Practice Test 7 - AP Calculus BC

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

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1. For a growth curve, compute $\lim_{x\to 0}\frac{(1+x)^7-1}{x}$ to estimate initial slope.
2. Water depth is $d(t)=5+\sin t$ (cm) for $0\le t\le\pi$. What is the average depth over this interval?
3. Rotate the region between $y=2x$ and $y=x^2$ on $0\le x\le2$ about the $x$-axis; which volume integral is correct?
4. A roller coaster's position is $\mathbf{r}(t)=\langle \sec t,\ \csc t,\ t\rangle$ for $0<t<\pi/2$; find $\mathbf{r}'(t)$.
5. For $y=\cos x$ and $y=0$ on $[0,2\pi]$, which setup gives the total area between the curves?
6. Which integral represents area between $x=10-y$ (right) and $x=2y^2$ (left) on $0\le y\le 2$?
7. A particle moves with $r(t)=\langle t^2-1,\,3t\rangle$ for $0\le t\le2$; what is its speed at $t=1$?
8. If $f(x)=\sqrt{7-3x}$ gives concentration, what is $f'(x)$?
9. A decay model uses $h(x)=e^{-x}$ for $x\ge0$; does $\int_{0}^{\infty} h(x)\,dx$ converge, and to what value?
10. Let $f(x)=x^3$ and $g(x)=x^2$. Which setup gives total area between the curves on $[0,1]$?
11. Given $L(x)=\dfrac{x^3-5}{2x^4+1}$, what is the horizontal asymptote as $x\to\infty$?
12. Compute $\displaystyle\lim_{x\to 0}\dfrac{\sin(2x)-2\sin x}{x^3}$ for a third-order comparison near zero.
13. Let $u(x)=\begin{cases}\ln(x),&x>0\\0,&x\le0\end{cases}$. What type of discontinuity occurs at $x=0$?
14. Which integral gives the volume when the region between $y=e^x$ and $y=1$ on $[0,\ln 3]$ is revolved about the $x$-axis?
15. Which of the following is the best estimate of the growth rate $H'(12)$ in centimeters per day?
16. A velocity integral is $\int \frac{\sec^2 x}{\tan x}\,dx$; which technique is most appropriate?
17. For $s(x)=\frac{x^3-8}{x-2}$ when $x\ne2$, which expression represents $\lim_{x\to2}s(x)$?
18. In a physics model, accumulated drift is $\sum_{n=1}^{\infty}\frac{1}{(n+1)\sqrt{n+1}}$. Use the integral test for convergence.
19. The cost function is $C(x)=7x^4-5x^2+9$. What is the third derivative $C^{(3)}(x)$?
20. If $\int_{-3}^{1} f(x)\,dx=2$ and $\int_{-3}^{1} g(x)\,dx=-5$, what is $\int_{-3}^{1} \big(3f(x)-2g(x)\big)\,dx$?

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