Practice Test 8 - AP Calculus AB

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

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1. Let $n(x)=\sin\!\left(\ln(7-4x)\right)$. What is $n'(x)$?
2. Which of the following is the most appropriate method to find the limit $\lim_{x \to \infty} \frac{\sqrt{9x^2 + x}}{2x+1}$?
3. For $0<x<9$, maximize $f(x)=x(9-x)^{1/2}$; which $x$ gives the maximum?
4. Let $f$ be continuous on $[1,5]$ with $f(1)=-2$ and $f(5)=6$. What must be true?
5. In a heat-transfer model, find $\displaystyle \int \frac{1}{x^2+6x+13}\,dx$.​
6. Given $r(x)=\begin{cases}x+4,&x< -1\\2,&x=-1\\x^2,&x>-1\end{cases}$, is $r$ continuous at $x=-1$, and why?
7. A cyclist's speed is $s(t)$ miles/hour at time $t$ hours. What does $\int_{2}^{4} s(t)\,dt$ represent?
8. For $u(x)=\begin{cases}1,&x<2\\frac{1}{x-2},&x>2\\0,&x=2\end{cases}$, what type of discontinuity is at $x=2$?
9. A signal is $S(t)=(e^{-t})(\sin t)$. What is $S'(t)$?
10. A particle's velocity is $v(t)$ with antiderivative $V(t)=t^3-6t$. Find $\int_{-2}^{1} v(t)\,dt$.
11. To determine the population's growth rate at the exact moment it reaches 50,000 cells, what is the most appropriate calculus-based approach?
12. Let $A(x)=\ln\!\big((x-2)^2+5\big)$. What is $A'(x)$?
13. A function $u$ satisfies $\lim_{x\to5}u(x)=7$, but $u(5)$ is not defined. Is $u$ continuous at $5$, and why?​
14. A bacteria culture grows at rate $B'(t)=6$ mg/hr for $0\le t\le1$ and $B'(t)=0$ for $1\le t\le4$. Total change?
15. Given $\int_{5}^{8} s(x)\,dx=-3$, what is $\int_{8}^{5} \left(\tfrac12 s(x)\right)\,dx$?​
16. At $x=-14$, $g'(-14)=0$ and $g''(-14)<0$; what does the Second Derivative Test imply?
17. A particle has acceleration $a(t)=2t$ with $v(0)=0$ and $x(0)=0$. What is the displacement on $[0,2]$?
18. A function is $d(x)=\sin\!\left(\sqrt{3x+7}\right)$. What is $d'(x)$?
19. A differentiable function $S(t)$ models savings. Which expression represents the instantaneous rate savings changes at $t=0$?
20. What is the correct volume integral setup if the base is bounded by $y=\sin x$ and $y=0$ on $[0,\pi]$ with square cross sections perpendicular to the $x$-axis?

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