Mini Exam 8 — Timed Mini Exams

Questions in this quiz (10)

1. For what values of $$x$$ does the series $$\sum_{n=1}^{\infty}\frac{(3x-2)^n}{n}$$ converge?

   • $$-\frac{1}{3}<x<1$$
   • $$-\frac{1}{3}\le x<1$$
   • $$\frac{1}{3}\le x<1$$
   • $$-\frac{1}{3}\le x\le1$$

2. If $$f(x)=\begin{cases} x^2, & x<1\\ 2x-1, & x\ge1 \end{cases}$$ then $$\lim_{x\to1}f(x)=$$

   • $$0$$
   • $$1$$
   • $$2$$
   • It does not exist

3. Evaluate $$\lim_{x\to2}\frac{\ln x^2-\ln4}{x-2}$$

   • $$0$$
   • $$1$$
   • $$2$$
   • $$\frac{1}{2}$$

4. A population $$P(t)$$ increases at a rate proportional to its size. If $$P(0)=50$$ and $$P(1)=61.070$$, how many years will it take the population to triple?

   • $$4.394$$
   • $$5.493$$
   • $$6.931$$
   • $$8.047$$

5. A rectangle with one side on the $$x$$-axis is inscribed in the triangle formed by the lines $$y=x,\quad y=0,\quad 2x+y=12$$ The area of the largest such rectangle is

   • $$6$$
   • $$3$$
   • $$\frac{5}{2}$$
   • $$5$$

6. The number of values of $$k$$ for which $$f(x)=e^x$$ and $$g(x)=k\sin x$$ have a common point of tangency is

   • $$0$$
   • $$1$$
   • large but finite
   • infinite

7. Evaluate $$\int\cos^3x\,dx$$

   • $$\frac{\cos^4x}{4}+C$$
   • $$\sin x-\frac{\sin^3x}{3}+C$$
   • $$\sin x+\frac{\sin^3x}{3}+C$$
   • $$\cos x-\frac{\cos^3x}{3}+C$$

8. The volume, in cubic feet, of an inner tube with inner diameter 4 feet and outer diameter 8 feet is

   • $$6\pi^2$$
   • $$12\pi^2$$
   • $$24\pi^2$$
   • $$48\pi^2$$

9. Suppose the function $$f$$ is continuous on $$1\le x\le2$$, that $$f'(x)$$ exists on $$1<x<2$$, that $$f(1)=3$$, and that $$f(2)=0$$. Which of the following statements is not necessarily true?

   • Integral exists
   • There exists c such that f''(c)=0
   • Intermediate value property
   • Limit exists

10. An ellipse has major axis 20 and minor axis 10. Rounded off to the nearest integer, the maximum area of an inscribed rectangle is

    • 50
    • 79
    • 82
    • 100