Practice Test 12 - AP Calculus BC

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

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1. Let $g(x) = \int_a^x f(t) \, dt$ and $h(x) = \int_b^x f(t) \, dt$, where $a$ and $b$ are constants. Which of the following statements must be true for all $x$ in the domain of $f$?
2. A lens model uses $\int \dfrac{dx}{x^2-4x+13}$; which antiderivative is correct?
3. For $x^2+ y^2=\ln(xy)$, find $\dfrac{dy}{dx}$ at a general point where $x>0,y>0$.
4. An alternating adjustment is $\sum_{n=1}^{\infty}(-1)^{n-1}\frac{n}{n^2+1}$. Does the series converge?
5. Let $f(x)=\sqrt{1+x}$. Find the degree-3 Taylor polynomial centered at $x=0$.​
6. For signal processing, select a method for $\int \ln(x)\,dx$ that is most appropriate.
7. What is the correct washer-method setup when the region bounded by $y=x^2+1$ and $y=5$ is revolved about $y=0$?
8. A quantity satisfies $\frac{dy}{dt}=\frac{1}{5}y$ with $y(0)=25$. Find $y(5)$.
9. A rate is $F(x)=\dfrac{\sqrt{x}+1}{x-3}$. Find $F'(x)$ for $x>0$.
10. Let $u$ be continuous on $[3,7]$ and differentiable on $(3,7)$ with $u(3)=8$ and $u(7)=0$. Does MVT guarantee some $c$ with $u'(c)=-2$?
11. Let $g(x)=\begin{cases}x+2,&x<1\\4,&x=1\\x^2+3,&x>1\end{cases}$. Is $g$ continuous at $x=1$, and why?
12. A lake's pollution changes at rate $p(t)=t\cos t$. If $M(x)=\int_{3}^{x} p(t)\,dt$, what is $M'(x)$?
13. If $p(t)=t^2+3t$, what is the instantaneous rate of change at $t=2$?
14. For $x=\ln(1+t)$ and $y=t^2$ on $0\le t\le3$, which integral sets up the arc length?
15. Base region is bounded by $x=\sqrt{y}$ and $x=0$ for $0\le y\le4$; cross sections perpendicular to the $y$-axis are equilateral triangles. Which integral gives the volume?
16. Approximating $\sum_{k=1}^{\infty}(-1)^k\frac{7}{k^{5/2}}$ by $S_{9}$, what is the maximum possible error?
17. The revenue model is $R(x)=9x^4-2x^3+6$. What is $R'(x)$?
18. For $h(x)=\dfrac{7x^3+x}{-2x^3+5}$, find the horizontal asymptote as $x\to-\infty$.
19. The region under $y=\frac{3}{2}x$ above the x-axis on $0\le x\le 2$ is revolved about the x-axis; which integral gives volume?
20. In a damping model, find $\displaystyle \lim_{x\to3}\frac{x-3}{\sqrt{x+6}-3}$ by algebraic manipulation.​

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