Practice Test 11 - AP Calculus BC

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

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1. If $f$ is concave up on $(a,b)$, which statement about $f'$ must be true on $(a,b)$?
2. A hiker's elevation gains sum $\sum_{n=1}^{\infty} \frac{1}{n^p}$ meters; which $p$ keeps total gain finite?
3. A signal model uses $S(t)=\dfrac{\sin t}{t^2+1}$. What is $S'(t)$?
4. For $f(x)=\frac{x^2-9}{x-3}$ when $x\ne3$ and $f(3)=1$, which expression represents the limit as $x$ approaches $3$?
5. What is the correct washer-method integral for the volume when the region between $y=x$ and $y=x^2$ on $[0,1]$ is revolved about the $x$-axis?
6. A force function is $F(x)=5x^4+11x-9$. Which function is an antiderivative of $F(x)$?​
7. For $f(x)=\sqrt{1+x}$, find the degree-3 Taylor polynomial centered at $x=0$.
8. If $F'(x)=3x^2-4x+1$ and $F(x)=x^3-2x^2+x$, what is $\int_{1}^{4} (3x^2-4x+1)\,dx$?
9. Using Euler's method with $h=0.1$ for $y' = y-x^2$, $y(0)=1$, approximate $y(0.2)$.
10. The region bounded by $x=2-y$ and the y-axis for $0\le y\le 2$ is revolved about the y-axis; which integral gives the volume?
11. A game awards points totaling $\sum_{n=1}^{\infty} \frac{1}{n^p}$; for which $p$ is the total score bounded?
12. Which slope field matches $\dfrac{dy}{dx}=\sin(x)$ for $-\pi\le x\le \pi$?
13. Evaluate $\int_{0}^{2}\langle t^2-1,\,4t\rangle\,dt$ as a vector.
14. For the area computation $\int_{0}^{\pi/2} \sin(3x)\cos(3x)\,dx$, which technique is most appropriate?
15. Find the Maclaurin series for $f(x)=\arctan(x)$, expressed as an infinite power series.​
16. For differentiable $h$, if $h(-1)=3$ and $h'(-1)=-2$, find the tangent line to $y=h(x)$ at $x=-1$.​
17. The temperature is modeled by $T(t)=50+10\sin t$ for $0\le t\le\pi$; what is the average temperature?
18. Let $t(x)=\dfrac{\sin(x)}{x-\pi}$. Which statement best describes the behavior as $x\to\pi$?
19. Find $\vec r(t)$ if $\vec r'(t)=\langle 4,\,3t^2,\,2\sin t\rangle$ and $\vec r(0)=\langle -1,5,0\rangle$.
20. A tank's temperature is $T(t)=70+5\,\ln(t+1)$; what does $T'(2)$ mean?

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