Section I Part A — Full Practice Exam 2

Questions in this quiz (30)

1. Consider the function $$f(x)$$, where $$b$$ is a constant.$$f(x)=2e^{x-2}+1$$ for $$x\leq2$$$$f(x)=3x^2+bx-7$$ for $$x$$>$$2$$What is the value of $$b$$ for which the function is continuous at $$x=2$$?

   • $$-5$$
   • $$-1$$
   • $$3$$
   • $$5$$

2. Function $$h$$ is twice differentiable. The table above gives selected values of $$h$$. Which of the following must be true?

   • $$h$$ has no critical points in the interval $$-2$$<$$x$$<$$3$$
   • $$h'(x)=8$$ for some value of $$x$$ in the interval $$-2$$<$$x$$<$$3$$
   • The graph of $$h$$ has no points of inflection in the interval $$-2$$<$$x$$<$$3$$
   • $$h'(x)$$>$$0$$ for all values of $$x$$ in the interval $$-2$$<$$x$$<$$3$$

3. If $$\sum_{n=1}^{\infty}b_n$$ is a geometric series of all positive terms with $$b_1=60$$ and $$b_3=15$$, then $$\sum_{n=1}^{\infty}b_n=$$?

   • diverges
   • $$80$$
   • $$120$$
   • converges to a sum that cannot be determined

4. If $$x=\sqrt{1+t^2}$$ and $$y=\tan^{-1}(t)$$, then $$\frac{dy}{dx}=$$?

   • $$\frac{t}{\sqrt{1+t^2}}$$
   • $$\frac{1}{t}$$
   • $$\frac{1}{\sqrt{1+t^2}}$$
   • $$\frac{1}{t\sqrt{1+t^2}}$$

5. The table above shows values of differentiable functions $$f$$ and $$g$$.If $$h(x)=g(f(x)),$$ then $$h'(2)=$$

   • $$6$$
   • $$8$$
   • $$10$$
   • $$12$$

6. $$\int_{0}^{1}(2x+1)^3dx=$$?

   • $$10$$
   • $$\frac{17}{2}$$
   • $$\frac{15}{2}$$
   • $$4$$

7. Given the parametric equations $$x(t)=t^2+1$$ and $$y(t)=2t^{3/2},$$ which expression gives the length of the curve from $$t=1$$ to $$t=4$$?

   • $$\int_{1}^{4}\sqrt{4t^2+9t}dt$$
   • $$\int_{1}^{4}\sqrt{4t+9t}dt$$
   • $$\int_{1}^{4}\sqrt{(2t)^2+(3t^{1/2})^2}dt$$
   • $$\int_{1}^{4}\sqrt{4t+3t}dt$$

8. If $$\sum_{n=0}^{\infty}\frac{(x+2)^n}{2^n(n+1)^3}$$, the radius of convergence is:

   • $$\frac{1}{2}$$
   • $$1$$
   • $$2$$
   • $$3$$

9. Which equation has the slope field shown below?

   • $$\frac{dy}{dx}=y$$
   • $$\frac{dy}{dx}=-y$$
   • $$\frac{dy}{dx}=\frac{1}{y}$$
   • $$\frac{dy}{dx}=x$$

10. Consider the function $$h$$ that is continuous on the interval $$[0,4]$$ such that $$h(1)=2$$ and $$h(4)=14$$. Which of the following must be true?

    • $$h$$ is increasing on $$[1,4]$$
    • $$h'(x)=4$$ has at least one solution in $$\left(1,4\right)$$
    • $$2\leq h(3)\leq14$$
    • $$h(x)=9$$ has at least one solution in $$[1,4]$$

11. Which of the following series converges conditionally?$$I.$$ $$\sum_{n=1}^{\infty}(-1)^n\frac{2^n}{3^n}$$$$II.$$ $$\sum_{n=1}^{\infty}\frac{(-1)^n}{\ln(n+1)}$$$$III.$$ $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n^3}$$

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

12. If $$x=3\sin\theta,$$ $$0\leq\theta\leq\frac{\pi}{2},$$ then$$\int_{0}^{3}\frac{x^2}{\sqrt{9-x^2}}dx$$is equivalent to:

    • $$9\int_{0}^{\frac{\pi}{2}}\sin^2\theta d\theta$$
    • $$\int_{0}^{\frac{\pi}{2}}9\sin^2\theta\cos\theta d\theta$$
    • $$\int_{0}^{\frac{\pi}{2}}3\sin\theta\tan\theta d\theta$$
    • $$\int_{0}^{\frac{\pi}{2}}\frac{3\sin^2\theta}{\cos\theta}d\theta$$

13. $$\int_{-2}^{2}|x-1|dx=$$?

    • $$2$$
    • $$3$$
    • $$4$$
    • $$5$$

14. The table gives the following values at $$x=2$$:Find $$\lim_{x\to2}\frac{f(x)-2g(x)}{x^2-4}$$.

    • $$-1$$
    • $$0$$
    • $$\frac{1}{4}$$
    • $$2$$

15. What is the volume of the solid generated by rotating the region enclosed by $$y=\frac{2}{x}$$ and the $$x$$-axis on $$[2,\infty)$$ about the $$x$$-axis?

    • $$\pi$$
    • $$2\pi$$
    • $$4\pi$$
    • divergent

16. If $$A=\int_{1}^{4}\ln xdx$$ is approximated using equal subdivisions, and $$L,$$ $$R,$$ $$T$$ represent left, right, and trapezoidal sums, then:

    • $$L\leq A\leq T\leq R$$
    • $$R\leq T\leq A\leq L$$
    • $$L\leq T\leq A\leq R$$
    • $$R\leq A\leq T\leq L$$

17. If $$\frac{dy}{dx}=y\cos x$$ and $$y=2$$ when $$x=0,$$ then when $$x=\frac{\pi}{2},$$ $$y=$$?

    • $$2e$$
    • $$2e^1$$
    • $$2e^{\sin x}$$
    • $$2e$$ evaluated at $$\sin\frac{\pi}{2}$$

18. The parametric equations are:$$x(t)=\cos(t^2),\quad y(t)=t^2$$Find the slope of the tangent line when $$t=1$$.

    • $$-2\sin(1)$$
    • $$\frac{-1}{\sin(1)}$$
    • $$\frac{-\sin(1)}{\sin(1)}$$
    • $$\frac{-\sin(1)}{2}$$

19. In which of the following series can convergence be determined using comparison with $$\sum\frac{1}{n^2}$$?

    • $$\sum\frac{4}{n^2+5}$$
    • $$\sum\frac{3n}{n^2+1}$$
    • $$\sum\frac{n^2}{n+1}$$
    • $$\sum\frac{5n}{n}$$

20. The graph of $$g'$$ is shown in the figure above. Of the following statements, which one is true about $$g$$ at $$x=1$$?

    • $$g$$ is not differentiable at $$x=1$$
    • $$g$$ is not continuous at $$x=1$$
    • $$g$$ attains an absolute maximum at $$x=1$$
    • There is an inflection point on the graph of $$g$$ at $$x=1$$

21. Find the slope of the curve $$r=2\cos\theta$$ at $$\theta=\frac{\pi}{4}$$

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

22. A particle has velocity $$v(t)=t^2-2t$$ and initial position $$s(0)=1$$. What is the total distance traveled from $$t=0$$ to $$t=2$$?

    • $$\frac{2}{3}$$
    • $$1$$
    • $$\frac{4}{3}$$
    • $$2$$

23. The Maclaurin series for a function $$f$$ is given by$$\sum_{n=1}^{\infty}\frac{(-1)^{n-1}x^{2n}}{4^nn}$$and converges to $$f(x)$$ for $$-2\leq x\leq2$$. If $$f\left(\frac{1}{2}\right)$$ is approximated using the fourth-degree Maclaurin polynomial, what is the alternating series error bound?

    • $$\frac{1}{4^3\cdot3}$$
    • $$\frac{1}{4^4\cdot4}$$
    • $$\frac{1}{4^4\cdot5}$$
    • $$\frac{1}{4^6\cdot3}$$

24. $$\int xe^xdx=$$?

    • $$xe^x-e^x+C$$
    • $$xe^x-e^{-x}+C$$
    • $$\frac{x^2}{2}e^x+C$$
    • $$e^{x^2}+C$$

25. Which one of the following series converges?

    • $$\sum_{n=1}^{\infty}\frac{1}{\sqrt{n}}$$
    • $$\sum_{n=1}^{\infty}\frac{1}{3n+1}$$
    • $$\sum_{n=1}^{\infty}\frac{n}{n^2+2}$$
    • $$\sum_{n=1}^{\infty}\frac{1}{n^2+1}$$

26. The coefficient of the $$(x-1)^2$$ term in the Taylor polynomial for $$f(x)=\ln x$$ centered at $$x=1$$ is:

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

27. If $$f'(x)=k(x)$$ and $$g(x)=x^4,$$ then $$\frac{d}{dx}f(g(x))=$$?

    • $$k(x^4)$$
    • $$4x^3k(x)$$
    • $$4x^3k(x^4)$$
    • $$k(4x^3)$$

28. Choose the integral that is the limit of the Riemann sum:$$\lim_{n\to\infty}\sum_{k=1}^{n}\left(\left(\frac{2k}{n}+1\right)^2\cdot\frac{2}{n}\right)$$

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

29. $$\int_{0}^{\infty}e^{-x/3}dx=$$?

    • $$1$$
    • $$3$$
    • $$\infty$$
    • $$-3$$

30. The graph of $$g(x)$$ consists of two-line segments as shown above. If $$h(x)=g^{-1}(x),$$ the inverse function of $$g(x),$$ find $$h'(4).$$

    • $$\frac{1}{5}$$
    • $$\frac{2}{3}$$
    • $$\frac{3}{2}$$
    • $$5$$