Practice Test 3 - AP Calculus BC

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

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1. Compute $\int_{-\infty}^{0} \frac{1}{1+x^2}\,dx$; does it converge, and what is the value?
2. A machine's output rate is $q(t)=\ln(t+5)$. Let $O(x)=\int_{0}^{x} q(t)\,dt$. What is $O'(x)$?
3. Let $F(x) = \int_0^x \frac{t^2 - 3}{t^2 + 1} \, dt$. For what positive value of $x$ does $F(x)$ have a local minimum?
4. For $x^2e^y+y=4$, what is $\dfrac{dy}{dx}$ at a general point $(x,y)$?
5. Classify $\sum_{n=1}^{\infty} (-1)^{n}\frac{\ln(n+1)}{\sqrt{n}}$ as absolutely convergent, conditionally convergent, or divergent.
6. A sum is $\sum_{n=1}^{\infty} \frac{1}{n}+\frac{1}{n^2}$. What does the nth-term test conclude?
7. The polar curve $r = 1+2\cos\theta$ has a vertical tangent line at which of the following values of $\theta$?
8. What is the interval of convergence of $\sum_{n=1}^{\infty}\frac{(x-2)^n}{n\,4^n}$?​
9. Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{(1+4x)^{1/2}-1}{x}$?
10. A population changes at rate $P'(t)=6t-2$ for $0 \le t \le 3$. What is the average rate of change on $[0,3]$?
11. A numerical approximation uses $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{n}{2^n}$. Does it converge?
12. For $h(x)=\dfrac{1}{(x-2)^2}$, what type of discontinuity occurs at $x=2$?
13. A damping kernel is $\int_{0}^{1} \ln(x)\,dx$; does the improper integral converge, and what is its value?
14. A curve is defined implicitly by $\sin(xy)+x=0$; find $\dfrac{dy}{dx}$ at a general point $(x,y)$.
15. Let $q(x)=\dfrac{|x|}{x}$ for $x\ne0$, and $q(0)=0$. Is $q$ continuous at $x=0$, and why?
16. For $r(x)=\sin x$, what is the instantaneous rate of change at $x=\frac{\pi}{3}$?
17. In the slope field for $\frac{dy}{dx}=\frac{1}{x-y}$, where are the solution curves' slopes undefined?
18. Let $u(x)=\begin{cases}\sin x,&x\ne\pi\\0,&x=\pi\end{cases}$. Is $u$ continuous at $x=\pi$, and why?
19. A rectangle is to be placed inside a circle of radius 5. To maximize area, what should be optimized?
20. A continuous function $r$ on $[-4,2]$ has critical points $x=-3,-1$ with $r(-4)=-2,r(-3)=1,r(-1)=-3,r(2)=0$. Where is the absolute minimum?

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