IB Mathematics Analysis & Approaches HL

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:

  1. Replace f(x) with y.

  2. Swap x and y.

  3. Solve for y.

  4. 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:

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

  2. 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:

  • 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.

  • 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.