Practice Test 5 - 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 $f(x)=\arctan x$, using $T_3(x)=x-\frac{x^3}{3}$, if $|f^{(4)}(x)|\le 24$ on $[0,0.3]$, what is the maximum error at $x=0.3$?
2. Given $R(x)=\dfrac{2-x}{x+8}$, what is the horizontal asymptote as $x\to\infty$?
3. Does $y=\dfrac{1}{x}$ satisfy $\dfrac{dy}{dx}=-y^2$ for $x\ne 0$?
4. A function $g$ satisfies $g'(x)>0$ on $(2,5)$ and $g'(x)<0$ on $(5,9)$; where is a local maximum?
5. An antiderivative of $m(x)$ is $M(x)=\dfrac{x^5}{5}-2x^2$. Compute $\int_{2}^{0} m(x)\,dx$.
6. A quantity $y(x)$ satisfies $\frac{dy}{dx}=\frac{x}{y+2}$. What is the general solution for $y(x)$?​
7. Consider $v(x)=\dfrac{\cos x}{x}$ for $x\ne0$ and $v(0)=1$. What type of discontinuity is at $x=0$?
8. Find $\displaystyle \lim_{x\to 0}\frac{(1+x)^5-1}{x}$ by expanding or factoring to remove $0/0$.​
9. Which integral gives the volume when the region bounded by $y=\sqrt{x}$, $y=0$, and $x=4$ is revolved about $y=-2$?
10. A model uses $f(x)=4x^{3/2}-7$. Which of the following is an antiderivative of $f(x)$?​
11. Choose the correct volume setup when the region between $y=x$ and $y=0$ for $0\le x\le 2$ is revolved about $y=3$.
12. A series for approximation is $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{1}{n^{0.9}}$. Does it converge?
13. Compute $\displaystyle\lim_{x\to 0}\dfrac{e^{x}-1-x}{x^2}$, appearing in a Taylor remainder bound.
14. A concentration satisfies $\frac{dC}{dt}=\frac{t}{1+C}$ with $C(0)=1$; using Euler's method with $h=0.5$, approximate $C(1)$.
15. Let $f(x)=\cos x$. What is the degree $6$ Taylor polynomial centered at $x=0$?​
16. A population satisfies $\dfrac{dP}{dt}=3P$. Is $P(t)=5e^{3t}$ a solution to the differential equation?
17. Given $H(x)=\int_{-2}^{x}(t^2+4t+3)\,dt$, on which interval is $H$ increasing?
18. If $y = 2\sin(t) + 3\cos(t)$, find the value of $\frac{d^2y}{dt^2}$ at $t = \frac{\pi}{2}$.
19. For a decay process, evaluate $\int_{0}^{\infty} e^{-3x}\,dx$; does it converge, and to what value?
20. A growth model uses $g(x)=\dfrac{4}{x\ln x}$; compute $\int \dfrac{4}{x\ln x}\,dx$.

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