Practice Test 3 - AP Calculus AB

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

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1. For a waveform, evaluate $\int \sec^2(5x-2)\,dx$ to find the phase accumulation.
2. A continuous profit function $P(x)$ satisfies $P(1)=-5$ and $P(3)=7$. What must exist?
3. Suppose $S(x)=\int_{3}^{x}\big(\cos t+\dfrac{1}{t^2}\big)\,dt$. What is $S'(x)$?
4. Select the washer-method integral for revolving the region between $x=\frac{1}{y+1}+3$ and $x=\tfrac{y}{2}+2$ on $[0,2]$ about the y-axis.
5. Let $f(x)=\begin{cases}\dfrac{x^2-2x-15}{x-5},&x\ne5\\12,&x=5\end{cases}$. What value should replace $12$ to remove the discontinuity at $x=5$?
6. A continuous function $f$ on $[-1,2]$ satisfies $f(-1)=0$ and $f(2)=5$. What must be true?
7. A rate is given by $u(x)=\tan\!\left(\ln(x^2+4)\right)$. What is $u'(x)$?
8. If $f(x)=(2x^3-1)^2$, what is the second derivative $f''(x)$?
9. Find the volume setup: base bounded by $y=x^2$ and $y=4$, semicircular cross sections perpendicular to the $x$-axis.
10. For $Y(x)=\begin{cases}\frac{x^2-49}{x-7},&x\ne7\\15,&x=7\end{cases}$, what type of discontinuity occurs at $x=7$?
11. A particle's velocity is $v(t)=\sin t$ for $-\pi \le t \le \pi$. What is its net displacement?
12. A machine's defect count $D(t)$ rises from 14 to 20 over 3 shifts. What is the average rate of change?
13. If $S'(x)=s(x)$ and $S(x)=\sqrt{x}+x$, evaluate $\int_{4}^{9} s(x)\,dx$.​
14. A function $f$ is continuous on $(0,4)$ but not defined at $x=0$ or $x=4$; which statement is true?
15. A savings account changes at rate $S'(t)=-2$ dollars/week for $0\le t\le5$. What is the net change in savings?
16. For $0<x<8$, the profit is $P(x)= -x^2+8x+3$; which $x$ maximizes profit?
17. What is the correct volume integral setup if the base is bounded by $y=\ln x$ and $y=0$ on $[1,e]$ with rectangular cross sections perpendicular to the $x$-axis of height $5$?
18. A pool's depth varies as $d(x)=3+\tfrac{x}{2}$ for $0\le x\le 6$. What is the average depth?
19. A car's speed $v(t)$ increases at a constant rate of $3$ m/s$^2$. Which differential equation models this?​​
20. Let $f(x)=\dfrac{2}{x(x-4)}$. Which limit notation matches the behavior as $x\to 4^+$?

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