Practice Test 7 - AP Calculus AB

20 questions. Answer them in any order, then submit to see your score and every explanation.

Start practice test

1. A runner's distance $s(t)$ increases with speed proportional to $\sqrt{s(t)}$. Which differential equation models $s$?
2. Select the washer-method integral when the region between $y=\tfrac{3}{2}x+2$ and $y=\tfrac{1}{2}x+1$ on $[0,4]$ is revolved about the x-axis.
3. Let $E(x)=\int_{-2}^{x}\dfrac{t^2+1}{t^2+4}\,dt$. What is $E'(x)$?
4. A function is $B(x)=\sin x+\cos x+\tan x$. Which differentiation procedure is sufficient to find $B'(x)$?
5. Let $k(x)=\begin{cases}x^2,&x<2\\4,&x=2\\5,&x>2\end{cases}$. Is $k$ continuous at $x=2$, and why?
6. What integral gives the volume if the base is bounded by $x=0$ and $x=\sqrt{9-y^2}$ for $-3\le y\le 3$, with semicircular cross sections perpendicular to the $y$-axis?
7. For $f(x)=\frac{|x|}{x}$ with $f(0)$ undefined, does $f'(0)$ exist, and why?
8. For $s(x)=\begin{cases}\sin x,&x\ne0\\1,&x=0\end{cases}$, is $s$ continuous at $x=0$, and why?
9. Define $F(x)=\int_{0}^{x} g(t)\,dt$. If $g$ is positive on $(0,2)$ and negative on $(2,4)$, what must be true about $F$ at $x=2$?
10. An antiderivative of $p(x)$ is $P(x)=\frac{1}{x}-x^2$. Evaluate $\int_{1}^{3} p(x)\,dx$ using FTC.
11. The derivative $f'(x)$ decreases on $(-3,0)$ and increases on $(0,5)$. Where is $f$ concave up and down?
12. A physics integral is $\int \frac{x^2+8x+20}{x+4}\,dx$. Which is an antiderivative?
13. A function satisfies $f'(x)=4-4x$ for $0\le x\le 2$. What is $f(2)-f(0)$?
14. A bank account balance changes at $-15$ dollars/week due to fees; what does this mean?
15. A rectangle with perimeter $60$ has length $x$ and width $30-x$; which $x$ maximizes area $A=x(30-x)$?
16. Does $y=x^2+1$ satisfy $\dfrac{dy}{dx}=2x$ at every $x$?
17. A particle's average velocity is $\dfrac{(x+1)^2-4}{x-1}$; find $\lim\limits_{x\to 1}\dfrac{(x+1)^2-4}{x-1}$.
18. For $0<x<5$, maximize $f(x)=x(5-x)^2$; which $x$ gives the maximum?
19. A product's demand price is $p(x)=50-2x$ dollars; revenue is $R(x)=xp(x)$ for $0<x<25$. Which $x$ maximizes revenue?
20. A differentiable one-to-one function $p$ satisfies $p(0)=7$ and $p'(0)=5$; compute $(p^{-1})'(7)$.

Practice tests