🎯⭐ INTERACTIVE LESSON

AP Exam Review

Learn step-by-step with interactive practice!

← Back to Standard Lesson

AP Exam Review - Complete Interactive Lesson

Part 1: Core Concepts

AP Calculus AB Exam Review

Part 1 of 7 — Limits & Continuity Review

Topic Overview

Part	Review Topic
1	Limits & Continuity
2	Differentiation Rules
3	Applications of Derivatives
4	Integration Techniques
5	Applications of Integration
6	Differential Equations & Modeling
7	Full Practice Exam

Exam Weight: Units 1–2 (Limits & Continuity) = 10–12% of the AP exam.

Limit Evaluation Techniques

Technique	When to Use	Example
Direct substitution	No indeterminate form	$\lim_{x\to 3}(x^2+1) = 10$

Essential Trig Limits

$\boxed{\lim_{x\to 0}\frac{\sin x}{x} = 1 \qquad \lim_{x\to 0}\frac{1-\cos x}{x} = 0}$

Limits at Infinity — Rational Functions

Degrees	Limit	HA
deg(top) $<$ deg(bottom)	$0$	$y=0$
deg(top) $=$ deg(bottom)

Continuity Conditions

$f$ is continuous at $x = a$ if ALL THREE hold:

$f(a)$ is defined
$\lim_{x\to a} f(x)$ exists
$\lim_{x\to a} f(x) = f(a)$

Limits Review Quiz 🎯

Identify the concept. 🔍

Compute the limit. ✍️

Key Takeaways — Part 1

Always try direct substitution first
$\frac{0}{0}$ requires algebraic manipulation
Compare degrees for limits at infinity
Continuity needs all three conditions verified

Part 2: Worked Examples

AP Exam Review — Differentiation Rules

Part 2 of 7

Differentiation Rules Reference

Rule	Formula
Power	$\frac{d}{dx}[x^n] = nx^{n-1}$

Part 3: Problem-Solving Patterns

AP Exam Review — Applications of Derivatives

Part 3 of 7

Applications of Derivatives Overview

Application	Key Idea
Related Rates	Differentiate an equation involving changing quantities with respect to time
Optimization	Find absolute max/min on a domain
Curve Sketching	Use $f'$ and $f''$ to determine behavior

Part 4: Graphs and Interpretation

AP Exam Review — Integration Techniques

Part 4 of 7

Integration Formulas Reference

Integral	Result
$\int x^n\,dx$	$\frac{x^{n+1}}{n+1} + C$ ()

Part 5: Applications

AP Exam Review — Applications of Integration

Part 5 of 7

Applications of Integration Summary

Application	Formula
Area under curve	$\int_a^b f(x)\,dx$
Area between curves	()

Part 6: Exam Strategy

AP Exam Review — Differential Equations & Modeling

Part 6 of 7

Differential Equations on the AP Exam

Type	Form	Method
Separable	$\frac{dy}{dx} = f(x)g(y)$

Part 7: Mixed Review

AP Exam Review — Full Practice Exam

Part 7 of 7

AP Calculus AB Exam Format

Section	Questions	Time	Calculator
MC Part A	30 questions	60 min	No
MC Part B	15 questions	45 min	Yes
FRQ Part A	2 questions	30 min	Yes
FRQ Part B	4 questions	60 min	No

Formula Quick Reference

Category	Key Formula
Derivative	$\frac{d}{d x} [f (g (x))$

\frac{leading coeff.}{leading coeff.}

lim_{x \to a} f (x) = f (a)

When to Use Each Rule

Scenario	Rule	Example
Two functions multiplied	Product	$x^2 \sin x$
One function divided by another	Quotient	$\frac{\ln x}{x}$
Function inside a function	Chain	$\sin(x^3)$
Product inside a composition	Chain + Product	$e^{x\sin x}$

Key Fact: The chain rule is the most-tested differentiation rule on the AP exam.

Worked Example — Multi-Rule Problem

Find $f'(x)$ for $f(x) = \frac{x^2 e^{3x}}{\cos x}$ .

Step 1: Identify — quotient rule with numerator = product.

Numerator: $h(x) = x^2 e^{3x}$

$h'(x) = 2x\,e^{3x} + x^2\cdot 3e^{3x} = e^{3x}(2x + 3x^2)$

Step 2: Quotient rule:

$f'(x) = \frac{e^{3x}(2x+3x^2)\cos x - x^2 e^{3x}(-\sin x)}{\cos^2 x}$

$= \frac{x\,e^{3x}[(2+3x)\cos x + x\sin x]}{\cos^2 x}$

Differentiation Rules Quiz 🎯

Implicit Differentiation Review

For equations not solved for $y$ , differentiate both sides with respect to $x$ , treating $y$ as a function of $x$ .

$\boxed{\text{Every } y \text{ term gets a } \frac{dy}{dx} \text{ factor (chain rule)}}$

Example: $x^2 + y^2 = 25$

$2x + 2y\frac{dy}{dx} = 0 \implies \frac{dy}{dx} = -\frac{x}{y}$

AP Tips for Derivative Problems

Tip	Why It Matters
Simplify BEFORE differentiating	Avoids unnecessary product/quotient rules
Check for chain rule	Most common error is forgetting inner derivative
Read carefully for $f'(a)$ vs $f(a)$	Question may ask for the derivative at a point
Know trig derivatives cold	These appear frequently in MC

More Practice 📝

Match the rule. 🔍

Compute the derivative. ✍️

Key Takeaways — Part 2

Master the product, quotient, and chain rules
Chain rule applies whenever a function is composed with another
Implicit differentiation: every $y$ gets $\frac{dy}{dx}$
Simplify before differentiating when possible

First & Second Derivative Analysis

Sign of $f'$	Sign of $f''$	Behavior of $f$
$+$	$+$	Increasing, concave up
$+$	$-$	Increasing, concave down
$-$	$+$	Decreasing, concave up
$-$	$-$	Decreasing, concave down
$0$	$+$	Local minimum
$0$	$-$	Local maximum

Key Fact: The second derivative test fails when $f''(c) = 0$ . Use the first derivative test instead.

Related Rates Checklist

Draw a diagram and label variables
Write an equation relating the variables
Differentiate both sides with respect to $t$
Substitute known values and solve

Worked Example — Related Rates

A balloon’s radius increases at $\frac{dr}{dt} = 2$ cm/s. Find $\frac{dV}{dt}$ when $r = 5$ .

$V = \frac{4}{3}\pi r^3 \implies \frac{dV}{dt} = 4\pi r^2 \frac{dr}{dt} = 4\pi(25)(2) = 200\pi$ cm³/s.

Applications Quiz 🎯

Optimization Strategy

$\boxed{\text{Absolute extrema on } [a,b]: \text{ compare } f(\text{critical pts}) \text{ and } f(a), f(b)}$

Example: Maximize $f(x) = -x^2 + 6x - 5$ on $[0, 5]$ .

$f'(x) = -2x + 6 = 0 \implies x = 3$ (critical point).

$x$	$f(x)$
$0$	$-5$
$3$

Absolute max = $4$ at $x=3$ . Absolute min = $-5$ at $x=0$ .

Identify the scenario. 🔍

Solve the optimization problem. ✍️

Key Takeaways — Part 3

Use $f'$ for increasing/decreasing and local extrema
Use $f''$ for concavity and inflection points
Related rates: differentiate an equation with respect to $t$
Optimization on closed intervals: check critical points AND endpoints

Fundamental Theorem of Calculus

$\boxed{\text{FTC Part 1: } \frac{d}{dx}\int_a^x f(t)\,dt = f(x)}$

$\boxed{\text{FTC Part 2: } \int_a^b f(x)\,dx = F(b) - F(a)}$

Chain Rule Variant: $\frac{d}{dx}\int_a^{g(x)} f(t)\,dt = f(g(x))\cdot g'(x)$

$u$ -Substitution Strategy

Step	Action
1	Identify inner function $u = g(x)$
2	Compute $du = g'(x)\,dx$
3	Rewrite integral entirely in terms of $u$
4	Integrate and substitute back

Worked Example — $u$ -Substitution

$\int x\cos(x^2)\,dx$

Let $u = x^2$ , $du = 2x\,dx$ , so $x\,dx = \frac{du}{2}$ .

$\frac{1}{2}\int \cos u\,du = \frac{1}{2}\sin u + C = \frac{1}{2}\sin(x^2) + C$

Integration Quiz 🎯

Definite Integral Properties

Property	Formula
Additivity	$\int_a^b f + \int_b^c f = \int_a^c f$
Constant multiple	$\int_a^b kf = k\int_a^b f$
Reverse limits	$\int_a^b f = -\int_b^a f$
Zero width	$\int_a^a f = 0$
Sum/Difference	$\int_a^b (f \pm g) = \int_a^b f \pm \int_a^b g$

Average Value Formula

$\boxed{f_{\text{avg}} = \frac{1}{b-a}\int_a^b f(x)\,dx}$

Example: Average value of $f(x) = x^2$ on $[0,3]$ :

$f_{\text{avg}} = \frac{1}{3}\int_0^3 x^2\,dx = \frac{1}{3}\cdot\frac{27}{3} = 3$

Match the technique. 🔍

Compute the integral. ✍️

Key Takeaways — Part 4

Know antiderivative formulas for power, exponential, trig, and $\ln$
FTC Part 1 connects derivatives and integrals
$u$ -substitution reverses the chain rule
Average value = $\frac{1}{b-a}\int_a^b f(x)\,dx$

\int_{a}^{b} [f (x) - g (x)] d x

Area Between Curves — Setup

$\boxed{A = \int_a^b [\text{top} - \text{bottom}]\,dx \quad\text{or}\quad \int_c^d [\text{right} - \text{left}]\,dy}$

Key Fact: When curves cross, split the integral at intersection points.

Worked Example — Area Between Curves

Find the area between $y = x^2$ and $y = x$ on $[0,1]$ .

Intersection: $x^2 = x \implies x=0, x=1$ . On $[0,1]$ : $x \ge x^2$ .

$A = \int_0^1 (x - x^2)\,dx = \left[\frac{x^2}{2} - \frac{x^3}{3}\right]_0^1 = \frac{1}{2} - \frac{1}{3} = \frac{1}{6}$

Applications of Integration Quiz 🎯

Volume Methods Comparison

Method	Axis	Slice Shape	Formula
Disk	$x$	Circle	$\pi\int [R(x)]^2\,dx$
Washer	$x$	Ring	$\pi\int ([R]^2 - [r]^2)\,dx$
Disk ( $y$ -axis)	$y$	Circle	$\pi\int [R(y)]^2\,dy$

Worked Example — Washer Method

Region between $y = x$ and $y = x^2$ rotated about the $x$ -axis on $[0,1]$ .

Outer radius: $R = x$ . Inner radius: $r = x^2$ .

$V = \pi\int_0^1 (x^2 - x^4)\,dx = \pi\left[\frac{x^3}{3} - \frac{x^5}{5}\right]_0^1 = \pi\left(\frac{1}{3} - \frac{1}{5}\right) = \frac{2\pi}{15}$

Accumulation Functions

$\boxed{F(x) = F(a) + \int_a^x f(t)\,dt}$

This says: starting value + net accumulation = current value.

Choose the correct setup. 🔍

Compute the volume. ✍️

Key Takeaways — Part 5

Area between curves: $\int [\text{top} - \text{bottom}]\,dx$
Disk: one curve, no hole. Washer: two curves (outer - inner)
Accumulation: initial value + integral of rate = total
Average value = $\frac{1}{b-a}\int_a^b f$

Separable Equations — Steps

$\boxed{\frac{dy}{dx} = f(x)g(y) \implies \frac{dy}{g(y)} = f(x)\,dx \implies \int \frac{dy}{g(y)} = \int f(x)\,dx}$

Worked Example — Separable DE

Solve $\frac{dy}{dx} = 2xy$ , $y(0) = 3$ .

Step 1: Separate: $\frac{dy}{y} = 2x\,dx$

Step 2: Integrate: $\ln|y| = x^2 + C$

Step 3: Solve for $y$ : $y = Ae^{x^2}$ where $A = e^C$

Step 4: Apply IC: $y(0) = A = 3$

Exponential Growth & Decay

$\boxed{\frac{dy}{dt} = ky \implies y(t) = y_0 e^{kt}}$

$k > 0$	$k < 0$
Exponential growth	Exponential decay
Population growth	Radioactive decay
Compound interest	Cooling (Newton’s Law)

Key Fact: Half-life formula: $t_{1/2} = \frac{\ln 2}{|k|}$

Differential Equations Quiz 🎯

Slope Fields — Reading Guide

Observation	Meaning
All segments same slope in a row	DE depends only on $y$
All segments same slope in a column	DE depends only on $x$
Segments get steeper as you move right	DE is increasing in $x$
Horizontal segments along a line	That line is an equilibrium ( $dy/dx=0$ )

AP Slope Field Tips

Matching: Plug in specific $(x,y)$ values to check if the slope matches
Sketching solutions: Follow the slopes like a river
Equilibrium: $\frac{dy}{dx} = 0$ lines are equilibrium solutions

Classify the differential equation. 🔍

Solve the IVP. ✍️

Key Takeaways — Part 6

Separable DEs: move $y$ terms to one side, $x$ terms to the other
Always apply the initial condition AFTER integrating
Exponential model: $\frac{dy}{dt} = ky$ has solution $y = y_0 e^{kt}$
Slope fields: plug in points to verify slopes, look for equilibrium lines

\frac{d}{d x} [f (g (x))] = f^{'} (g (x)) \cdot g^{'} (x)

Key Fact: You must show ALL work on FRQs. An answer without justification earns 0 points.

Practice Exam — No Calculator 🎯

Practice Exam — Calculator Active 📱

Quick-fire theorem check. 🔍

FRQ-style computation. ✍️

AP Exam Day Strategies

Strategy	Details
Time management	~2 min/MC question, 15 min/FRQ
MC tips	Eliminate obviously wrong answers first
FRQ tips	Show all work; label answers with units
Common mistakes	Forgetting $+C$ , sign errors, chain rule omission
Calculator section	Use it for graphing and numerical integration

Common AP Mistakes to Avoid

Mistake	Correction
Writing $\int f(x)$ without $dx$	Always include $dx$
Forgetting $+C$ on indefinite integrals	Points deducted every time
Not justifying with theorems	Name the theorem (IVT, MVT, etc.)
Plugging in before differentiating	Differentiate first, THEN substitute
Confusing displacement vs. distance	Distance uses $

Completion Checklist

Unit	Review Topic	Status
1–2	Limits & Continuity	✅
3–4	Differentiation Rules	✅
5	Applications of Derivatives	✅
6	Integration Techniques	✅
7–8	Applications of Integration	✅
7	Differential Equations	✅
—	Full Practice Exam	✅

You’ve completed the AP Calculus AB Exam Review! Good luck on exam day! 🎉