# AP Tests•AP Calculus AB•Questions

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#### 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}