Practice Test 1 - AP Calculus AB

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

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1. A population satisfies $\dfrac{dP}{dt}=4P$. What is the general solution for $P$ as a function of $t$?​
2. A rate model uses $w(x)=(2x+1)\,3^{x}$. What is $w'(x)$?
3. A savings account balance is $B(t)$. Which expression represents the instantaneous earning rate at $t=12$ months?​​
4. A continuous function $g$ on $[0,6]$ has critical points $x=2,5$ and values $g(0)=1,g(2)=4,g(5)=0,g(6)=3$; where is the absolute minimum?
5. A measurement model is $r(t)=\arcsin\!\left(\frac{t-4}{3}\right)$. What is $r'(t)$?
6. A tank contains $V(t)$ liters; it leaks at a rate proportional to the amount present. Which differential equation models $V$?
7. The table suggests $c(x)$ approaches $-1$ as $x$ approaches $4$. Which limit expression correctly represents this behavior?
8. For $q(x)=\begin{cases}\frac{x^2-16}{x-4},&x\ne4\\9,&x=4\end{cases}$, what value should replace $q(4)$ to remove the discontinuity?
9. A model for revenue is $R(t)=5t^4-3t^2+7$. What is $R'(t)$?
10. A rectangle has one side on the $x$-axis and top corners on $y=9-x^2$; if the right top corner is at $x$, which $x$ maximizes area $A=2x(9-x^2)$?
11. Let $h(x)=x^2+2x$ (restricted to $x\ge -1$) and $h(3)=15$. What is $(h^{-1})'(15)$?
12. For $f(x)=\sin\!\left(\ln(3x^2-5x+4)\right)$, what is $f'(x)$?
13. The graph of $f$ is increasing and concave down on $(0,5)$. Which graph could represent $f'$ on $(0,5)$?
14. Let $Q(x)=\int_{5}^{x}\ln(1+t)\,dt$ for $x>-1$. What is $Q'(x)$?
15. An object's position is $s(x)=9x^4+2x^3-5$. Find $s'(x)$.
16. For $h(x)=\ln\!\left(\dfrac{x^2+1}{\sqrt{5x-2}}\right)$, which differentiation rules should be applied to find $h'(x)$?
17. A bacterium culture grows at rate $g(t)$ cells/hour, $t$ hours after start. What does $\int_{2}^{5} g(t)\,dt$ represent?
18. Let $X(x)=\int_{-2}^{x}\dfrac{1}{(t^2+1)^{1/2}}\,dt$. What is $X'(x)$?
19. If $M(x)=\int_{4}^{x}\left(2^{t}+\sqrt{t}\right)dt$ for $x\ge 0$, what is $M'(x)$?
20. Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$; which statement correctly describes where absolute extrema can occur?

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