Section I Part B — Full Practice Exam 1

Questions in this quiz (15)

1. Let $$f(x)=\int_{0}^{x^2-1}e^{t^2}\,dt.$$At what value of $$x$$ does $$f(x)$$ have a relative maximum?

   • $$x=-1$$
   • $$x=0$$
   • $$x=1$$
   • No value of $$x$$

2. The length of the path described by the parametric equations $$x=t^2+1$$ and $$y=2t-3,$$ when $$0\le t\le2,$$ is:

   • $$3.771$$
   • $$4.472$$
   • $$5.236$$
   • $$5.916$$

3. The length of the curve determined by the equations $$x=t^2-1$$ and $$y=\ln(t+1)$$ from $$t=0$$ to $$t=3$$ is:

   • $$4.215$$
   • $$5.873$$
   • $$6.944$$
   • $$9.344$$

4. What is the volume of the solid generated by rotating about the $$x$$-axis the region in the first quadrant enclosed by the curve $$y=\cos x$$ and the $$x$$- and $$y$$-axes?

   • $$0.785$$
   • $$1.233$$
   • $$1.571$$
   • $$2.467$$

5. The $$x$$-coordinate of the point on the curve $$y=x^2-2$$ closest to the point $$(1,0)$$ is:

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

6. The area of the region inside the polar curve $$r=3+3\cos\theta$$ and outside the polar curve $$r=3$$ is:

   • $$4.712$$
   • $$7.069$$
   • $$9.425$$
   • $$25.069$$

7. $$\lim_{x\to1}\frac{\int_{1}^{x}e^{-t^2}\,dt}{x-1}$$ is:

   • $$e$$
   • $$e^{-1}$$
   • $$-e^{-1}$$
   • $$0$$

8. What is the approximation of the value of $$e^1$$ using the third-degree Taylor polynomial about $$x=0$$ for $$e^x$$?

   • $$2.5$$
   • $$2.667$$
   • $$2.833$$
   • $$3$$

9. $$\int_{0}^{1}\frac{1+\sqrt{1-x^2}}{\sqrt{1-x^2}}\,dx=$$?

   • $$1$$
   • $$\frac{\pi}{2}$$
   • $$1+\frac{\pi}{2}$$
   • $$\pi$$

10. The region bounded by the $$x$$-axis and the graph of $$y=\sin x$$ is divided by the vertical line $$x=k$$. If the area of the region $$0\le x\le k$$ is twice the area of the region $$k\le x\le\pi,$$ then $$k=$$?

    • $$\arccos{\frac{\pi}{4}}$$
    • $$\frac{\pi}{3}$$
    • $$-\arccos\left(-\frac{1}{2}\right)$$
    • $$\arccos\left(-\frac{1}{3}\right)$$

11. If a particle moves in the $$xy$$-plane so that at time $$t>0$$ its position vector is $$(\cos t,\sin3t),$$ then at time $$t=1,$$ its acceleration vector is:

    • $$(-\cos1,-9\sin3)$$
    • $$(-\sin1,-9\cos3)$$
    • $$(-\cos1,-9\cos3)$$
    • $$(\sin1,9\cos3)$$

12. If $$f(x)=(x-3)^2$$ for $$x\leq4$$$$f(x)=frac{1}{x-3}$$ for $$x$$>$$4$$ then $$\int_{3}^{5}f(x)\,dx=$$?

    • $$1.026$$
    • $$3.693$$
    • $$4.386$$
    • $$5.099$$

13. The slope of the line normal to the curve $$x^2y+y^2=5$$ at $$(1,2)$$ is:

    • $$-\frac{1}{2}$$
    • $$\frac{1}{2}$$
    • $$-2$$
    • $$\frac{5}{4}$$

14. A $$15$$-foot ladder rests against a vertical wall. If the ladder is sliding down the wall, how far is the foot of the ladder from the base of the wall at the moment when the top of the ladder is sliding down three times as fast as the foot of the ladder is moving away?

    • $$7.115$$
    • $$8.743$$
    • $$10.115$$
    • $$14.230$$

15. What is the height of the cone with maximum volume if the sum of the radius and the slant height is $$10$$ inches?

    • $$2$$
    • $$2\sqrt{2}$$
    • $$2\sqrt{5}$$
    • $$5$$