Section I Part A — Full Practice Exam 2

Questions in this quiz (30)

1. If $$f(x)=\frac{1}{16}x^4-3x^2$$, find $$f'(2)$$.

   • $$-16$$
   • $$-10$$
   • $$8$$
   • $$16$$

2. Find $$\lim_{x\to0}\frac{5x^2}{\cos x-1}$$.

   • $$-10$$
   • $$-5$$
   • $$5$$
   • $$10$$

3. Find $$\lim_{x\to4}\frac{x^2-16}{x-4}$$.

   • $$0$$
   • $$4$$
   • $$8$$
   • The limit does not exist

4. If $$f(x)=\frac{x^4-2x+1}{x^2+5}$$, find $$f'(x)$$.

   • $$\frac{4x^3-2}{2x}$$
   • $$\frac{(x^2+5)(4x^3-2)-(x^4-2x+1)(2x)}{x^2+5}$$
   • $$\frac{(x^4-2x+1)(2x)-(x^2+5)(4x^3-2)}{(x^2+5)^2}$$
   • $$\frac{(x^2+5)(4x^3-2)-(x^4-2x+1)(2x)}{(x^2+5)^2}$$

5. Evaluate $$\lim_{h\to0}\frac{3\left(\frac{1}{3}+h\right)^4-3\left(\frac{1}{3}\right)^4}{h}$$.

   • $$\frac{4}{9}$$
   • $$\frac{1}{9}$$
   • $$\frac{4}{3}$$
   • The limit does not exist

6. Evaluate $$\int x\sqrt{5x}dx$$.

   • $$\frac{2\sqrt{5}}{7}x^{7/2}+C$$
   • $$\frac{5\sqrt{5}}{2}x^{3/2}+C$$
   • $$\frac{\sqrt{5}}{2}x^{5/2}+C$$
   • $$\frac{2\sqrt{5}}{5}x^{5/2}+C$$

7. Find $$k$$ so that $$f(x)$$ is continuous for all $$x$$.$$f(x)= \frac{x^2-9}{x-3}$$, $$x\ne3$$$$f(x)= k$$, $$x=3$$

   • $$0$$
   • $$3$$
   • $$6$$
   • There is no real value of $$k$$ that makes $$f(x)$$ continuous for all $$x$$.

8. Which of the following integrals correctly gives the area of the region above the $$x$$-axis and below the curve $$y=6+3x-x^2$$?

   • $$\int_{-1}^{6}(x^2-3x-6)dx$$
   • $$\int_{-2}^{3}(6+3x-x^2)dx$$
   • $$\int_{-1}^{6}(6+3x-x^2)dx$$
   • $$\int_{-2}^{3}(x^2-3x-6)dx$$

9. If $$f(x)=x^2\sin3x$$, find $$f'(x)$$.

   • $$2x\sin3x+3x^2\cos3x$$
   • $$2x\sin3x-3x^2\cos3x$$
   • $$x^2\cos3x+2x\sin3x$$
   • $$2x-3\cos3x$$

10. An equation of the line tangent to $$y=3x^4-5x^2$$ at $$x=2$$ is:

    • $$y-8=44(x-2)$$
    • $$y-8=20(x-2)$$
    • $$y-28=76(x-2)$$
    • $$y+8=\frac{1}{20}(x-2)$$

11. Evaluate $$\int_{0}^{\frac{\sqrt{3}}{2}}\frac{1}{\sqrt{1-x^2}}dx$$.

    • $$\frac{\pi}{6}$$
    • $$\frac{\pi}{3}$$
    • $$\frac{2\pi}{3}$$
    • $$\pi$$

12. Find a positive value $$c$$ that satisfies the conclusion of the Mean Value Theorem for Derivatives for $$f(x)=2x^2-3x+4$$ on the interval $$[1,4]$$.

    • $$\frac{3}{2}$$
    • $$2$$
    • $$\frac{5}{2}$$
    • $$\frac{11}{2}$$

13. Given $$f(x)=x^2-6x+5$$, find the absolute minimum of $$f(x)$$ on $$[0,4]$$.

    • $$-4$$
    • $$-3$$
    • $$-1$$
    • $$0$$

14. Find $$\frac{dy}{dx}$$ if $$x^2y+y^2x=6$$.

    • $$2xy+y^2$$
    • $$\frac{2xy+y^2}{x^2+2xy}$$
    • $$-\frac{2xy+y^2}{x^2+2xy}$$
    • $$-\frac{x^2+2xy}{2xy+y^2}$$

15. Find $$\lim_{x\to0}\frac{3^x-1}{x}$$.

    • $$\ln3$$
    • $$1$$
    • $$3$$
    • $$\frac{1}{\ln3}$$

16. Evaluate $$\int5xe^{x^2}dx$$.

    • $$\frac{5}{2}e^{x^2}+C$$
    • $$\frac{1}{2}e^{x^2}+C$$
    • $$5e^{x^2}+C$$
    • $$\frac{1}{5}e^{x^2}+C$$

17. Find the equation of the tangent line to $$4x^2+9y^2=36$$ through the point $$(3,2)$$.

    • $$4x+9y-30=0$$
    • $$8x+9y-36=0$$
    • $$8x-9y-12=0$$
    • $$6x+9y-36=0$$

18. A particle's position is given by $$s(t)=t^3-3t^2+6t$$. What is its acceleration at $$t=2$$?

    • $$0$$
    • $$6$$
    • $$-6$$
    • $$12$$

19. If $$f(x)=2^x$$, then $$f'(x)=$$

    • $$\ln2\cdot2^x$$
    • $$\frac{2^x}{\ln2}$$
    • $$2^{x-1}$$
    • $$\ln x\cdot2^x$$

20. The average value of $$f(x)=\frac{1}{x}$$ from $$x=2$$ to $$x=6$$ is:

    • $$\frac{\ln3}{2}$$
    • $$\frac{\ln3}{4}$$
    • $$\ln3$$
    • $$\frac{1}{\ln3}$$

21. If $$f(x)=\cos x$$, find $$f'''(x)$$.

    • $$\sin x$$
    • $$-\sin x$$
    • $$-\cos x$$
    • $$\cos x$$

22. Find the slope of the normal line to $$y=x+\sin x$$ at $$(0,0)$$.

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

23. Evaluate $$\int e^{4x}dx$$.

    • $$\frac{1}{5}e^{5x}+C$$
    • $$e^{4x}+C$$
    • $$4e^{4x}+C$$
    • $$\frac{1}{4}e^{4x}+C$$

24. Find $$\lim_{x\to0}\frac{\sin^3(3x)}{x^3}$$.

    • $$-27$$
    • $$3$$
    • $$27$$
    • The limit does not exist

25. A solid is generated when the region in the first quadrant bounded by the graph $$y=2+\cos^2x$$, the line $$x=\frac{\pi}{3}$$, the $$x$$-axis, and the $$y$$-axis is revolved about the $$x$$-axis. Its volume is found by evaluating which of the following integrals?

    • $$\pi\int_{0}^{\pi/3}(2+\cos^2x)dx$$
    • $$\pi\int_{0}^{\pi/3}(2+\cos^2x)^2dx$$
    • $$\pi\int_{0}^{1}(2+\cos^2x)^2dx$$
    • $$\pi\int_{0}^{1}(2+\cos^2x)dx$$

26. If $$y=\left(\frac{x^2-3}{2x^3-1}\right)^4$$, find $$\frac{dy}{dx}$$ at $$x=1$$.

    • $$-448$$
    • $$-16$$
    • $$16$$
    • $$20$$

27. Evaluate $$\int x\sqrt{4-x}dx$$.

    • $$-\frac{8}{3}(4-x)^{3/2}+C$$
    • $$\frac{8}{3}(4-x)^{3/2}-\frac{2}{5}(4-x)^{5/2}+C$$
    • $$\frac{8}{3}(4-x)^{3/2}+\frac{2}{5}(4-x)^{5/2}+C$$
    • $$-\frac{8}{3}(4-x)^{3/2}+\frac{2}{5}(4-x)^{5/2}+C$$

28. If $$\frac{dy}{dx}=\frac{x^2}{y}$$ and $$y=1$$ when $$x=1$$, then when $$x=2$$, $$y=$$?

    • $$\sqrt{3}$$
    • $$\sqrt{5}$$
    • $$\sqrt{\frac{17}{3}}$$
    • $$\pm\sqrt{3}$$

29. The graph of $$y=4x^4-2x^2$$ has an inflection point (or points) at:

    • $$x=0$$ only
    • $$x=\pm\frac{1}{\sqrt{3}}$$
    • $$x=\pm\sqrt{3}$$
    • $$x=\pm\frac{1}{2\sqrt{3}}$$

30. Evaluate $$\int_{0}^{\pi/4}\tan xdx$$.

    • $$0$$
    • $$\ln(\cos\frac{\pi}{4})$$
    • $$\ln(\sec\frac{\pi}{4})$$
    • $$\ln(\sec\frac{\pi}{4})-1$$