'(x)

• Implicit Differentiation:
  Used when the function is not explicitly solved for one variable. Differentiate both sides of the equation with respect to x and solve for dy/dx.

• Higher-Order Derivatives:
  These refer to the derivatives of the derivative, like the second derivative (f''(x)) or third derivative (f'''(x)), which give information about concavity, acceleration, etc.

Applications of Differentiation

• Tangents:
  The derivative of a function at a point gives the slope of the tangent line at that point.
  Equation of tangent line:

  • y - f(a) = f'(a)(x - a)

• Normal Lines:
  A normal line is perpendicular to the tangent line. Its slope is the negative reciprocal of the tangent slope.

• Motion:
  Derivatives describe the rate of change in motion, such as velocity and acceleration.

  • Position: s(t)

  • Velocity: v(t) = s'(t)

  • Acceleration: a(t) = v'(t) = s''(t)

• Optimization:
  Derivatives are used to find maximum or minimum values of a function.

  • Critical points occur when f'(x) = 0 or does not exist.

  • Use the second derivative test to determine concavity and confirm if it’s a maximum or minimum.

Integration

• Fundamental Theorem of Calculus:

  • Part 1: If f is continuous on [a, b] and F is an antiderivative of f, then ∫[a, b] f(x)dx = F(b) - F(a).

  • Part 2: If f is continuous on [a, b], then the derivative of the integral function is f(x), i.e., d/dx ∫[a, x] f(t) dt = f(x).

• Techniques:

  • Substitution:
    Useful when the integral contains a composite function. Let u = g(x), then du = g'(x)dx, and the integral becomes easier to solve.

  • Integration by Parts:
    Based on the product rule: ∫u dv = uv - ∫v du.
    Useful when the integrand is a product of two functions.

• Definite and Indefinite Integrals:

  • Definite Integral:
    Represents the area under a curve between two points. ∫[a, b] f(x)dx gives the signed area between the curve and the x-axis from x = a to x = b.

  • Indefinite Integral:
    Represents a family of antiderivatives, written as ∫f(x)dx = F(x) + C.

Applications of Integration

• Area under a Curve:
  The integral ∫[a, b] f(x)dx gives the area between the function f(x) and the x-axis from x = a to x = b.
  For curves above and below the x-axis, break the integral into parts and take the absolute value if needed.

• Area Between Curves:
  The area between two curves is given by ∫[a, b] [f(x) - g(x)] dx, where f(x) is the upper curve and g(x) is the lower curve.

• Volume of Revolution:
  The volume of a solid formed by rotating a region about an axis is given by:

  • Disk method: V = π∫[a, b] [f(x)]² dx (for rotation about the x-axis).

  • Washer method: V = π∫[a, b] [f(x)² - g(x)²] dx (for hollow objects).

Vectors

Vector Operations

Addition and Subtraction

Vectors are added or subtracted by adding or subtracting their corresponding components.

• u + v = (u₁ + v₁, u₂ + v₂, u₃ + v₃)

• u - v = (u₁ - v₁, u₂ - v₂, u₃ - v₃)

Scalar Multiplication

A vector is multiplied by a scalar by multiplying each component of the vector by the scalar.

• k * u = (k u₁, k u₂, k u₃)

Dot Product

The dot product of two vectors is a scalar and is calculated as:

• u · v = u₁v₁ + u₂v₂ + u₃v₃
  The dot product gives the magnitude of the projection of one vector onto another.

Cross Product

The cross product of two vectors results in a vector that is perpendicular to both vectors. It is calculated as:

• u × v = (u₂v₃ - u₃v₂, u₃v₁ - u₁v₃, u₁v₂ - u₂v₁)
  The magnitude of the cross product gives the area of the parallelogram formed by the two vectors.

Equations of Lines and Planes

Parametric Form of a Line

A line in space can be represented parametrically as:

• r(t) = r₀ + t * v,
  where r₀ is a point on the line, v is the direction vector, and t is a scalar parameter.

Cartesian Form of a Line

A line can also be represented in Cartesian form as:

• (x - x₀) / a = (y - y₀) / b = (z - z₀) / c,
  where (x₀, y₀, z₀) is a point on the line and (a, b, c) is the direction vector.

Equation of a Plane

The equation of a plane can be written as:

• Ax + By + Cz = D,
  where (A, B, C) is a normal vector to the plane, and (x, y, z) are the coordinates of points on the plane.

Parametric Form of a Plane

A plane can also be written parametrically as:

• r(u, v) = r₀ + u v₁ + v v₂,
  where r₀ is a point on the plane, and v₁ and v₂ are two non-parallel vectors on the plane.

Intersection of Lines and Planes

The intersection of a line and a plane can be found by substituting the parametric equation of the line into the equation of the plane.
The result is a point where the line intersects the plane.

Applications of Vectors

Geometry Problems

Vectors are widely used to solve geometric problems, including finding distances between points, angles between lines or planes, and determining the areas and volumes of geometric shapes like triangles and parallelograms.

Motion in Three Dimensions

Vectors are used to describe motion in 3D space, including velocity and acceleration.

• Position Vector: Describes the position of a point in space.

• Velocity Vector: Describes the rate of change of position.

• Acceleration Vector: Describes the rate of change of velocity.
  Vectors help in modeling and analyzing physical systems, such as projectile motion or the movement of objects in 3D space.

Probability & Statistics

Probability Rules

Basic Probability Rules

• Addition Rule: P(A ∪ B) = P(A) + P(B) − P(A ∩ B)

• Multiplication Rule: P(A ∩ B) = P(A) P(B | A)
  For independent events, P(A ∩ B) = P(A) P(B).

Conditional Probability

Conditional probability is the probability of event A occurring given that event B has occurred.

• P(A | B) = P(A ∩ B) / P(B)
  Where P(A | B) is the conditional probability of A given B.

Bayes’ Theorem

Bayes’ Theorem relates conditional probabilities and is useful for updating probabilities with new information.

• P(A | B) = [P(B | A) * P(A)] / P(B)

Distributions

Binomial Distribution

A binomial distribution models the number of successes in a fixed number of independent trials, each with two possible outcomes (success or failure).

• P(X = k) = (n × C × k) × p^k × (1 − p)^{(n − k)}
  Where n is the number of trials, k is the number of successes, and p is the probability of success.

Normal Distribution

The normal distribution is a continuous probability distribution characterized by its bell-shaped curve.

• It is fully defined by its mean (μ) and standard deviation (σ).

• The standard normal distribution has a mean of 0 and a standard deviation of 1.

Mean, Variance, Standard Deviation

• Mean (μ): The average of a set of values.

• Variance (σ²): The average of the squared differences from the mean.

• Standard Deviation (σ): The square root of the variance, representing the spread of data points.

Correlation and Regression

Pearson’s Correlation Coefficient

Pearson’s correlation coefficient measures the linear relationship between two variables.

• r = (Σ(xi − x̄)(yi − ȳ)) / (√(Σ(xi − x̄)² * Σ(yi − ȳ)²))

• r ranges from -1 (perfect negative correlation) to +1 (perfect positive correlation), with 0 indicating no linear correlation.

Least Squares Regression Line

The least squares regression line is the line that best fits a set of data points, minimizing the sum of the squared differences between the observed values and the predicted values.

• The equation of the line is:
  y = mx + b,
  where m is the slope and b is the y-intercept.
  The slope m is calculated as:
  m = Σ[(xi − x̄)(yi − ȳ)] / Σ(xi − x̄)²
  The intercept b is:
  b = ȳ − m * x̄

Complex Numbers

Algebra of Complex Numbers

Addition, Multiplication, Division

• Addition: (a + bi) + (c + di) = (a + c) + (b + d)i

• Multiplication: (a + bi)(c + di) = (ac − bd) + (ad + bc)i

• Division: (a + bi) / (c + di) = [(a + bi)(c − di)] / (c² + d²)

Polar and Exponential Forms

Modulus and Argument

• Modulus (r): The distance of a complex number from the origin.

  • r = √(a² + b²)

• Argument (θ): The angle the complex number makes with the positive real axis.

  • θ = tan⁻¹(b / a)

Euler’s Formula

Euler's formula connects complex numbers in polar form with trigonometric functions.

• e^(iθ) = cos(θ) + i sin(θ)
  This is useful for converting between exponential and trigonometric forms of complex numbers.

De Moivre’s Theorem

De Moivre's Theorem allows for the power of a complex number in polar form.

• (r (cos θ + i sin θ))ⁿ = rⁿ (cos(nθ) + i sin(nθ))
  This is helpful for finding powers and roots of complex numbers.

Finding Roots and Powers

• To find nth roots of a complex number, use De Moivre’s Theorem.

• Root Formula:

  • r^(1/n) (cos(θ/n + 2kπ/n) + i sin(θ/n + 2kπ/n)), for k = 0, 1, 2, ..., n−1

Differential Equations

Solving First-Order Differential Equations

Separation of Variables

• Separation of variables is used when a differential equation can be written in the form:

  • (dy/dx) = f(x)g(y)

• The variables are separated:

  • (1/g(y)) dy = f(x) dx

• Both sides are integrated to solve for y.

Integrating Factor Method

• Used for linear first-order equations:

  • dy/dx + P(x)y = Q(x)

• Multiply through by the integrating factor:

  • e^(∫P(x) dx)

• This transforms the equation into an exact differential, making it easier to integrate and solve for y.

Applications in Growth and Decay

• Exponential Growth/Decay:

  • dy/dt = ky, where k is a constant (positive for growth, negative for decay).

  • Solution: y(t) = y₀ e^(kt)

• Population growth and radioactive decay are common real-life applications of these models.

Mathematical Reasoning and Proof

Direct and Indirect Proof

Direct Proof

• Direct proof involves assuming the hypothesis is true and using logical steps to directly show that the conclusion follows.

• Example: To prove "If p, then q," assume p is true and use reasoning to show q must also be true.

Indirect Proof

• Indirect proof assumes the opposite of what you want to prove and shows that this assumption leads to a contradiction.

• Example: To prove "If p, then q," assume p is true and q is false, and derive a contradiction.

Proof by Contradiction

• Proof by contradiction is a specific type of indirect proof.

• You assume that the statement you want to prove is false, and then show that this assumption leads to an inconsistency or contradiction, thereby proving the statement must be true.

• Example: To prove "There is no smallest positive rational number," assume the opposite (that such a number exists) and show this leads to a contradiction.

Unit 4: Statistics and Probability

4. 1 Basics

Sample Space

• A collection of all possible outcomes of a random experiment

• Example: a sample space contains 36 outcomes

Sample Size

• Sample size refers to the number of observations or individuals in a sample selected from a larger population. In statistical analysis and research, determining an appropriate sample size is crucial as it directly impacts the reliability and validity of the study's results and conclusions.
  The choice of sample size depends on several factors:

  1. Population Variability: Higher variability within the population usually requires a larger sample size to capture the diversity of the population accurately.

  2. Margin of Error: Smaller margins of error or narrower confidence intervals require larger sample sizes.

  3. Confidence Level: Higher confidence levels (such as 95% or 99% confidence) typically require larger sample sizes to achieve the desired precision in estimating population parameters.

  4. Desired Power: In experiments or hypothesis testing, higher statistical power often necessitates larger sample sizes to detect smaller effects reliably.

  5. Resource Constraints: Practical considerations, including time, cost, and accessibility to participants, may limit the sample size that can be obtained.

• Sample size determination involves statistical calculations and considerations specific to the study design, the research question, and the statistical methods being used. Techniques like power analysis, formulas based on desired confidence levels and margin of error, or sample size calculators aid researchers in determining an appropriate sample size for their study.
  A larger sample size generally leads to more precise estimates and increased statistical power, but it's essential to balance this with practical constraints and the study's objectives to ensure an optimal sample size that best represents the population of interest.

Discrete Random Variables

• a random variable that has either a finite or countable number of values.

• The values can be plotted on a number line with space between each point.

Continuous Random Variables

• A variable that has infinitely many values.

• The values can be plotted on a line in an uninterrupted fashion.

Sampling Methods

• Simple

• Convenience

• Systematic

• Quota

• Stratified

4.2 B and W and Cumulative Frequency

• Grouped Frequency Distributions (4.2 B):

  • Grouped frequency distributions are used when dealing with large sets of data. Instead of listing each individual data point, the data are grouped into intervals or classes.

  • Intervals or classes should be mutually exclusive (no overlap) and collectively exhaustive (covering all data).

  • Example: if you have test scores ranging from 0 to 100, you might group them into intervals such as 0-10, 11-20, 21-30, and so on.

• Cumulative Frequency (4.2 W):

  • Cumulative frequency is the running total of frequencies in a frequency distribution.

  • It is obtained by adding up the frequencies of all intervals up to a certain point.

  • Example: if you have grouped data showing the number of students scoring within different score ranges:

    • Interval 0-10: Frequency = 15

    • Interval 11-20: Frequency = 25

    • Interval 21-30: Frequency = 30

    • Cumulative Frequency for 0-10: 15

    • Cumulative Frequency for 0-20: 15 (0-10) + 25 (11-20) = 40

    • Cumulative Frequency for 0-30: 15 (0-10) + 25 (11-20) + 30 (21-30) = 70, and so on.

• Cumulative Frequency Curve:

  • A cumulative frequency curve (also known as an Ogive) is a graphical representation of cumulative frequencies.

  • On the x-axis, you plot the upper-class boundaries or midpoints of the intervals, and on the y-axis, you plot the cumulative frequencies.

  • Connecting the points in a cumulative frequency curve shows the overall pattern of the data's distribution and helps in visualizing cumulative frequencies.

4.3 Averages and Dispersion

• There are 4 different types of graphs

  1. Perfect or strong Correlation ( r>0.7)

  2. Moderate Correlation (0.5 < r < 0.7 )

  3. Weak Correlation ( 0.3 < r < 0.5)

  4. No Correlation ( r < 0.3()

4.4 Correlation and Regression

• Correlation: Correlation measures the relationship between two variables and the strength and direction of their association. In IB studies, students might use correlation analysis to explore relationships between different data sets. For instance, in an Economics class, students might analyze the correlation between factors like inflation rates and unemployment rates to understand their relationship within an economy. In Mathematics, students might study correlation coefficients and use them to determine the strength of the linear relationship between variables.

• Regression: Regression analysis helps in understanding and modeling the relationship between a dependent variable and one or more independent variables. In IB studies, regression analysis could be used to predict outcomes based on certain factors or variables. For instance, in a Biology class, students might perform linear regression to model the relationship between the concentration of a substance and the rate of a reaction. In Economics, regression analysis might be used to predict the impact of factors like government spending on GDP growth.

4.5 Probability Basics

• There are several probability rules that are commonly used in probability theory. Here are a few of them:

  1. Addition Rule: P(A or B) = P(A) + P(B) - P(A and B) This rule is used to calculate the probability of either event A or event B occurring, or both.

  2. Multiplication Rule: P(A and B) = P(A) * P(B|A) This rule is used to calculate the probability of both event A and event B occurring.

  3. Complement Rule: P(A') = 1 - P(A) This rule is used to calculate the probability of the complement of event A (i.e., the probability of event A not occurring).

• These rules are fundamental in probability theory and are used to solve various probability problems.

4.7 D.R.V

• known as Discrete Random Variables

• random experiments where they are assigned a probability

• Examples: a dice roll and the outcomes or a spinner and seeing the chance of an outcome in a specific section and percentage.

4.8 Binomial Distribution

• The binomial distribution applies to events that can be described as a "success" if one outcome occurs or a "failure" if any other outcome occurs. There can be more than 2 outcomes, but it needs to be black and white in terms of success or failure.

• The formula for the binomial distribution is
  
  Where:

  • P(X=k) is the probability of getting exactly k successes in n independent Bernoulli trials.

  • n is the number of trials.

  • k is the number of successes.

  • p is the probability of success in a single trial.

• Note: The binomial distribution assumes that each trial is independent and has the same probability of success.
  nr)(1−p)n−rpr

4.9 Normal Distribution

• The formula for the normal distribution, also known as the Gaussian distribution, is:
  
  Where:

  • f(x) is the probability density function

  • x is the random variable

  • μ (mu) is the mean of the distribution

  • σ (sigma) is the standard deviation of the distribution

• This formula describes the shape of a bell curve, with the mean at the center and the standard deviation determining the spread of the data.

5.0 X on Y Regression

• another way of describing a linear regression

5.1 Conditional Independence

• A concept in probability theory and statistics. It refers to the independence of two random variables given the value of a third random variable. In other words, two variables X and Y are conditionally independent given Z if the probability distribution of X and Y does not change when the value of Z is known. This can be denoted as X ⊥ Y | Z. Conditional independence is an important concept in various fields, including machine learning, graphical models, and Bayesian networks.

5.2 Normal Standardisation

• Standardization refers to the process of establishing a set of guidelines or criteria to ensure consistency and uniformity in various contexts. It can be applied to different areas such as education, industry, and measurement. In education, standardization often involves the development of curriculum standards, assessment criteria, and grading systems to ensure that students are evaluated fairly and consistently. In industry, standardization aims to establish common practices, specifications, and protocols to promote interoperability and efficiency. Overall, standardization plays a crucial role in maintaining quality, reliability, and comparability in various fields.

5.3 Bayes Theorem ( HL)

• Bayes' theorem is a fundamental concept in probability theory and statistics. It describes how to update the probability of a hypothesis based on new evidence. The theorem is named after Thomas Bayes, an 18th-century mathematician. In mathematical notation, Bayes' theorem can be expressed as:
  P(A|B) = (P(B|A) * P(A)) / P(B)
  Where:

  • P(A|B) is the probability of hypothesis A given evidence B.

  • P(B|A) is the probability of evidence B given hypothesis A.

  • P(A) is the prior probability of hypothesis A.

  • P(B) is the prior probability of evidence B.

• Bayes' theorem is widely used in various fields, including machine learning, medical diagnosis, and spam filtering. It allows for the updating of beliefs or probabilities based on new information, making it a powerful tool for decision-making and inference.

5.4 C.R.V

• CRV stands for Continuous Random Variable in statistics. It refers to a random variable that can take on any value within a certain range or interval. Unlike discrete random variables, which can only take on specific values, continuous random variables have an infinite number of possible values. Examples: height, weight, and time. The probability distribution of a continuous random variable is described by a probability density function (PDF) rather than a probability mass function (PMF) as in the case of discrete random variables.

• The formula for a continuous random variable is the probability density function (PDF). The PDF represents the probability distribution of the random variable over a continuous range of values. It is denoted as f(x) and satisfies the following properties:

  1. f(x) ≥ 0 for all x in the range of the random variable.

  2. The total area under the curve of the PDF is equal to 1.

• The probability of a continuous random variable falling within a specific interval [a, b] can be calculated by integrating the PDF over that interval:
  P(a ≤ X ≤ b) = ∫[a, b] f(x) dx
  Where X is the continuous random variable and f(x) is its PDF.

Terms

Statistics

• a branch of math that goes specifically into crunching math

Mean

• The average of a population

• You add all of the numbers together and then dive by the amount of numbers that there are

Median

• The middle

• you cross the numbers out and when you get two middle numbers you add them and divide by two because this means that you are averaging them out 

Mode

• the value that occurs the most often

• Bimodal: Two modes

• Multimodal: More than 2 modes

• When looking at a group of data you want to find one with the highest frequency called the modal class.

Range

• the difference between the highest and lowest data values.

Probability

• The likelihood that something will happen.

• it can be measured numerically

Random Experiment 

• An experiment in which there is no way to determine the outcome beforehand 

• Example: is a dice game

Trial

• An action in a random experiment 

• Example: Rolling Dice

Outcome

• Possible result of a trial

• Example: rolling a 2-4 on a dice

Event

• Set of possible outcomes 

• Example: A dice is six so the outcomes of that will be 1-5, 2-4, 3-3, 4-2 and 5-1

Multiplication Rule

1. Mutually Exclusive Events: These are events that cannot occur simultaneously. For mutually exclusive events, the addition rule states that the probability of either one event OR another event happening is calculated by simply adding their individual probabilities.
   Mathematically: P(A or B) = P(A) + P(B)

2. Non-Mutually Exclusive Events: These are events that can occur together. When dealing with non-mutually exclusive events, the addition rule accounts for the possibility of their overlap (the events occurring together). It's expressed as:
   Mathematically: P(A or B) = P(A) + P(B) - P(A and B)
   Here, P(A and B) represents the probability of both events A and B occurring simultaneously. The subtraction of P(A and B) adjusts for the double counting of the overlapping probability when summing individual probabilities.

The addition rule is fundamental in computing probabilities when considering the occurrence of events independently or jointly and helps in understanding the combined likelihood of different outcomes in various scenarios.

Multiplication Rule

• the rule can apply to both independent and dependent events

• two events within one sample space

Dependent Events

• P(A ∩ B) = P(B). P(A|B)

Independent Events

• P(A ∩ B) = P(A). P(B)

Frequency distributions

• a list of each category and the number of occurrences for each of the categories of data

Standard Deviation

• measures the deviation between scores and the mean.

• It’s the square root of the variance

• A sort of average of differences between the values and the mean.

• The symbol for Standard Deviation is σ

• It is normally distributed

Deviation

• The difference between the data values of X and the mean.

Binomial Distribution

It applies to the following:

• When a distribution is discrete

• In a fixed number of trials

• When there is a success and a failure

• When the probability of success is the same for all of the trials which is when the trials are independent 

Random Variables

• Random variables are a key concept in probability and statistics, representing numerical outcomes of random phenomena. They are used to quantify uncertain events in a mathematical framework.
  A random variable, often denoted by a capital letter such as X, Y, or Z, assigns a numerical value to each outcome of a random experiment. There are two types of random variables: discrete and continuous.

Discrete Random Variables

• a random variable that has either a finite or countable number of values.

• The values can be plotted on a number line with space between each point.

Continuous Random Variables

• A variable that has infinitely many values.

• The values can be plotted on a line in an uninterrupted fashion.

Binomial Distribution and Probability

Chi-Square

Goodness of Fit

• a statistical test that tries to determine whether a set of observed values match those expected under the applicable model.

Cumulative Frequency Distribution

• The graphs can be used to estimate the median and to find other properties of the data.

• A distribution that displays the aggregate frequency of a category can also be known as discrete data. It displays the total number of observations which are less than or equal to the categories.

• For the continuous data: they observe the total number of observations that are less than or equal to the upper class limit of a class.

Cumulative Relative Frequency Distribution

• A distribution that displays a proportion or a percentage of observations that are less than or equal to the categories of the discrete data. This can also be the proportion or percentage of observations that are less than or equal to the upper-class limits of the continuous data.

Hypothesis Testing

• An assumption about a population parameter such as a mean or a proportion. This assumption may or may not be true

• A hypothetical estimate of what an outcome may be

• If one hypothesis is true then the other one must be false

• Stating either the null or the alternative hypothesis depends on the nature of the test as well as the motivation of the person who is conducting the specific test

Alternative Hypothesis ( H0)

• new drug on a different effect on average compared to that of a new and current drug or that the new drug is better on average than the current drug

• opposite of the null hypothesis

• it is believed to be true if the null hypothesis is found to be false

• never use the equal sign

• used in the research hypothesis as well as the claim that the researchers wish to support

Null Hypothesis(H1)

• always use an equal sign

• Can be used with standard deviation

• tested using the sample data

• represents the status quo

• belief that the population parameter is greater than or equal to a specific value.

• beloved to be true unless there is an overwhelming evidence

• might be that the new drug is no better, on average than the current drug.

Logic of Hypothesis

1. Reject

2. Fail to reject

Types of Testing

1. Left Tailed Test

   • population mean is less than a specific value K

2. Right Tailed Test

   • population mean is greater than a specific value K

3. One tail test

   • left or right-tailed

4. Two-Tailed Test

   • two critical values

   • reject or the other one

T-test

Z- Score

R-value

Different Types of Graphs

Vocabulary For Graphs

Outlier

• a data point whose values is significantly greater than or less than the other values

Cluster

• an isolated group of points

Gap

• a large space between the data points

Histograms

• Definition: a drawing of each class using rectangles. The height of each rectangle is the frequency or relative frequency of the class. The width of the rectangles are all the same and the rectangles also do not touch each other.

• contain a commonality or what they are testing which is the height in feet of how tall the Black Cherry Trees grow

• contain the frequency which is how often something happens which is in this case how often the specific Black Cherries grow on trees

Stem and Leaf Plots

• Stem: the digits on the left side of the rightmost digits of the raw data

• Leaf: the rightmost digit in the plot

• it is similar to a dot plot but the number that the line is vertical and there are numbers used instead of X’s

• A representation of graphical quantitative data in which the data itself is used to create a certain graph. The raw data gets retrieved from the graph.

Box and Whisker Plots also known as a Box Plot

• A type of graph that shows the dispersion of data

• a quick and simple way of organizing numerical data.

• Normally used when there is only one group of data with less than 50 values

• Five number summary

• Upper Quartile(Q3):

• Lower Quartile(Q1):

Interquartile Range (IQR): The difference between upper and lower quartiles Q3-Q1.

Pareto Chart

• a bar graph whose bars are drawn in decreasing order of frequency or relative frequency

Pie Chart

• a circle divided into sectors where each of the sectors represents a category of a certain data. The area of each of the sectors is proportional to the frequency of each of the categories.

Dot Plot

• they are drawn by placing each of the observations horizontally in the increasing order in which a dot is placed above the observation every time that it is observed

Time Series Plot

• they are obtained by plotting time in which a variable is measured on a horizontal axis and where the corresponding value of the variable is located on the vertical axis. The line segments

Ogive

• a graph that represents the cumulative or cumulative relative frequency of a class. They are constructed by plotting points which

Degree of Confidence or the Confidence level

• The confidence interval actually does ensure that the population parameter

Margin of Error

dC

Population Proportion

The population proportion refers to the proportion or percentage of individuals in a population that exhibits a certain characteristic or have a specific attribute of interest. It represents the ratio of the number of individuals possessing a particular trait to the total population size.

• For instance, imagine a population of 1000 people, and out of these, 300 individuals have brown hair. The population proportion of individuals with brown hair would be calculated by dividing the number of individuals with brown hair by the total population size:
  Population Proportion of Brown-Haired Individuals = Number of Brown-Haired Individuals / Total Population Size Population Proportion of Brown-Haired Individuals = 300 / 1000 = 0.3 or 30%

• in statistical terms, if you were to take multiple random samples from this population and calculate the proportion of brown-haired individuals in each sample, the average of these sample proportions would typically converge toward the population proportion as the sample size increases, following the principles of the law of large numbers and the central limit theorem.
  Estimating population proportions and understanding their variability through statistical inference is important in various fields such as public health, sociology, market research, and more. Techniques such as confidence intervals and hypothesis testing are used to make inferences about population proportions based on sample data.

Unit 5: Calculus

Differentiation

1.1. Definition of the Derivative:

• The derivative of a function represents the rate of change of that function at a given point.

• It measures how the function's output changes concerning its input.

1.2. Rules of Differentiation:

• Power Rule: If f(x) = x^n, then f(x) = x^{n-1}

• Product Rule: If f(x) = g(x)*h(x), then f’(x) = g’(x)*h’(x) + g(x)*h(x)

• Quotient Rule: If \frac{g(x)}{h(x)} , then f’(x) = \frac{g’(x)*h(x) - g(x)*h’(x)}{[h(x)]²}

• Chain Rule: If f(x)=g(h(x)), then f'(x)=g'(h(x))*h'(x)

1.3. Finding Derivatives of Elementary Functions:

• Derivatives of basic functions like polynomials, trigonometric functions, exponential functions, and logarithmic functions.

1.4. Applications of Derivatives:

• Finding Maxima/Minima: Use derivatives to find critical points and determine whether they correspond to maxima, minima, or points of inflection.

• Optimization: Use derivatives to optimize functions, such as maximizing profit or minimizing cost.

• Related Rates: Use derivatives to solve problems involving rates of change in related variables.

1.5. Notable Derivatives:

• \frac{d}{dx}(e^x)= e^x

• \frac{d}{dx}(ln(x))=\frac{1}{x}

• \frac{d}{dx}(sin(x))=cos(x)

• \frac{d}{dx}(cos(x))=-sin(x)

1.6. Implicit Differentiation:

• Technique used to find the derivative of an implicitly defined function.

1.7. Higher Order Derivatives:

• Second and higher derivatives denote rates of change of rates of change (acceleration, jerk, etc.).

1.8. Derivatives in Graphical Analysis:

• Interpretation of the slope of the tangent line as the derivative at a point.

• Derivative as a measure of the rate of change of a function's graph.

Integration:

2.1. Indefinite Integrals:

• The indefinite integral of a function represents a family of antiderivatives or primitives.

• Denoted as \int f(x)dx, where f(x) is the integrand and d(x) indicates the variable of integration.

2.2. Basic Definite Integrals and Their Properties:

• The definite integral of a function over an interval represents the signed area between the graph of the function and the x-axis.

• Denoted as \int_{a}^{b}f(x)dx, where a and b are the limits of integration.

• Properties include linearity, the additive property, and the constant multiple property.

2.3. Integration Techniques:

• Substitution: Involves substituting a new variable to simplify the integrand.

• Integration by Parts: Involves applying the product rule for differentiation in reverse.

• Partial Fractions: Used to integrate rational functions by decomposing them into simpler fractions.

2.4. Applications of Integration:

• Area under Curves: Use definite integrals to find the area between the curve and the x-axis over a given interval.

• Finding Volumes: Use definite integrals to find volumes of solids of revolution using methods like disk/washer and shell methods.

2.5. Fundamental Theorem of Calculus (FTC):

• Part 1: If f is continuous on [a,b] then the function F defined by F(x)=\int_{a}^{x}f(t)dt is continuous on (a,b) and differentiable on (a,b), and F'(x)=f(x)

• Part 2: If F is an antiderivative of f on [a,b] then \int_{b}^{a}f(x)dx = F(b)-F(a)

2.6. Applications of Integration in Geometry:

• Arc Length: Use integrals to find the length of a curve.

• Surface Area: Use integrals to find the surface area of a solid of revolution.

2.7. Numerical Integration:

• Techniques such as the Trapezoidal Rule for approximating definite integrals when an analytic solution is not feasible.

2.8. Integration in Graphical Analysis:

• Interpretation of the integral as the accumulation of quantities represented by the function's graph.

Applications of Differentiation and Integration:

3.1. Rate of Change Problems:

• Definition: Use derivatives to analyze how one quantity changes concerning another.

• Example: Speed of a moving object at a specific time, rate of change of population growth, etc.

3.2. Optimization Problems:

• Definition: Use derivatives to find maximum or minimum values of a function.

• Example: Maximizing profit, minimizing cost, optimizing dimensions of a container, etc.

3.3. Area and Volume Problems:

• Definition: Use integrals to find the area under a curve or the volume of a solid.

• Example: Finding the area enclosed by a curve and the x-axis, calculating the volume of a three-dimensional shape like a cylinder or cone.

3.4. Related Rates Problems:

• Use derivatives to find the rate of change of one quantity with respect to another related quantity.

• Example: Rate at which the area of a circle is changing concerning its radius, rate of change of the volume of a cone concerning its height, etc.

3.5. Motion Problems:

• Definition: Use derivatives to analyze the motion of objects.

• Example: Position, velocity, acceleration of an object at a given time, distance traveled over a certain time interval, etc.