Section I Part B — Full Practice Exam 1

Questions in this quiz (15)

1. Which of the following equations corresponds to the tangent line to the graph of $$f(x)=e^{2x}$$ at the point where the derivative $$f'(x)$$ equals $$10$$?

   • $$y=10x-6.61$$
   • $$y=10x-1.61$$
   • $$y=x-1.61$$
   • $$y=10x+6.61$$

2. The graph shown represents the derivative of a function $$f$$. Which of the following statements correctly compares the values of $$f(0)$$, $$f(3)$$, $$f(7)$$, and $$f(10)$$?

   • $$f(10)<f(0)<f(3)<f(7)$$
   • $$f(0)<f(3)<f(10)<f(7)$$
   • $$f(3)<f(0)<f(10)<f(7)$$
   • $$f(0)<f(7)<f(3)<f(10)$$

3. The derivative of a function is $$f'(x)=2x^3-8x+1.$$ At which value of $$x$$ does $$f$$ have a local minimum?

   • $$-1.32$$
   • $$-0.13$$
   • $$0.13$$
   • $$1.32<x<2$$

4. Car $$A$$ is traveling south at $$50$$ mph toward a checkpoint, while Car $$B$$ is traveling west at $$40$$ mph, also toward the checkpoint. At a certain moment, both cars are $$80$$ miles from the checkpoint. How fast is the distance between the two cars changing at that moment, in $$\text{mph}$$?

   • $$32.0$$
   • $$40.0$$
   • $$48.0$$
   • $$64.0$$

5. Let $$f(x)=\frac{x^2-9}{x-3}.$$ Which of the following statements is true?$$I$$. $$f$$ is continuous at $$x=-3$$$$II$$. $$f$$ is differentiable at $$x=3$$$$III$$. $$f$$ has a local extremum at $$x=-3$$

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

6. If $$y=3x-4$$ and $$x\ge0$$, determine the minimum value of the product $$x^2y.$$

   • $$-1.78$$
   • $$-1.05$$
   • $$0$$
   • $$2.25$$

7. Determine the area of the region enclosed by$$y=\sin x,$$ $$y=\frac{1}{2}x,$$ and the $$y$$-axis.

   • $$0.42$$
   • $$1.22$$
   • $$1.57$$
   • $$1.88$$

8. A region $$R$$ in the first quadrant is enclosed by the $$x$$-axis, the curve $$y=\sin x,$$ and the line $$y=\frac{1}{2}x.$$ Calculate the volume of the solid generated when this region is revolved about the $$y$$-axis.

   • $$0.98$$
   • $$1.67$$
   • $$2.14$$
   • $$2.62$$

9. The graph of $$f$$ is shown below. Let $$g(x)=\int_{0}^{x}f(t)dt.$$ For which value of $$x$$ does $$g(x)$$ reach a relative maximum?

   • $$a$$
   • $$b$$
   • $$c$$
   • $$d$$

10. For the function $$y=x^3-2x+\sin x,$$ the graph changes concavity at a certain value of $$x$$. At which value of $$x$$ does this inflection point occur?

    • $$-0.52$$
    • $$-0.18$$
    • $$0$$
    • $$0.52$$

11. The table below gives values of a continuous function $$g$$.Using four equal subintervals over the interval $$[0,8]$$, which of the following represents a right-hand Riemann sum approximation for $$\int_{0}^{8}g(x)dx?$$

    • $$24$$
    • $$28$$
    • $$32$$
    • $$36$$

12. Define the function $$f$$ by $$f(x)=x^2$$ for $$x<0$$ and $$f(x)=\sqrt{x}$$ for $$x>0$$. Let $$g(x)=\int_{-4}^{x}f(t)dt.$$ For which value of $$x>-4$$ does $$g(x)=0?$$

    • $$4$$
    • $$6.25$$
    • $$8$$
    • $$12.25$$

13. Above is the graph of a function $$f$$. Which of the following statements about $$f$$ is false?

    • $$f$$ is continuous but not differentiable at $$x=a.$$
    • $$\lim_{x\to a^-}\left(\frac{f(x+a)-f(x)}{a}\right)\ne\lim_{x\to a^+}\left(\frac{f(x+a)-f(x)}{a}\right).$$
    • $$f(a)$$ is defined, but $$f(c)$$ is not.
    • $$\lim_{x\to c^-}(f(x))\ne\lim_{x\to c^+}(f(x))$$

14. The graph below shows an object's acceleration in $$ft/sec^2.$$ It consists of a quarter-circle and two line segments. If the object was at rest at $$t=5$$ seconds, what was its initial velocity?

    • $$-2$$ $$ft/sec$$
    • $$3-\pi$$ $$ft/sec$$
    • $$\pi-3$$ $$ft/sec$$
    • $$\pi+3$$ $$ft/sec$$

15. For what value(s) of $$a$$ is it true that $$\lim_{x\to a}f(x)$$ exists and $$f(a)$$ exists, but $$\lim_{x\to a}f(x)\ne f(a)$$? It is possible that $$a=$$?

    • $$-1$$ only
    • $$2$$ only
    • $$-1$$ or $$1$$ only
    • $$-1$$ or $$2$$ only