Section I Part A — Full Practice Exam 5

Questions in this quiz (23)

1. What is the instantaneous rate of change for $$f(x)=\frac{x^3-1}{x-1}$$ at $$x=2$$?

   • $$3$$
   • $$4$$
   • $$5$$
   • $$6$$

2. Evaluate $$\int_1^4 \frac{2}{x^2}\,dx$$

   • $$\frac12$$
   • $$\frac34$$
   • $$1$$
   • $$\frac32$$

3. What is the slope of the curve defined by $$x^2+xy+y^2-3x+2y=0$$ at the point $$(1,1)$$?

   • $$-1$$
   • $$0$$
   • $$1$$
   • $$2$$

4. Evaluate $$\int_0^1 \sqrt{x}(x^2-2x+1)\,dx$$

   • $$\frac{2}{7}$$
   • $$\frac{8}{15}$$
   • $$\frac{16}{105}$$
   • $$\frac49$$

5. The radius of a sphere is increasing at a rate of $$3$$ inches per minute. At what rate is the volume increasing when the surface area is $$16\pi$$?

   • $$12$$
   • $$24$$
   • $$36$$
   • $$48$$

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

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

7. What is the average rate of change of $$f(x)=x^3-2x^2+x+2$$ over $$[0,3]$$?

   • $$4$$
   • $$5$$
   • $$6$$
   • $$7$$

8. If the graph of $$f''(x)$$ is a line with slope $$4$$, then $$f$$ could be which type of function?

   • linear
   • quadratic
   • cubic
   • quartic

9. Let $$f(x)=\begin{cases}\sqrt{x+2},&x\ge-2\\x+4,&x<-2\end{cases}$$. What is the average value of $$f$$ over $$[-4,2]$$?

   • $$\frac{5}{3}$$
   • $$\frac{7}{3}$$
   • $$3$$
   • $$\frac{11}{9}$$

10. Let $$f$$ be continuous on $$[1,9]$$. If $$f(1)=2$$ and $$f(9)=10$$, then the Mean Value Theorem guarantees that

    • $$f(5)=6$$
    • $$f''(5)=1$$
    • $$f''(c)=1$$ for at least one $$c$$ between $$1$$ and $$9$$
    • $$f(c)=0$$ for at least one $$c$$ between $$1$$ and $$9$$

11. Evaluate $$\frac{d}{dx}\int_0^{x^2}e^t\,dt$$.

    • $$e^{x^2}$$
    • $$2xe^{x^2}$$
    • $$e^x$$
    • $$2xe^x$$

12. Let $$f(x)=e^{2x}$$. If the rate of change of $$f$$ at $$x=c$$ is twice its rate of change at $$x=1$$, then $$c=$$?

    • $$1$$
    • $$\frac32$$
    • $$2$$
    • $$3$$

13. Let $$f$$ be differentiable on $$[0,8]$$ such that $$f(0)=1$$ and $$f(8)=5$$. If there are exactly two solutions to $$f(x)=3$$ over $$(0,8)$$, which must be true?

    • $$f'(c)=0$$
    • local maximum
    • $$f''(c)=0$$
    • strictly increasing

14. The normal line to the curve $$y=\sqrt{9-x^2}$$ at $$(0,3)$$ has slope

    • $$0$$
    • $$1$$
    • $$-1$$
    • undefined

15. What are all values for $$k$$ such that $$\int_{-1}^{k}x^2dx=1$$?

    • $$0$$
    • $$1$$
    • $$-1\ and\ 1$$
    • $$2$$

16. If the rate of change of $$y$$ is directly proportional to $$y$$, then it is possible that

    • $$y=3e^{2t}$$
    • $$y=2t^2$$
    • $$y=\ln(t+1)$$
    • $$y=\sqrt t$$

17. The graph of $$y=2x^3-3x^2+1$$ is decreasing for which interval?

    • $$(-\infty,0)$$
    • $$(0,1)$$
    • $$(1,\infty)$$
    • $$(-\infty,\infty)$$

18. Determine the value for $$c$$ on $$[1,4]$$ that satisfies the mean value theorem for $$f(x)=x^2-2x$$

    • $$1$$
    • $$2$$
    • $$3$$
    • $$4$$

19. The function $$f$$ is continuous on the closed interval $$[0,2]$$ and has values as defined by the table above. Which of the following statements must be true?

    • $$f$$ must be increasing on $$[0,2]$$
    • $$f$$ must be concave down on $$\left(0,2\right)$$
    • $$f'\left(1\right)$$>$$0$$
    • The average rate of change of $$f$$ over $$[0,2]$$ is $$2$$

20. Let $$f,g$$ and their derivatives be defined by the table above. If $$h(x)=f(g(x))$$, for what value $$c$$ is $$h'(c)=0.$$

    • $$1$$
    • $$2$$
    • $$0$$
    • $$3$$
    • $$1$$
    • $$4$$
    • $$2$$
    • $$3$$

21. The function $$f$$ is continuous on the closed interval $$[0,2]$$. It is given that $$f(0)=0$$ and $$f(2)=4$$. If $$f'(x)>0$$ for all $$x$$ on $$[0,2]$$ and $$f''(x)<0$$ for all $$x$$ on $$(0,2)$$, then $$f(1)$$ could be

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

22. The water level in a cylindrical tank is rising at a rate of $$2$$ inches per minute. If the radius of the tank is $$6$$ inches, what is the rate that water is entering the tank, in cubic inches per minute, when the volume is $$144\pi$$ cubic inches?

    • $$24\pi$$
    • $$48\pi$$
    • $$72\pi$$
    • $$96\pi$$

23. If $$f(x)=\arctan(x^3)$$, then $$f'(1)=$$

    • $$\frac12$$
    • $$\frac32$$
    • $$\frac34$$
    • $$\frac14$$