AP Exam Review

AP Calculus AB Exam Review

Part 1 of 7 — Limits & Continuity Review

Topic Overview

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

Limit Evaluation Techniques

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

Continuity Conditions

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

1. $f(a)$ is defined
2. $\lim_{x\to a} f(x)$ exists
3. $\lim_{x\to a} f(x) = f(a)$

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

AP Exam Review — Differentiation Rules

Part 2 of 7

Differentiation Rules Reference

When to Use Each Rule

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

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

AP Exam Review — Applications of Derivatives

Part 3 of 7

Applications of Derivatives Overview

First & Second Derivative Analysis

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

Related Rates Checklist

1. Draw a diagram and label variables
2. Write an equation relating the variables
3. Differentiate both sides with respect to $t$
4. 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).

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

AP Exam Review — Integration Techniques

Part 4 of 7

Integration Formulas Reference

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

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

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

AP Exam Review — Applications of Integration

Part 5 of 7

Applications of Integration Summary

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

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 = $\frac{1}{b-a}\int_a^b f$

AP Exam Review — Differential Equations & Modeling

Part 6 of 7

Differential Equations on the AP Exam

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: $y(0) = A = 3$

$y = 3e^{x^2}$

Exponential Growth & Decay

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

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

Differential Equations Quiz 🎯

Slope Fields — Reading Guide

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

AP Exam Review — Full Practice Exam

Part 7 of 7

AP Calculus AB Exam Format

Formula Quick Reference

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

Common AP Mistakes to Avoid

Completion Checklist

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