Practice Test 6 - 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 rectangle has area $A=x(18-x)$ for $0<x<18$; which $x$ maximizes $A$?
2. Given $g'(5)=0$ and $g''(5)>0$, what does the Second Derivative Test conclude at $x=5$?
3. A differentiable function $F$ has $F(6)=1$ and $F'(6)=3$. Approximate $F(5.9)$ using local linearity.
4. If $f(x)=\frac{x^2+1}{3}$, find $\lim_{x\to -2} f(x)$ using limit laws.
5. Compute $\int (3x^2-4x)(x^3-2x^2+7)^5\,dx$ from a composite response function.
6. Direct substitution into the limit $\lim_{x \to 0} \frac{e^x - 1}{\sin(x)}$ yields $\frac{0}{0}$. Which procedure is most effective for evaluating this limit without using L'Hôpital's Rule?
7. If $T(x)=\int_{-3}^{x}\left(e^{2t}-\cos t\right)dt$, what is $T'(x)$?
8. Let $X(x)=\begin{cases}2x+3,&x< -1\\1,&x=-1\\2x+3,&x>-1\end{cases}$. Is $X$ continuous at $x=-1$, and why?
9. Let $\theta(x)=\begin{cases}x+5,&x< -1\\x^2,&x\ge -1\end{cases}$. What type of discontinuity does $\theta$ have at $x=-1$?
10. A graph of $f'$ touches the $x$-axis at $x=3$ and stays positive on both sides; which could be $f$ at $x=3$?​
11. A lake's pollutant concentration $C(t)$ decreases at a rate proportional to $C(t)$ due to natural cleanup. Which differential equation models $C$?
12. A continuous function $p$ on $[1,5]$ has critical points $x=2,4$. If $p(1)=0,p(2)=3,p(4)=1,p(5)=2$, where is the absolute maximum?
13. Water flows into a tank at rate $r(t)=3t-6$ liters/min for $0\le t\le4$. What is the net change in volume?
14. A function $v$ is continuous on $[0,4]$ with $v(0)=5$ and $v(4)=1$. Does the Mean Value Theorem guarantee a $c$ with $v'(c)=-1$?
15. If $p$ has a vertical tangent at $x=5$, does $p'(5)$ exist, and why?
16. A slope field corresponds to $\frac{dy}{dx}=\frac{y-x}{y+x}$. At points where $y=x$, slope segments are
17. A population of bacteria is modeled by the logistic function $P(t) = \frac{2000}{1 + 49e^{-0.2t}}$, where $t$ is time in days. What number does the population approach as time increases without bound?
18. From the graph, $H(x)$ approaches $-2$ as $x$ approaches $-1$ from the right only. Which limit notation represents this?
19. A probability density uses $\int \dfrac{10x}{\sqrt{5x^2+9}}\,dx$. Find an antiderivative.
20. For $r(x)=\begin{cases}3,&x<0\\1,&x=0\\-3,&x>0\end{cases}$, what type of discontinuity occurs at $x=0$?

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