Mini Exam 4 — Timed Mini Exams

Questions in this quiz (9)

1. What are all the values of $$x$$ for which the series $$\sum_{n=1}^{\infty}\frac{(x-1)^n}{n2^n}$$ converges?

   • $$-3\le x\le1$$
   • $$-1\le x<3$$
   • $$-3\le x\le-1$$
   • $$1\le x\le3$$

2. Let $$f(x)=|x^2-1|$$. Let $$R$$ be the region bounded by $$f$$, the $$x$$-axis, and the vertical lines $$x=-2$$ and $$x=2$$. Let $$T_4$$ represent the approximation of the area of $$R$$ using the trapezoidal rule with $$n=4$$. The quotient $$\frac{T_4}{\int_{-2}^{2}f(x)\,dx}$$ is approximately

   • $$0.912$$
   • $$0.987$$
   • $$1$$
   • $$1.284$$

3. If $$f(x)=\tan x$$, then $$f'(x)$$ has how many zeros over the closed interval $$[0,2\pi]$$?

   • $$0$$
   • $$1$$
   • $$2$$
   • $$3$$

4. Consider the region in the first quadrant bounded by $$y=x^3$$ over $$[0,2]$$. Let $$L_2$$ represent the Riemann approximation using left endpoints, $$R_2$$ using right endpoints, $$M_2$$ using midpoints, and $$T_2$$ the trapezoidal approximation. Which of the following statements is true?

   • $$R_2<T_2<\int_0^2x^3\,dx<M_2<L_2$$
   • $$L_2<M_2<\int_0^2x^3\,dx<T_2<R_2$$
   • $$M_2<L_2<\int_0^2x^3\,dx<R_2<T_2$$
   • $$L_2<M_2<\int_0^2x^3\,dx<R_2<T_2$$

5. The sum of the infinite geometric series $$\frac{3}{4}+\frac{3}{16}+\frac{3}{64}+\frac{3}{256}+\cdots$$ is

   • $$0.875$$
   • $$1.000$$
   • $$1.125$$
   • $$1.250$$

6. Let $$f$$ be a strictly increasing differentiable function on the closed interval $$[2,6]$$ such that $$f(2)=5$$ and $$f(6)=21$$. Which of the following must be true for the function $$f$$ on the interval $$[2,6]$$? I. The average rate of change of $$f$$ is $$4$$ II. The absolute maximum value of $$f$$ is $$21$$ III. $$f'(4)>0$$

   • I only
   • II only
   • I and II
   • I, II, and III

7. Let $$F(x)$$ be an antiderivative of $$f(x)=e^{3x}$$. If $$F(0)=1$$, then $$F(2)=$$

   • $$\frac{e^6}{3}$$
   • $$\frac{e^6}{3}+1$$
   • $$\frac{e^6+2}{3}$$
   • $$\frac{e^6-1}{3}$$

8. The base of a solid is the region in the first quadrant bounded by $$y=4-x^2$$ The cross sections perpendicular to the $$x$$-axis are squares. Find the volume of the solid.

   • $$4.267$$
   • $$5.333$$
   • $$8.533$$
   • $$17.067$$

9. Evaluate $$\int_0^{\pi/2}\sin x\cos x\,dx$$

   • $$\frac{1}{4}$$
   • $$\frac{1}{2}$$
   • $$\frac{1}{8}$$
   • $$1$$