Practice Test 5 - 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 function $f$ is continuous on $[0,2]$ and differentiable on $(0,2)$ with $f(0)=1$ and $f(2)=1$. Does MVT guarantee some $c$ with $f'(c)=0$?
2. A function is $K(x)=\sin\!\left(\dfrac{1}{1+\cos x}\right)$. What is $K'(x)$?
3. A continuous function $b$ has $b(6)=1$ and $b(9)=9$. Which is guaranteed for some $c\in(6,9)$?
4. A chemical amount changes at rate $A'(t)=4t^9+11$. Which expression is a general antiderivative of $A'(t)$?​
5. In the context of motion, calculus allows us to define instantaneous velocity. This is a profound shift from algebra, which can only be used to compute what type of velocity?
6. A function is $E(x)=\csc(x^2-x+4)$. What is $E'(x)$?
7. For $f(x)=|x-4|+|x-4|^2$, does $f'(4)$ exist, and why?
8. A graph of $f$ shows a smooth curve except a jump at $x=0$; does $f'(0)$ exist?
9. If the cost to produce 50 units is $C(50) = \$1500$ and the cost to produce 52 units is $C(52) = \$1580$, which of the following is the best estimate for the marginal cost when 50 units are produced?
10. Given $C(3)=4$ and $C'(3)=-1$, approximate $C(2.8)$ using local linearity at $3$.
11. Which integral gives the volume when the region under $y=2-x$ from $x=0$ to $x=2$ is revolved about the $x$-axis?
12. Let $A(x)=\int_{0}^{x} f(t)\,dt$. If $f(x)<0$ and $f$ is increasing on $(1,4)$, what describes $A$ on $(1,4)$?
13. A company's cost is $C(x)=7x^5-4x^2+9$. What is $C'(x)$?
14. The derivative of $f(x) = \frac{4x^5 - 6x^3}{2x^2}$ for $x \neq 0$ is
15. For $y=\arcsec(5x)$, what is $\dfrac{dy}{dx}$?
16. What integral gives the volume if the base is bounded by $y=\tan x$ and $y=0$ on $0\le x\le \frac{\pi}{4}$, with semicircular cross sections perpendicular to the $x$-axis?
17. A model gives $f'(x)=9x^{2}-\frac{5}{2}x^{4}$. Which is an antiderivative of $f'(x)$?
18. The derivative satisfies $k'(x)<0$ on $( -9,-4)$ and $k'(x)>0$ on $(-4,-1)$. Where does $k$ have a local minimum?
19. If $r$ is continuous on $[0,5]$ and differentiable on $(0,5)$ with $r(0)=-6$ and $r(5)=-1$, does MVT guarantee $c$ with $r'(c)=1$?​
20. For $s$ with $s'(x)=\dfrac{x^2-9}{x^2+1}$, on which interval is $s$ increasing?

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