Survey of Calculus Exam 1 Study Guide

Functions

Basic Properties of Functions

Understanding functions is fundamental in calculus. A function relates each input (domain) to exactly one output (range).

• Domain: The set of all possible input values (x-values) for which the function is defined.

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

• Finding Function Values: Function values can be determined using a graph, table, or equation.

• Example: For $f(x) = x^2$, the domain is all real numbers, and the range is all non-negative real numbers.

Lines

Linear functions are represented by straight lines and are characterized by their slope and intercepts.

• Slope: Measures the steepness of the line. Calculated as $m = \frac{y_2 - y_1}{x_2 - x_1}$.

• Finding a Line Between Two Points: Use the slope formula and point-slope form.

• Finding a Line from Slope and Point: Use point-slope form: $y - y_1 = m(x - x_1)$.

• Slope-Intercept Form: $y = mx + b$, where $m$ is the slope and $b$ is the y-intercept.

• Example: Find the equation of a line passing through (2,3) with slope 4: $y - 3 = 4(x - 2)$.

Quadratic Functions

Quadratic functions have the form $f(x) = ax^2 + bx + c$ and their graphs are parabolas.

• Vertex: The point $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f(h)$.

• y-intercept: The value of $f(0)$.

• x-intercepts: Solve $ax^2 + bx + c = 0$ for $x$.

• Example: For $f(x) = x^2 - 4x + 3$, vertex at $(2, -1)$, y-intercept at $3$, x-intercepts at$x=1$and$x=3$.

Exponential Functions and Compound Interest

Exponential functions model growth and decay, including financial applications like compound interest.

• Continuous Compound Interest Formula: $A_t = P e^{rt}$, where $P$ is principal, $r$ is rate, $t$ is time.

• Example: If $P = 1000$, $r = 0.05$, $t = 3$, then $A_t = 1000 e^{0.15}$.

Limits

Existence of Limits

Limits describe the behavior of functions as inputs approach a specific value.

• Limit at a Point: $\lim_{x \to a} f(x)$ exists if $f(x)$ approaches a single value as $x$ approaches $a$.

• Polynomial and Rational Functions: Limits exist at all points except where the denominator is zero.

• Piecewise Functions: Check limits from both sides at points where the formula changes.

• Example: $\lim_{x \to 2} (x^2 - 3x + 2) = 0$.

Limits at Infinity and Horizontal Asymptotes

Limits at infinity help identify horizontal asymptotes in rational functions.

• Horizontal Asymptote: If $\lim_{x \to \infty} f(x) = L$, then $y = L$ is a horizontal asymptote.

• Example: $\lim_{x \to \infty} \frac{2x}{x+1} = 2$.

One-Sided and Two-Sided Limits

Limits can be approached from the left ($x \to a^-$) or right ($x \to a^+$).

• One-Sided Limit: $\lim_{x \to a^-} f(x)$ or $\lim_{x \to a^+} f(x)$.

• Two-Sided Limit: $\lim_{x \to a} f(x)$ exists if both one-sided limits are equal.

• Example: For $f(x)$ defined differently on each side of $x=1$, check both limits.

Limit Definition of Instantaneous Rate of Change

The derivative is defined as the limit of the average rate of change as the interval shrinks to zero.

• Definition: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: For $f(x) = x^2$, $f'(a) = \lim_{h \to 0} \frac{(a+h)^2 - a^2}{h} = 2a$.

Derivatives

Computing Derivatives

Derivatives measure the instantaneous rate of change of a function.

• Polynomial Functions: Use the power rule: $\frac{d}{dx} x^n = n x^{n-1}$.

• Exponential Functions: $\frac{d}{dx} e^{x} = e^{x}$; $\frac{d}{dx} a^{x} = a^{x} \ln a$.

• Example: $\frac{d}{dx} x^3 = 3x^2$; $\frac{d}{dx} e^{2x} = 2e^{2x}$.

Definition of Derivative

The derivative at a point is the instantaneous rate of change, given by the limit definition.

• Instantaneous Rate of Change: $f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$

• Example: For $f(x) = x^2$, $f'(2) = \lim_{h \to 0} \frac{(2+h)^2 - 4}{h} = 4$.

Non-Existence of Derivative

The derivative may not exist at points where the function is not continuous or has a sharp corner.

• Discontinuity: If the function is not continuous at a point, the derivative does not exist.

• Sharp Corners: At points where the graph has a corner or cusp, the derivative is undefined.

• Example: $f(x) = |x|$ at $x=0$ has no derivative.

Applications

Cost, Revenue, Profit, and Marginal Analysis

Calculus is used in business to analyze cost, revenue, and profit functions.

• Cost Function ($C(x)$): Total cost to produce $x$ units.

• Revenue Function ($R(x)$): Total revenue from selling $x$ units.

• Profit Function ($P(x)$): $P(x) = R(x) - C(x)$.

• Marginal Cost/Revenue/Profit: The derivative of each function, representing the rate of change per unit.

• Example: If $C(x) = 5x + 100$, then marginal cost is $C'(x) = 5$.

Position, Velocity, and Acceleration

In physics, derivatives describe motion.

• Position ($s(t)$): Location at time $t$.

• Velocity ($v(t)$): $v(t) = s'(t)$, the rate of change of position.

• Acceleration ($a(t)$): $a(t) = v'(t) = s''(t)$, the rate of change of velocity.

• Example: If $s(t) = t^2$, then $v(t) = 2t$, $a(t) = 2$.

Compound Interest Applications

Exponential functions are used to model compound interest in finance.

• Continuous Compound Interest: $A_t = P e^{rt}$

• Example: $P = 500$, $r = 0.08$, $t = 5$ years: $A_t = 500 e^{0.4}$