MA100 Study Notes - Week 2

MA100 - Introduction to Calculus for the Natural Sciences

Exponents

• For any $a > 0$ and any positive integers $n, k$:

  • $a^n$ means multiply $n$ instances of $a$ together.

  • Example:

    • $2^3 = 2 imes 2 imes 2 = 8$

    • $a^2 = a imes a$

    • $a^1 = a$

    • $5^3 = 5 imes 5 imes 5 = 125$

    • $5^2 = 5 imes 5 = 25$

    • $5^{-2} = rac{1}{5^2} = rac{1}{25}$

• If $a,b > 0$ and $n, k$ are positive integers:

  • $a^{n+k} = a^n imes a^k$

  • Example:

    • $2^{13} = 2^{10 + 3} = 2^{10} imes 2^3$

• Some other properties:

  • $a^{n-k} = rac{a^n}{a^k}$

  • $a^{(n)}_k = a^{nk}$

  • $(ab)^n = a^n b^n$

  • Example:

    • For $n = 3$:

    • $10^3 = 10 imes 10 imes 10$

• $a^0 = 1$ for any $a <br>eq 0$.

Radicals

• $ ext{√}a$ refers to the positive square root of $a$:

  • Example:

  • $ ext{√}9 = 3$

  • $ ext{√}4 = 2$

  • $ ext{√}(-3)$ is not defined in real numbers.

• Important property of radicals:

  • $ ext{√}(ab) = ext{√}a imes ext{√}b$

  • Example:

    • $ ext{√}36 = ext{√}(4 imes 9) = ext{√}4 imes ext{√}9 = 2 imes 3 = 6$

• Note:

  • $ ext{√}(a + b)$ is different from $ ext{√}a + ext{√}b$.

  • Example:

    • $ ext{√}(16 + 9)
      eq ext{√}16 + ext{√}9$.

Rationalizing Denominators or Numerators

• Goal: Rewrite the expression to eliminate the square root from the denominator.

• Example:

  • To simplify $ rac{3}{ ext{√}x + 9}$, multiply the numerator and denominator by $ ext{√}x - 9$ to rationalize:

    • $ rac{3( ext{√}x - 3)}{( ext{√}x + 9)( ext{√}x - 9)} = rac{3( ext{√}x - 3)}{x - 9}$.

Inequalities Rules

• Basic properties of inequalities:

  • If $a < b$, then $a + c < b + c$.

  • If $a < b$ and $c < d$, then $a + c < b + d$.

  • If $a < b$ and $c > 0$, then $ac < bc$.

  • If $a < b$ and $c < 0$, then $ac > bc$ (multiplication by negative flips the inequality).

  • If $0 < a < b$, then:

  • $ rac{1}{b} < rac{1}{a}$.

• Example of evaluating an inequality:

  • Solve $3 + 2x < 10x + 5$.

Absolute Value

• Absolute value represents the distance from 0:

  • $|a| = a$ if $a ext{ is non-negative}$;

  • $|a| = -a$ if $a ext{ is negative}$.

• Notation overview:

  • $|3| = 3$;

  • $|-3| = 3$;

  • $|0| = 0$.

• Rules:

  • $|ab| = |a||b|$;

  • $|a^n| = |a|^n$ for $n ext{ is any integer}$.

Cartesian Coordinates in the xy-plane

• Named after René Descartes.

• An ordered pair $(x, y)$:

  • The $x$ coordinate determines horizontal position:

  • $x > 0$ is to the right,

  • $x < 0$ is to the left.

  • The $y$ coordinate determines vertical position:

  • $y > 0$ is up,

  • $y < 0$ is down.

• This is often referred to as $ ext{R}^2$, meaning pairs of real numbers.

Distance Formula

• The distance between points $(c,d)$ and $(a,b)$ is given by: $ext{Distance} = ext{√}((c - a)^2 + (d - b)^2)$

  • Example: For points $(6,5)$ and $(2,-3)$:

  • Distance = $ext{√}((6 - 2)^2 + (5 - (-3))^2)$.

Lines and Circles

• Equation for a circle of radius $R$ centered at $(a,b)$:
  $(x-a)^2 + (y-b)^2 = R^2$.

• Non-vertical lines can be written as:
  $y = mx + b$ where $m$ is the slope and $b$ is the y-intercept.

• Example: Line through $(1,4)$ and $(8,5)$ where the slope $m$ can be found by:
  $m = \frac{y<em>2 - y</em>1}{x<em>2 - x</em>1}$.

Parallel and Perpendicular Lines

• Two lines are parallel if they have the same slope:

  • Example: Lines $y = 3x + 5$ and $y = 3x - 4$.

• Two lines are perpendicular if the product of their slopes is $-1$:

  • Example: Lines $y = 4x + 5$ and $y = -\frac{1}{4}x + 2$.

Interval Notation

• Interval notation notation uses square and round brackets:

  • $(a,b):$ set of real numbers $x$ satisfying $a < x < b$;

  • $[a,b]:$ set of real numbers $x$ satisfying $a \leq x \leq b$.

• Unions: $A \bigcup B$ is all points in either $A$ or $B$ or both.

• Intersections: $A \cap B$ is all points in both $A$ and $B$.

Functions - Domain and Range

• A function assigns exactly one output for each input (domain).

• The domain is the set of inputs and the range is the set of outputs.

• The Vertical Line Test: A graph represents $y$ as a function of $x$ if no two points lie on the same vertical line.

Example Functions

• Determine the domain and range of:

  • $f(x) = x^2$:

  • Domain: $(-\infty, \infty)$; Range: $[0, \infty)$.

  • $g(x) = \frac{x^2 + 4}{x^2 - 9}$:

  • Domain excludes points making the denominator zero; thus, $x^2 - 9 <br>eq 0$ (avoid $x = 3$, $x = -3$).

  • Range varies based on simplifications and exclusions of values leading to vertical asymptotes.