AP Calculus AB/BC — Premium Practice Exam

Core Concepts & Memorization

01 Limits & Continuity

\(\lim_{x\to a}f(x)=L\) iff \(\lim_{x\to a^-}f(x)=\lim_{x\to a^+}f(x)=L\)

MEMORIZE Special limits:
\(\displaystyle\lim_{x\to 0}\frac{\sin x}{x}=1\), \(\displaystyle\lim_{x\to 0}\frac{1-\cos x}{x}=0\), \(\displaystyle\lim_{x\to\infty}\!\left(1+\tfrac{1}{n}\right)^n=e\)

KEY A function \(f\) is continuous at \(a\) iff (1) \(f(a)\) exists, (2) the limit exists, (3) they are equal. L'Hôpital's Rule: if \(\tfrac{0}{0}\) or \(\tfrac{\infty}{\infty}\), then \(\displaystyle\lim\frac{f}{g}=\lim\frac{f'}{g'}\).

EXAMPLE

Evaluate \(\displaystyle\lim_{x\to 0}\frac{\sin 3x}{5x}\).

Answer: \(\dfrac{3}{5}\) (multiply and divide: \(\dfrac{\sin 3x}{3x}\cdot\dfrac{3}{5}\to 1\cdot\tfrac{3}{5}\))

02 Differentiation Rules

Chain: \([f(g(x))]'=f'(g(x))\cdot g'(x)\) Product: \([fg]'=f'g+fg'\)

MEMORIZE \(\frac{d}{dx}\sin x=\cos x\), \(\frac{d}{dx}\cos x=-\sin x\), \(\frac{d}{dx}\tan x=\sec^2 x\), \(\frac{d}{dx}\ln x=\frac{1}{x}\), \(\frac{d}{dx}e^x=e^x\)

KEY Implicit differentiation: differentiate both sides w.r.t. \(x\); treat \(y\) as a function of \(x\) and apply chain rule → \(\frac{dy}{dx}\) appears, then solve algebraically.

EXAMPLE

If \(y=\ln(\sin x)\), find \(\dfrac{dy}{dx}\).

Answer: \(\dfrac{\cos x}{\sin x}=\cot x\)

03 Mean Value Theorem (MVT)

If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), then \(\exists\, c\in(a,b)\) such that \(f'(c)=\dfrac{f(b)-f(a)}{b-a}\)

KEY Rolle's Theorem: if additionally \(f(a)=f(b)\), then \(\exists\, c\) with \(f'(c)=0\).

IVT Intermediate Value Theorem: if \(f\) is continuous on \([a,b]\) and \(k\) is between \(f(a)\) and \(f(b)\), then \(\exists\, c\) with \(f(c)=k\).

04 Integration & FTC

FTC Part 1: \(\displaystyle\frac{d}{dx}\int_a^x f(t)\,dt = f(x)\) FTC Part 2: \(\displaystyle\int_a^b f(x)\,dx = F(b)-F(a)\)

MEMORIZE \(\int x^n dx=\dfrac{x^{n+1}}{n+1}+C\), \(\int e^x dx=e^x+C\), \(\int\frac{1}{x}dx=\ln|x|+C\)
\(\int\sin x\,dx=-\cos x+C\), \(\int\cos x\,dx=\sin x+C\), \(\int\sec^2 x\,dx=\tan x+C\)

KEY u-substitution: let \(u=g(x)\), then \(du=g'(x)dx\). Change limits when computing definite integrals.

05 Applications of Integration

Area between curves: \(\displaystyle\int_a^b[f(x)-g(x)]\,dx\) where \(f(x)\ge g(x)\)

MEMORIZE Disk method: \(V=\pi\int_a^b[f(x)]^2dx\) Washer: \(V=\pi\int_a^b\!\left([R(x)]^2-[r(x)]^2\right)dx\)

KEY Particle motion: position \(s(t)\), velocity \(v(t)=s'(t)\), acceleration \(a(t)=v'(t)\). Total distance \(=\int_a^b|v(t)|dt\). Net displacement \(=\int_a^b v(t)dt\).

06 BC Topics: Series & Parametric

Geometric series: \(\displaystyle\sum_{n=0}^{\infty}ar^n=\frac{a}{1-r}\) for \(|r|<1\) Taylor: \(f(x)=\sum_{n=0}^{\infty}\frac{f^{(n)}(a)}{n!}(x-a)^n\)

MEMORIZE \(e^x=\sum\frac{x^n}{n!}\), \(\sin x=\sum\frac{(-1)^n x^{2n+1}}{(2n+1)!}\), \(\cos x=\sum\frac{(-1)^n x^{2n}}{(2n)!}\), \(\frac{1}{1-x}=\sum x^n\)

PARAMETRIC If \(x=x(t),\,y=y(t)\): \(\dfrac{dy}{dx}=\dfrac{dy/dt}{dx/dt}\), Arc length \(=\int_a^b\!\sqrt{\left(\tfrac{dx}{dt}\right)^2+\left(\tfrac{dy}{dt}\right)^2}\,dt\)

POLAR Area \(=\dfrac{1}{2}\int_\alpha^\beta[r(\theta)]^2\,d\theta\)

EXAMPLE

Find the 3rd-degree Taylor polynomial for \(e^x\) centered at 0.

Answer: \(1+x+\dfrac{x^2}{2}+\dfrac{x^3}{6}\)

Practice Exam — 20 Questions

PROGRESS 0 / 20 answered

Q 01

Limits

★★☆ Medium

What is \(\displaystyle\lim_{x \to 0} \frac{\sin(5x)}{3x}\)?

Q 02

Continuity

★★☆ Medium

Let \(f(x) = \begin{cases} x^2 - 1 & x < 2 \\ k & x = 2 \\ 3x - 5 & x > 2 \end{cases}\). For what value of \(k\) is \(f\) continuous at \(x = 2\)?

Q 03

L'Hôpital's Rule

★★☆ Medium

Evaluate \(\displaystyle\lim_{x \to 0} \frac{e^{2x} - 1 - 2x}{x^2}\).

Q 04

IVT / Squeeze Theorem

★☆☆ Easy

If \(f\) is continuous on \([1,5]\), \(f(1)=-3\), and \(f(5)=7\), which of the following is guaranteed by the Intermediate Value Theorem?

Q 05

Chain Rule

★★☆ Medium

If \(f(x) = \sin(x^3)\), what is \(f'(x)\)?

Q 06

Implicit Differentiation

★★★ Hard

If \(x^2 + y^2 = 25\), what is \(\dfrac{dy}{dx}\)?

Q 07

Mean Value Theorem

★★☆ Medium

Let \(f(x)=x^2\) on \([1,3]\). By the Mean Value Theorem, there exists \(c\in(1,3)\) such that \(f'(c)\) equals which of the following?

Q 08

Related Rates

★★★ Hard

The radius of a circle is increasing at a rate of \(3\) cm/s. How fast is the area increasing when the radius is \(5\) cm? (Area \(A=\pi r^2\))

Q 09

FTC Part 1

★★☆ Medium

If \(g(x) = \displaystyle\int_1^x (t^2 + 1)\,dt\), what is \(g'(x)\)?

Q 10

u-Substitution

★★☆ Medium

Evaluate \(\displaystyle\int_0^1 2x\,e^{x^2}\,dx\).

Q 11

Definite Integral / FTC Part 2

★★☆ Medium

What is \(\displaystyle\int_0^{\pi/2} \cos x\,dx\)?

Q 12

Accumulation / Net Change

★★★ Hard

A particle moves along the \(x\)-axis with velocity \(v(t)=t^2-4\) for \(0\le t\le 3\). What is the total distance traveled from \(t=0\) to \(t=3\)?

Q 13

Area Between Curves

★★☆ Medium

What is the area of the region enclosed between \(y = x^2\) and \(y = x\)?

Q 14

Optimization

★★★ Hard

A rectangle has its base on the \(x\)-axis and its upper two vertices on the parabola \(y = 12 - x^2\). What is the maximum area of such a rectangle?

Q 15

Volume of Revolution (Disk)

★★★ Hard

The region bounded by \(y=\sqrt{x}\), \(y=0\), and \(x=4\) is revolved about the \(x\)-axis. What is the volume of the resulting solid?

Q 16

Particle Motion

★★☆ Medium

A particle moves along a line with position \(s(t) = t^3 - 6t^2 + 9t\). At what time \(t > 0\) does the particle change direction?

Q 17

BC — Geometric Series

★★☆ Medium

What is the sum of the series \(\displaystyle\sum_{n=0}^{\infty} \left(\frac{2}{3}\right)^n\)?

Q 18

BC — Taylor Series

★★★ Hard

Using the Maclaurin series for \(\sin x\), what is the coefficient of \(x^5\) in the series for \(\sin x\)?

Q 19

BC — Parametric Derivatives

★★★ Hard

If \(x = t^2\) and \(y = t^3\), what is \(\dfrac{dy}{dx}\) in terms of \(t\)?

Q 20

BC — Polar Area

★★★ Hard

What is the area enclosed by the polar curve \(r = 2\cos\theta\) for \(0 \le \theta \le \pi\)?

AP CalculusAB / BC

AP Calculus
AB / BC