AP Calculus AB Study Guide

Key Exam Details

The AP® Calculus AB exam has:
- Duration: 3 hours and 15 minutes
- Breakdown: 45 multiple-choice questions (50%) and 6 free-response questions (50%).

Content Categories:

Limits and Continuity: 10–12% of questions
Differentiation:
- Definition and Basic Derivative Rules: 10–12%
- Composite, Implicit, and Inverse Functions: 9–13%
- Contextual Applications of Differentiation: 10–15%
- Applying Derivatives to Analyze Functions: 15–18%
Integration and Accumulation of Change: 17–20%
Differential Equations: 6–12%
Applications of Integration: 10–15%

Limits and Continuity

Definition of Limits

The limit of a function f as x approaches c is L if f can be made arbitrarily close to L as x approaches c (but not equal to c).
Denoted as: $\lim_{x \to c} f(x) = L$ .
If no such value exists, the limit does not exist (DNE).
Calculating limits can be accomplished using:
- Tables
- Graphs
- Algebra

Example

If values of a function f are given:

| x     | 0.9   | 0.99  | 0.999 | 1.001 | 1.01  | 1.1  |
|-------|-------|-------|-------|-------|-------|------|
| f(x)  | 2.488 | 2.490 | 2.499 | 2.501 | 2.504 | 2.513|

The limit as x approaches 1 is $\lim_{x \to 1} f(x) = 2.5$ .

Techniques for Finding Limits:

Algebraic techniques like factoring and rationalizing radicals.
Special limits:
- $\lim_{x \to 0} \frac{\sin x}{x} = 1$
- $\lim_{x \to 0} (1 - \cos x) / x^2 = 0$

One-Sided Limits:

$\lim_{x \to c^+} f(x)$ approaches from the right
$\lim_{x \to c^-} f(x)$ approaches from the left
Existence conditions for $\lim_{x \to c} f(x)$ : both one-sided limits must exist and equal.

Continuity

A function f is continuous at c if:

$\lim_{x \to c} f(x)$ exists.
$f(c)$ exists.
$\lim_{x \to c} f(x) = f(c)$ .

Types of Discontinuity:

Jump Discontinuity: exists when limit approaches different values from left and right.
Removeable Discontinuity: exists when limit exists but does not equal the function's value.

Intermediate Value Theorem

If f is continuous on [a, b], and d is a value between f(a) and f(b), then there exists a c in (a, b) such that f(c) = d.

Differentiation: Definition and Fundamental Properties

Definition of the Derivative

The average rate of change of a function f from x=a to x=a+h is given by:
$\frac{f(a+h) - f(a)}{h}$ . The instantaneous rate of change as h approaches 0 is defined as:
$f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}$ .

If this limit exists, f is differentiable at x=a.

Important Derivative Rules:

Constant Rule: $f'(c) = 0$
Power Rule: $\frac{d}{dx} x^n = nx^{n-1}$
Sum Rule: $\frac{d}{dx}(f(x) + g(x)) = f'(x) + g'(x)$
Difference Rule: $\frac{d}{dx}(f(x) - g(x)) = f'(x) - g'(x)$
Product Rule: $\frac{d}{dx}(f(x)g(x)) = f'(x)g(x) + f(x)g'(x)$
Quotient Rule: $\frac{d}{dx}\left(\frac{f(x)}{g(x)}\right) = \frac{f'(x)g(x) - f(x)g'(x)}{(g(x))^2}$ .

Applications of Differentiation

Contextual Applications: Derivatives represent instantaneous rates of change; key in physics for motion (position, velocity, and acceleration).

Integration and Accumulation of Change

Definite Integral

The definite integral of a function on [a, b] represents the area under the curve between x=a and x=b:
$\int_{a}^{b} f(x) dx$

Positive area is above the x-axis; negative area below.

Fundamental Theorem of Calculus

Connects differentiation and integration, showing that:

If F is an antiderivative of f, then:
$\int_{a}^{b} f(x)dx = F(b) - F(a)$ .
$\frac{d}{dx}\int_{a}^{x} f(t) dt = f(x)$ (if f is continuous).

Differential Equations

Differential equations involve functions and their derivatives. Solutions to a differential equation describe the behavior of quantities:

Separable Equations: Can be solved by separating variables and integrating both sides.
Exponential Growth/Decay: Model scenarios where the rate of change of a quantity is proportional to the quantity itself, represented as:
$\frac{dy}{dt} = ky$ where k is a constant.