Practice Test 6 - 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 series in a control algorithm is $\sum_{n=1}^\infty (-1)^{n-1}\frac{1}{n^{3/2}}$; determine its convergence classification.
2. A particle moves with $x=2t-1$ and $y=t^2$ on $-1\le t\le 3$; which integral gives total distance traveled?
3. The graph of a function $h(x)$ is described as being identical to the graph of the line $y=x+3$, with the exception that there is a hole at the point $(1, 4)$. Which pair of mathematical statements accurately describes this function $h(x)$?
4. A particle's position uses $p(t)=\frac{t^2-16}{t-4}$ for $t\ne4$ and $p(4)=1$; what value makes $p$ continuous at $t=4$?
5. A particle's velocity is $v(t)=t^2\sin t$. If $A(x)=\int_0^x v(t)\,dt$, what is $A'(x)$?
6. Let $H'(x)=\cos x$ and $H(x)=\sin x$. What is the value of $\int_{-\pi/2}^{\pi/3} \cos x\,dx$?
7. A probability model uses weights $\sum_{n=1}^{\infty}\frac{1}{n^{3/2}}$; does this series converge or diverge?
8. A farmer has 200 m of fencing to enclose a rectangular pen; one side borders a river needing no fence. What should be maximized?
9. A spherical balloon is being inflated. The rate of change of its radius is given by the function $r'(t)$, measured in centimeters per second. Which of the following represents the total increase in the balloon's radius from $t=2$ to $t=6$ seconds?
10. A growth model uses $f(x)=\frac{1}{x(\ln x)}$ for $x\ge2$; does $\int_{2}^{\infty} \frac{1}{x\ln x}\,dx$ converge, and to what value?
11. If $t'(x)=|x-2|-1$, on which interval(s) is $t$ decreasing?
12. Let $t(x)=\dfrac{e^x}{e^x+5}$. What is the horizontal asymptote of $t$ as $x\to\infty$?
13. For $f(x)=\cos x$ and $g(x)=0$, which setup gives total area between the curve and $x$-axis on $[0,2\pi]$?
14. The expression $\lim_{n \to \infty} \sum_{k=1}^{n} \sin\left(\frac{\pi k}{n}\right) \frac{\pi}{n}$ is the limit of a Riemann sum for a certain definite integral. Which of the following definite integrals represents this limit?
15. A rectangle's sides $x$ and $y$ change with constant area 60: $xy=60$; which relation connects $\dfrac{dx}{dt}$ and $\dfrac{dy}{dt}$?
16. Find an antiderivative of $\displaystyle \int \frac{x+1}{x^2+2x+5}\,dx$ after completing the square in the denominator.​
17. A robot arm tip is $\mathbf{r}(t)=\langle (3t-1)^4,\ \cos t+\sin t,\ \ln(t^2+1)\rangle$; find $\mathbf{r}'(t)$.
18. If $f(x)=x^5+x$ and $f(0)=0$, what is $(f^{-1})'(0)$?
19. Determine the radius of convergence of $\sum_{n=0}^{\infty}\dfrac{(x-2)^n}{n!}$.​
20. In a model, $R(x)=\dfrac{\sin x}{x^2+1}$. What is $R'(x)$?

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