Practice Test 1 - AP Calculus BC

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

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1. A temperature model is $T(t)=3t^{-2}-5t^4+11$; which expression equals $T'(t)$?
2. A signal decay is modeled by $\sum_{n=1}^{\infty}\frac{1}{\sqrt{n^2+4}}$. Use the integral test to decide convergence.
3. A differentiable function $f$ has $f'(x)$ crossing the $x$-axis upward at $x=5$ (negative to positive); which could be true about $f$ at $x=5$?
4. For $t(x)=\dfrac{(x-2)^2}{(x-2)(x+3)}$, which statement correctly identifies any vertical asymptote?
5. If $q(4)=-1$ and $q'(4)=\tfrac{1}{3}$, which is the linearization of $q$ at $x=4$?​
6. A plant population satisfies $\frac{dy}{dt}=0.6y\left(1-\frac{y}{900}\right)$. Which best describes the long-term limiting population?
7. If $\dfrac{1}{2}-|x|\le f(x)\le \dfrac{1}{2}+|x|$ near $0$, what is $\lim_{x\to 0} f(x)$?
8. A numerical method uses $\sum_{n=1}^{\infty} (-1)^{n-1}\frac{1}{n+\sin n}$; classify its convergence.
9. Which integral gives the area of the region traced by $r=3\sin\theta$ on $0\le\theta\le\pi$?
10. Which of the following is the slope of a tangent line to the graph of the polar curve $r = 1 - 2\cos\theta$ at the pole?
11. Find the correct volume setup: base bounded by $y=1-x^2$ and $y=0$ on $[-1,1]$, semicircle cross sections perpendicular to $x$.
12. Given $f'(x)=(x+2)(x-5)$, on which interval(s) is $f$ decreasing?
13. For $f(x)=\sqrt{x}$ and $g(x)=x^2$ on $[0,1]$, which integral represents the area between them?
14. A box with square base has volume $108\text{ in}^3$. If surface area is to be minimized, what is the objective function?
15. For $f(x)=\begin{cases}3x,&x<2\\7,&x=2\\x^2+1,&x>2\end{cases}$, which expression represents $\lim_{x\to2}f(x)$?
16. Which method is most appropriate for evaluating $\lim_{x\to 0}\dfrac{\sin x}{x+\sin x}$?
17. A graph of $v(x)$ approaches the line $y=-2$ as $x\to\pm\infty$; what is $\lim_{x\to\infty} v(x)$?
18. A curve is given implicitly by $\sin(xy)=x+y$. What is $\dfrac{dy}{dx}$ at a general point $(x,y)$?
19. A field has a discontinuity at $x=2$ with total $\int_{0}^{4} \frac{1}{|x-2|}\,dx$; does it converge, and what value?
20. A curve is $x(t)=t^2+\frac{1}{2}t$ and $y(t)=t^2-\frac{1}{2}t$; find $\frac{dy}{dx}$.

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