Section I Part A — Full Practice Exam 1

Questions in this quiz (30)

1. What is the slope of the line tangent to the graph of $$y=\frac{x^2+1}{x^2+2}$$ when $$x=1$$?

   • $$-\frac{1}{2}$$
   • $$\frac{2}{9}$$
   • $$\frac{1}{2}$$
   • $$1$$

2. If $$y^2-3x^2y=10,$$ then $$\frac{dy}{dx}=$$?

   • $$\frac{6x}{y-3x^2}$$
   • $$\frac{6xy}{2y-3x^2}$$
   • $$\frac{6x+3xy}{y-3x^2}$$
   • $$\frac{3xy}{y+x^2}$$

3. $$\int x^3(x^4+2)^5\,dx=$$?

   • $$\frac{1}{24}(x^4+2)^6+C$$
   • $$\frac{1}{6}(x^4+2)^6+C$$
   • $$\frac{1}{6}(x^4+2)^5+C$$
   • $$\frac{1}{4}x^3(x^4+2)^6+C$$

4. The values of a continuous function $$f$$ for selected values of $$x$$ are given in the table below.What is the value of the left Riemann sum approximation to $$\int_{0}^{40}f(x)\,dx$$ using the subintervals $$[0,10],$$ $$[10,20],$$ and $$[20,40]$$?

   • $$190$$
   • $$210$$
   • $$220$$
   • $$250$$

5. Which of the following gives the length of the curve $$y=\ln x$$ over the closed interval $$[1,e]$$?

   • $$\int_{1}^{e}\sqrt{1+\frac{1}{x}}\,dx$$
   • $$\int_{1}^{e}\sqrt{1+\frac{1}{x^2}}\,dx$$
   • $$\int_{1}^{e}\sqrt{1-\frac{1}{x^2}}\,dx$$
   • $$\int_{1}^{e}\sqrt{1+\frac{1}{2x}}\,dx$$

6. $$\int\frac{4}{x^2+6x+8}\,dx=$$?

   • $$-\ln|(x+4)(x+2)|+C$$
   • $$\ln\left|\frac{x+2}{x+4}\right|+C$$
   • $$2\ln\left|\frac{x+2}{x+4}\right|+C$$
   • $$4\ln|(x+4)(x+2)|+C$$

7. If $$f(x)=x^3+2$$ and $$g$$ is a differentiable function of $$x,$$ what is the derivative of $$f(g(x))$$?

   • $$3g(x)^2$$
   • $$3g'(x)^2$$
   • $$3g(x)^2g'(x)$$
   • $$3x^2g'(x)$$

8. A particle moves in the $$xy$$-plane with position given by $$(x(t),y(t))=(2t-1,4-t^2)$$ at time $$t$$. In which direction is the particle moving as it passes through the point $$(3,0)$$?

   • Up and to the right
   • Up and to the left
   • Down and to the right
   • Down and to the left

9. Let $$y=f(x)$$ be the solution to the differential equation $$\frac{dy}{dx}=x+y$$ with initial condition $$f(0)=1.$$ What is the approximation for $$f(1)$$ obtained by using Euler's method with $$2$$ steps of equal length starting at $$x=0$$?

   • $$\frac{5}{2}$$
   • $$\frac{5}{2}$$
   • $$3$$
   • $$\frac{7}{2}$$

10. Which of the following series converges?

    • $$\sum_{n=1}^{\infty}\frac{2n}{n+3}$$
    • $$\sum_{n=1}^{\infty}\frac{2n}{n^2+3}$$
    • $$\sum_{n=1}^{\infty}\frac{2n}{n^2+3n}$$
    • None of the above

11. $$\int(3^t+e^5)\,dt=$$?

    • $$\frac{3^t}{\ln3}+e^5t+C$$
    • $$\frac{3^t}{\ln3}+e^5t+C$$
    • $$\frac{3^t}{\ln3}+e^5+C$$
    • $$3^t\ln3-\frac{e^6}{6}+C$$

12. $$\lim_{x\to0}\frac{\ln(1+x)}{x}$$ is:

    • $$0$$
    • $$1$$
    • $$\infty$$
    • $$-1$$

13. A population of bacteria is modeled by the function $$P$$ and grows according to the logistic differential equation$$\frac{dP}{dt}=4P\left(1-\frac{P}{8000}\right),$$where $$t$$ is time in hours and $$P(0)=500$$. Which of the following statements are true?$$I.$$ $$\lim_{t\to\infty}P(t)=8000$$$$II.$$ $$\frac{dP}{dt}$$ is positive for $$t$$>$$0$$$$III.$$ $$\frac{d^2P}{dt^2}$$ is negative for all $$t$$>$$0$$

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

14. Which of the following integrals gives the area of the region that is bounded by the graphs of the polar equations$$\theta=0,$$ $$\theta=\frac{\pi}{3},$$ and $$r=\frac{3}{\cos\theta+\sin\theta}$$?

    • $$\int_{0}^{\pi/3}\frac{3}{\cos\theta+\sin\theta}\,d\theta$$
    • $$\int_{0}^{\pi/3}\frac{9}{\cos\theta+\sin\theta}\,d\theta$$
    • $$\int_{0}^{\pi/3}\frac{9}{(\cos\theta+\sin\theta)^2}\,d\theta$$
    • $$\int_{0}^{\pi/3}\frac{9}{2(\cos\theta+\sin\theta)^2}\,d\theta$$

15. The sum of the series $$1+\frac{3^1}{1!}+\frac{3^2}{2!}+\frac{3^3}{3!}+\cdots+\frac{3^n}{n!}+\cdots$$ is:

    • $$e^3$$
    • $$e^3-1$$
    • $$3e^3$$
    • $$e$$

16. If $$x(t)=t^3+1$$ and $$y(t)=t^2+5$$ for $$t>0,$$ then in terms of $$t,$$ $$\frac{d^2y}{dx^2}=$$?

    • $$-\frac{2}{9t^4}$$
    • $$\frac{2}{3t^2}$$
    • $$\frac{2}{9t^3}$$
    • $$\frac{6t}{3t^2}$$

17. If $$\frac{dy}{dt}=-8e^{-t/4}$$ and $$y(0)=16,$$ what is the value of $$y(1)$$?

    • $$16e^{-1}$$
    • $$16e^{-2}$$
    • $$8e^{-1}$$
    • $$32e^{-1/4}-16$$

18. Let $$f$$ be a function with second derivative $$f''(x)=\sqrt{1+2x}$$. The coefficient of $$x^3$$ in the Taylor series for $$f$$ about $$x=0$$ is:

    • $$\frac{1}{24}$$
    • $$\frac{1}{12}$$
    • $$\frac{1}{6}$$
    • $$\frac{1}{4}$$

19. What is the radius of convergence for the power series:$$\sum_{n=0}^{\infty}\frac{(x-2)^n}{5\cdot4^n}?$$

    • $$\frac{1}{4}$$
    • $$4$$
    • $$2$$
    • $$5$$

20. $$\int_{2}^{\infty}\frac{1}{x^p}\,dx$$ and $$\int_{0}^{2}\frac{1}{x^p}\,dx$$ both converge when $$p=$$?

    • $$2$$
    • $$1$$
    • $$\frac{1}{2}$$
    • There is no value of $$p$$ that makes both integrals converge.

21. What are the equations of the horizontal asymptotes of the graph of $$y=\frac{3x}{\sqrt{x^2+4}}?$$

    • $$y=0$$ only
    • $$y=1$$ only
    • $$y=3$$ only
    • $$y=-3$$ and $$y=3$$ only

22. If $$F(x)=\int_{1}^{x^3}\sqrt{t+2}\,dt$$ for all real numbers $$x>0,$$ then $$F'(x)=$$?

    • $$\sqrt{x+2}$$
    • $$3x^2\sqrt{x^3+2}$$
    • $$x^2\sqrt{x^3+2}$$
    • $$3x\sqrt{x^3+2}$$

23. Which of the following is the solution to the differential equation $$\frac{dy}{dx}=3xy$$ with the initial condition $$y(0)=2?$$

    • $$y=2e^{\frac{3}{2}x^2}$$
    • $$y=e^{3x^2}+2$$
    • $$y=2e^{3x^2}$$
    • $$y=2e^{\frac{3}{2}x}$$

24. $$\lim_{h\to0}\frac{\cos\left(\frac{\pi}{4}+h\right)-\cos\left(\frac{\pi}{4}\right)}{h}$$ is:

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

25. Let $$g$$ be the function defined by:$$g(x)=\int_{0}^{x}\frac{t^3-2t^2-3t}{\sqrt{t^2+5}}\,dt$$On which of the following intervals is $$g$$ decreasing?

    • $$x\le-1$$ and $$0\le x\le3$$
    • $$-1\le x\le0$$ and $$x\ge3$$
    • $$x\le-1$$ and $$x\ge3$$
    • $$-1\le x\le3$$

26. If $$f(x)=\cos x+3x-2$$ and $$g$$ is the inverse function of $$f,$$ what is the value of $$g'(1)$$?

    • $$\frac{1}{3-\sin1}$$
    • $$3-\sin1$$
    • $$\frac{1}{3+\sin1}$$
    • $$3+\sin1$$

27. Let $$f$$ be a function that has derivatives of all orders for all real numbers, and let $$P_2(x)$$ be the second-degree Taylor polynomial for $$f$$ about $$x=0$$. The Taylor series for $$f$$ about $$x=0$$ converges at $$x=1,$$ and $$|f^{(n)}(x)|\le\frac{n+1}{n+2}$$ for $$1\le n\le3$$ and all values of $$x$$. Of the following, which is the smallest value of $$k$$ for which the Lagrange error bound guarantees that $$|f(1)-P_2(1)|\le k?$$

    • $$\frac{3}{4}$$
    • $$\frac{3}{4}\cdot\frac{1}{3!}$$
    • $$\frac{4}{5}\cdot\frac{1}{3!}$$
    • $$\frac{2}{3}\cdot\frac{1}{2!}$$

28. Let $$f$$ be a function defined and continuous on the closed interval $$[a,b].$$ If $$f$$ has a relative maximum at $$x=c$$ and $$a$$<$$c$$<$$b,$$ which of the following statements must be true?$$I.$$ $$f'(c)=0$$$$II.$$ $$f''(c)<0$$$$III.$$ $$f$$ is differentiable at $$x=c$$

    • $$I$$ only
    • $$II$$ only
    • $$I$$ and $$III$$ only
    • None of the above

29. If$$f(x)=(x-1)^2$$ for $$x\le2$$$$f(x)=\frac{1}{x-1}$$ for $$x$$>$$2$$then $$\int_{1}^{4}f(x)\,dx=$$?

    • $$5$$
    • $$4$$
    • $$\frac{1}{3}+\ln3$$
    • $$3$$

30. If $$\frac{dy}{dx}=\sin x$$, then the average rate of change of $$y$$ with respect to $$x$$ on the closed interval $$[0,\pi]$$ is:Average rate of change on $$[0,\pi]$$ is:$$\frac{y(\pi)-y(0)}{\pi-0}$$Using the Fundamental Theorem of Calculus,$$y(\pi)-y(0)=\int_{0}^{\pi}\sin x\,dx$$Compute the integral.$$\int_{0}^{\pi}\sin x\,dx$$$$=\left.-\cos x\right|_{0}^{\pi}$$$$=-\cos\pi-(-\cos0)$$$$=-(-1)-(-1)$$$$=1+1$$$$=2$$So, the average rate of change is: $$\frac{2}{\pi}$$Final answer: $$\frac{2}{\pi}$$

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