Practice Test 10 - AP Calculus AB

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

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1. If $f''(x)=-(x-5)^2$ for all $x$, on which intervals is $f$ concave up and concave down?
2. If $F(x)=\sqrt{x}=x^{1/2}$, what is the second derivative $F''(x)$ for $x>0$?
3. A function is $w(x)=\sin\!\left(\ln\!\left(x^3+2\right)\right)$. What is $w'(x)$?
4. Let $F(x)=\int_{1}^{x} g(t)\,dt$. If $g(x)<0$ but increasing on $(1,5)$, which describes $F$ on $(1,5)$?
5. A reservoir's volume changes at rate $V'(t)$ acre-feet/day, $t$ days after measurement begins. What does $\int_{2}^{9} V'(t)\,dt$ represent?
6. Given the implicit relation $\dfrac{x}{y}+y=3$, what is $\dfrac{dy}{dx}$ in terms of $x$ and $y$?
7. A temperature change uses $\dfrac{(x-1)^2}{x-1}$; find $\lim\limits_{x\to 1}\dfrac{(x-1)^2}{x-1}$.
8. A balloon's radius is $r(t)$ cm. Which describes the instantaneous rate of change of radius at $t=4$ seconds?
9. Let $S(x)=\int_{-2}^{x}\left(\frac{1}{t^2+4}\right)dt$. What is $S'(x)$?
10. A tank's inflow rate is $r(t)=2e^{t}$; with antiderivative $R(t)=2e^{t}$, compute $\int_{0}^{2} r(t)\,dt$.
11. Find $\displaystyle \lim_{x\to 0}\frac{\ln(1+\sin x)-\sin x}{x^2}$.
12. A continuous function $h$ satisfies $h(-5)=8$ and $h(-1)=2$. What must exist?
13. A population grows at rate $P'(t)=12t^5-5t^2$. Find an antiderivative of $P'(t)$.​
14. For $f(x)=\dfrac{1}{x-2}$, which limit notation represents the behavior as $x$ approaches $2$ from the right?
15. A function has $f'(x)>0$ on $( -\infty,0)$ and $(2,\infty)$, but $f'(x)<0$ on $(0,2)$. If $f''(x)>0$ on $( -\infty,\infty)$, what is true?
16. On $(2,8)$, $f'(x)<0$ for all $x$ and $f''(x)>0$ for all $x$. Which graph could be $f$?
17. A function is $\psi(x)=\ln\!\big(\ln(x+2)\big)$. What is $\psi'(x)$?
18. If $f''(x)=\dfrac{x}{(x-2)^2}$ for $x\ne2$, where is $f$ concave up and concave down?
19. For $p(x)=\dfrac{1}{(x-3)^2}$, which limit expression correctly describes the behavior as $x\to 3$?
20. For $r(x)=\begin{cases}\dfrac{x^2-25}{x+5},&x\ne-5\\9,&x=-5\end{cases}$, what value should replace $9$ to remove the discontinuity at $x=-5$?​

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