Section I Part A — Full Practice Exam 3

Questions in this quiz (26)

1. If $$f(x)=\frac{x^2+5x+6}{x+2}$$, then $$f^{\prime}(x)=$$

   • $$1$$
   • $$\frac{x^2+4x+2}{(x+2)^2}$$
   • $$\frac{x^2+4x+4}{(x+2)^2}$$
   • $$\frac{x^2+6x+8}{(x+2)^2}$$

2. $$\int 4x(\sqrt{x}-x^2)\,dx=$$

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

3. What is the value of $$\sum_{n=1}^{\infty}\frac{(-2)^{n+1}}{4^n}$$ ?

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

4. Which of the following is the equation of the tangent line to $$x^2+2xy=5$$ at $$(1,1)$$ ?

   • $$y-1=-\frac{3}{2}(x-1)$$
   • $$y-1=-\frac{1}{2}(x-1)$$
   • $$y-1=-2(x-1)$$
   • $$y-1=\frac{3}{2}(x-1)$$

5. If $$y=\frac{1}{3}x^{3/5}-\frac{2}{x^4}$$, then $$\frac{dy}{dx}=$$

   • $$\frac{1}{5x^{2/5}}+\frac{8}{x^5}$$
   • $$\frac{1}{5x^{2/5}}-\frac{8}{x^5}$$
   • $$\frac{3}{5x^{2/5}}+\frac{8}{x^5}$$
   • $$\frac{1}{5x^2}+\frac{2}{x^4}$$

6. A population $$y$$ changes at a rate modeled by $$\frac{dy}{dt}=0.3y(800-y)$$. For which values of $$y$$ is the population increasing at a decreasing rate?

   • $$0<y<400$$
   • $$400<y<800$$
   • $$y=400$$
   • $$0

7. Which of the following gives the length of the path described by $$x(t)=1+2t$$, $$y(t)=3+t^2$$ from $$t=0$$ to $$t=2$$ ?

   • $$\int_0^2 \sqrt{4+4t^2}\,dt$$
   • $$\int_0^2 \sqrt{1+4t^2}\,dt$$
   • $$\int_0^2 \sqrt{5+4t^2}\,dt$$
   • $$\int_0^2 \sqrt{4+t^2}\,dt$$

8. Let $$y=f(x)$$ be the solution to $$\frac{dy}{dx}=x+y$$ with $$f(0)=1$$. Using Euler's method with two equal steps from $$x=0$$ to $$x=2$$, the approximation of $$f(2)$$ is

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

9. If $$\int_0^k \frac{x}{x^2+9}\,dx=\frac{1}{2}\ln 3$$ where $$k>0$$, then $$k=$$

   • $$\sqrt{3}$$
   • $$3$$
   • $$\sqrt{6}$$
   • $$3\sqrt2$$

10. The third-degree Taylor polynomial for $$f$$ about $$x=2$$ is $$2+(x-2)-\frac{(x-2)^2}{4}+\frac{(x-2)^3}{24}$$. What is $$f^{\prime\prime\prime}(2)$$ ?

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

11. For which of the following does $$\lim_{x\to\infty} f(x)=0$$ ?

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

12. If $$a$$ and $$b$$ are positive constants, then $$\lim_{x\to\infty}\frac{\ln(ax+2)}{\ln(bx^2+5)}=$$

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

13. For which values of $$x$$ does the series $$\sum_{n=1}^{\infty}\frac{(-1)^n}{n}(x-1)^n$$ converge?

    • $$0<x<2$$
    • $$0\le x<2$$
    • $$0<x\le2$$
    • $$0\le x\le2$$

14. For $$0

    • $$\frac{1}{50}\ln P-\frac{1}{50}\ln(50-P)$$
    • $$\frac{1}{50}\ln P+\frac{1}{50}\ln(50-P)$$
    • $$50\ln P-50\ln(50-P)$$
    • $$\ln(50P-P^2)$$

15. If $$\lim_{h\to0}\frac{\arccos(a+h)-\arccos(a)}{h}=-2$$, which of the following could be the value of $$a$$ ?

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

16. Which of the following is the Maclaurin series for $$\frac{1}{(1+x)^2}$$ ?

    • $$1+x+x^2+x^3+\cdots$$
    • $$1-2x+3x^2-4x^3+\cdots$$
    • $$1+2x+3x^2+4x^3+\cdots$$
    • $$1-2x+3x^2-4x^3+\cdots$$

17. At time $$t\ge0$$, a sphere has volume $$V(t)$$ and radius $$r(t)$$. If the volume decreases at a rate proportional to its surface area, which differential equation could describe the rate of change of volume?

    • $$\frac{dV}{dt}=-kr^2$$
    • $$\frac{dV}{dt}=-kr^3$$
    • $$\frac{dV}{dt}=-kt^2$$
    • $$\frac{dV}{dt}=-krt$$

18. Which of the following is true about the curve $$x^2+xy+y^2=7$$ at the point $$(1,2)$$ ?

    • The tangent line is horizontal
    • The tangent line is vertical
    • The tangent line is neither horizontal nor vertical
    • The derivative does not exist

19. What is the coefficient of $$x^4$$ in the Taylor series for $$\frac{e^{2x^2}}{3}$$ about $$x=0$$ ?

    • $$\frac{2}{3}$$
    • $$\frac{4}{3}$$
    • $$\frac{6}{3}$$
    • $$\frac{8}{9}$$

20. The function $$g(x)=x^3-3x^2+2$$. What is the absolute minimum value of $$g$$ on the interval $$[0,3]$$ ?

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

21. Which of the following is the solution to $$\frac{dy}{dx}=e^{x-y}$$ with $$y(0)=\ln 2$$ ?

    • $$y=x+\ln 2$$
    • $$y=\ln(e^x+1)$$
    • $$y=\ln(2e^x)$$
    • $$y=-\ln(e^x+2)$$

22. Which of the following series converges? I. $$\sum \frac{1}{n^2}$$ II. $$\sum \frac{1}{n}$$III. $$\sum \frac{(-1)^n}{n}$$

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

23. If $$\int_2^x f(t)\,dt=\frac{10x}{\sqrt{x^2+4}}-5$$, then $$\int_2^{\infty} f(t)\,dt$$ is

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

24. If $$x=t^3+1$$ and $$y=\ln t$$, then $$\frac{d^2y}{dx^2}$$ in terms of $$t$$ is

    • $$-\frac{1}{3t^6}$$
    • $$-\frac{1}{3t^5}$$
    • $$\frac{1}{3t^4}$$
    • $$\frac{1}{3t^5}$$

25. Which of the following is equal to the area of the region inside the curve $$r=4\cos\theta$$ and outside the curve $$r=2\cos\theta$$ ?

    • $$2\int_0^{\pi/2}(16\cos^2\theta-4\cos^2\theta)\,d\theta$$
    • $$\int_0^{\pi/2}(16\cos^2\theta-4\cos^2\theta)\,d\theta$$
    • $$\frac{1}{2}\int_0^{\pi/2}(16\cos^2\theta-4\cos^2\theta)\,d\theta$$
    • $$\frac{1}{2}\int_0^{\pi}(16\cos^2\theta-4\cos^2\theta)\,d\theta$$

26. Let $$g$$ be a differentiable function, and let $$f$$ be defined by $$f(x)=\frac{g(2x+1)}{5}$$. Which of the following could be equal to $$g^{\prime}(5)$$ ?

    • $$f^{\prime}(2)$$
    • $$2f^{\prime}(2)$$
    • $$5f^{\prime}(2)$$
    • $$g^{\prime}(5)=\frac{5}{2}f^{\prime}(2)$$