AP Tests•AP Calculus AB•Questions

Try some AP Calculus AB exam practice questions

54 video review lessons
Review AP Calculus AB by watching and listening to over 11 hours worth of videos carefully coordinated to the AP Calculus AB syllabus.
182 AP Calculus AB practice questions
Test your understanding of each concept without having to take an entire AP Calculus AB practice exam.
Customized AP Calculus AB study plan
Once you're ready, take our diagnostic exam to see where you need to focus your efforts.
Access through June
You'll have unlimited access to our entire AP Calculus AB review suite through the AP exam.
Full-length AP Calculus AB practice test
Personalized progress

AP Calculus AB

1. 1.

Suppose the function $f$ has the graph as shown above. Which of the following limits exist?

1. $\lim\limits_{x \to a}f(x)$
2. $\lim\limits_{x \to b}f(x)$
3. $\lim\limits_{x \to c}f(x)$
1. Limit exists because the limit is independent of the actual output $f(a)$. It does not matter where the point actually is at $x=a$
2. Limit does not exist because $\lim\limits_{x\to b^-}f(x)\not=\lim\limits_{x\to b^+}f(x)$
3. Limit exists because it is a simple curve
2. 2.

Let $f$ be the function defined by $f(x)=x^3+x-4$. What is the value of $c$ for which the instantaneous rate of change of $f$ at $x=c$ is the same as the average rate of change of $f$ over $[-1,3]$?

\begin{aligned} f^\prime(x)&=3x^2+1\\ \text{Ave slope }&=\frac{f(3)-f(-1)}{3-(-1)} = \frac{26-(-6)}{4}=\frac{32}{4}=8\\\\ \text{We need } f^\prime(x)&=3x^2+1=8\\ x&=\sqrt{\frac{7}{3}} \end{aligned}
3. 3.

$$\int \frac{x}{(3x^2+5)^4}\,dx=$$

\begin{aligned} \int \frac{x}{(3x^2+5)^4}\,dx\\ \text{Use substitution: } u&=3x^2+5\\ du&=6x\,dx\\ \frac{1}{6}\,du&=x\,dx\\ \text{So }\int \frac{x}{(3x^2+5)^4}\,dx&= \frac{1}{6}\int \frac{1}{u^4}\,du\\ &=\frac{1}{6}\int u^{-4}\,du\\ &=\frac{1}{6}\frac{u^{-3}}{-3}+C\\ &=\frac{-1}{18(3x^2+5)^3+C} \end{aligned}