Section I Part B — Full Practice Exam 4

Questions in this quiz (12)

1. If $$f(x)=\frac{e^{2x}}{\cos x}$$, then $$f'(x)=$$

   • $$\frac{e^{2x}(2\cos x+\sin x)}{\cos^2 x}$$
   • $$\frac{e^{2x}(2\cos x-\sin x)}{\cos^2 x}$$
   • $$\frac{e^{2x}(\sin x-2\cos x)}{\cos^2 x}$$
   • $$\frac{2e^{2x}}{\cos x}$$

2. Let $$f$$ be a function such that $$\lim_{h\to0}\frac{f(2+h)-f(2)}{h}=5$$. Which of the following must be true? I. $$f'(2)=5$$ II. $$f$$ is continuous at $$x=2$$ III. $$f(2)=5$$

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

3. The function $$f$$ whose derivative is $$f'(x)=3x^2-6x+2$$ has a local minimum at $$x=$$

   • $$\frac13$$
   • $$1$$
   • $$1+\frac{\sqrt3}{3}$$
   • $$2$$

4. A car travels east at $$50$$ mph while another car travels north at $$40$$ mph. If both cars start from the same point at the same time, how fast is the distance between them changing after $$2$$ hours?

   • $$50\text{ mph}$$
   • $$64.03\text{ mph}$$
   • $$90\text{ mph}$$
   • $$70\text{ mph}$$

5. Which of the following is an equation for a line tangent to the graph of $$f(x)=e^{3x}$$ when $$f'(x)=12$$?

   • $$y=12x-5.59$$
   • $$y=4x-5.59$$
   • $$y=12x-3.59$$
   • $$y=12x-4\ln4+4$$

6. Let $$f(x)=\left|\frac{x^2-4}{x-2}\right|$$. Which of these statements is true? I. $$f$$ is continuous at $$x=2$$ II. $$f$$ is differentiable at $$x=2$$ III. $$f$$ has a local minimum at $$x=2$$

   • I only
   • I and III
   • II only
   • None of I, II, and III

7. What is the area of the region bounded by $$y=\cos x$$, $$y=\frac12 x$$, and the $$y$$-axis?

   • $$0.642$$
   • $$1.284$$
   • $$2.156$$
   • $$3.412$$

8. A region $$R$$ in the first quadrant is bounded by the x-axis, $$y=\cos x$$, and $$y=\frac13 x$$. Determine the volume when rotated about the y-axis.

   • $$1.842$$
   • $$2.913$$
   • $$4.276$$
   • $$6.105$$

9. Let $$f(x)=\frac{2x^3}{e^x}$$. For what value of $$x$$ is the slope equal to $$0.512$$?

   • $$-2.418$$
   • $$-1.732$$
   • $$0.864$$
   • $$1.245$$

10. The graph of $$y=x^4-x+\cos x$$ changes concavity at $$x=$$

    • $$0.412$$
    • $$0.528$$
    • $$0.603$$
    • $$0.741$$

11. Let $$f(x)=\begin{cases}-x,&x\le0\\x^2,&x>0\end{cases}$$ and $$g(x)=\int_{-1}^{x}f(t)\,dt$$. For what value of $$x\ne-1$$ would $$g(x)=0$$?

    • $$0$$
    • $$1$$
    • $$\sqrt2$$
    • $$2$$

12. If $$y=3x-7$$ and $$x\ge0$$, what is the minimum value of $$x^2y$$?

    • $$-5.646$$
    • $$0$$
    • $$1.555$$
    • $$2.813$$