Section I Part B — Full Practice Exam 3

Questions in this quiz (14)

1. For what value of $$k$$ does the series $$\sum_{n=1}^{\infty}\frac{k^n}{3^{2n+1}}$$ converge?

   • $$-3\le k\le 3$$
   • $$-9<k<9$$
   • $$-\frac{1}{3}\le k\le \frac{1}{3}$$
   • $$-9\le k\le 9$$

2. Let $$R$$ be the region enclosed by the graph of $$y=-x^2+6x-5$$ and the $$x$$-axis. What is the volume of the solid generated when $$R$$ is rotated about the $$y$$-axis?

   • 14.332
   • 17.278
   • 19.739
   • 201.062

3. Let $$f(x)=e^x\sin x$$ for $$0\le x\le 2\pi$$. Which of the following is the interval on which $$f$$ is decreasing?

   • $$0.785<x<3.927$$
   • $$0<x<1.571$$
   • $$1.571<x<4.712$$
   • $$2.356

4. Parametric equations are given by $$x=3t^2-t^3,\quad y=t^3-6t$$. Which of the following is the equation of the tangent line at $$t=1$$?

   • $$x-2y=5$$
   • $$x+y=-3$$
   • $$x-3y=4$$
   • $$2x-3y=5$$

5. The velocity of a particle moving in the plane is given by $$\frac{dx}{dt}=e^t\cos t,\quad \frac{dy}{dt}=e^t\sin t$$. Which of the following is the distance traveled by the particle from $$t=0$$ to $$t=1$$?

   • $$e-1$$
   • $$e$$
   • $$e^2-1$$
   • $$e^2$$

6. If the series $$\sum_{n=1}^{\infty}\frac{1}{n^p}$$ converges, where $$p>0$$, which of the following must also converge?

   • $$\sum_{n=1}^{\infty}\frac{n}{n^p}$$
   • $$\sum_{n=1}^{\infty}\frac{1}{n^{p+1}}$$
   • $$\sum_{n=1}^{\infty}\frac{\ln n}{n^2}$$
   • $$\sum_{n=1}^{\infty}\frac{1}{n^{p-1}}$$

7. A particle moves along the x-axis with velocity $$v(t)=t^3-6t^2+9t+1$$ for $$0\le t\le 5$$. Which of the following must be true on the interval $$1

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

8. The position of a particle is given by $$x(t)=t^2+3t,\quad y(t)=t^3-t$$. What is the value of $$\frac{dy}{dx}$$ when $$t=1$$?

   • $$\frac{2}{5}$$
   • $$2$$
   • $$\frac{7}{2}$$
   • $$4$$

9. Let $$R$$ be the region enclosed by the graphs $$y=e^{-x},\quad y=x^2,\quad y=0$$. If cross sections perpendicular to the x-axis are semicircles, which of the following gives the volume?

   • $$\frac{\pi}{8}\int_0^1(e^{-x}-x^2)^2\,dx$$
   • $$\frac{\pi}{4}\int_0^1(e^{-x}-x^2)^2\,dx$$
   • $$\frac{\pi}{2}\int_0^1(e^{-x}-x^2)^2\,dx$$
   • $$\pi\int_0^1(e^{-x}-x^2)^2\,dx$$

10. The Taylor series for $$f(x)$$ about $$x = 0$$ is $$1 + x^2 + x^{4/2!} + x^{6/3!} + ...$$ . What is the point of intersection of $$y = f(x)$$ and the line $$x = 0.4$$?

    • $$(0.4,1.083)$$
    • $$(0.4,1.173)$$
    • $$(0.4,1.197)$$
    • $$(0.4,1.256)$$

11. Let f be differentiable. If $$f'(x)$$>$$0$$ and $$f'(x)$$>$$0$$ and $$g(x)=\int_{1}^{x}f(t)\,dt$$ which must be true?

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

12. Let $$g(x)=\int_{1}^{x}e^{-t^2}\cos(t)\,dt$$. At what $$x$$ does $$g$$ have an inflection point?

    • 0.522
    • 0.684
    • 0.785
    • 0.902

13. Let $$f(x)=\sqrt{x}$$. If the tangent at $$x=4$$ is used to approximates $$f(4.5)$$, what is the error in the approximation??

    • 0.004
    • 0.011
    • 0.021
    • 0.034

14. Let $$F(x)=\int_{0}^{x}\cos(t^2)\,dt$$. Use trapezoidal rule with four subintervals of $$[0,1]$$ to approximate $$F(1)$$.

    • 0.759
    • 0.781
    • 0.802
    • 0.824