Section I Part A — Full Practice Exam 4

Questions in this quiz (22)

1. Evaluate $$\int_0^1 e^{3x}\,dx$$.

   • $$\frac{e^3-1}{1}$$
   • $$e^3-1$$
   • $$\frac{e^3}{3}$$
   • $$\frac{e^3-1}{3}$$

2. If $$f(x)=\tan(e^{\cos x})$$, then $$f'(x)=$$

   • $$-e^{\cos x}\sin x\sec^2(e^{\cos x})$$
   • $$e^{\cos x}\sin x\sec^2(e^{\cos x})$$
   • $$-\sin x\sec(e^{\cos x})\tan(e^{\cos x})$$
   • $$e^{\cos x}\sec^2(e^{\cos x})$$

3. If $$F(x)=\int_1^{x^3} t^2\,dt$$, then $$F'(x)=$$

   • $$3x^2(x^3)^2$$
   • $$3x^2$$
   • $$x^6$$
   • $$3x^8$$

4. If $$f(x)=\tan^2 x+\cos x$$, then $$f'\left(\frac{\pi}{4}\right)=$$

   • $$4-\frac{\sqrt2}{2}$$
   • $$4-\frac{\sqrt2}{2}$$
   • $$1+\frac{\sqrt2}{2}$$
   • $$1-\frac{\sqrt2}{2}$$

5. At which point is the graph of $$f(x)=x^4-4x^3+3x^2+2$$ decreasing and concave down?

   • $$(0,2)$$
   • $$(1,2)$$
   • $$(2,2)$$
   • $$(3,2)$$

6. Which of the following are antiderivatives of $$f(x)=\sin x\cos^2 x$$? I. $$F(x)=-\frac{\cos^3 x}{3}$$ II. $$F(x)=\frac{\sin^2 x}{2}-\frac{\sin^4 x}{4}$$ III. $$F(x)=\frac{1-\cos^3 x}{3}$$

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

7. Determine $$\frac{dy}{dx}$$ for the curve defined by $$x^2+y^3=3xy$$.

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

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

   • $$\frac12$$
   • $$\frac14$$
   • $$\frac{\sqrt3}{4}$$
   • $$\frac{\sqrt3}{2}$$

9. Find the average value of $$f(x)=\cos 3x$$ over $$\left[0,\frac{\pi}{3}\right]$$.

   • $$0$$
   • $$\frac13$$
   • $$\frac23$$
   • $$\frac{1}{\pi}$$

10. Evaluate $$\lim_{x\to\infty}\frac{4x^2-5x+1}{2x^2+3x-7}$$.

    • $$2$$
    • $$\frac12$$
    • $$1$$
    • $$0$$

11. Evaluate $$\int_1^e\frac{\ln x}{x}\,dx$$.

    • $$\frac12$$
    • $$1$$
    • $$\frac13$$
    • $$2$$

12. A particle's position is given by $$s(t)=\cos t+t$$. The average velocity over $$[0,\pi]$$ is

    • $$1$$
    • $$0$$
    • $$\frac{\pi-2}{\pi}$$
    • $$\frac1\pi$$

13. If $$f(x)=\begin{cases} e^{2x}, & x<\ln3 \\ 3, & x\ge\ln3 \end{cases}$$ then $$\lim_{x\to\ln3} f(x)=$$

    • $$1$$
    • $$3$$
    • $$\ln3$$
    • Limit does not exist

14. Evaluate $$\lim_{h\to0}\frac{\sin\left(\frac{\pi}{6}+h\right)-\sin\frac{\pi}{6}}{h}$$.

    • $$\frac{\sqrt3}{2}$$
    • $$\frac12$$
    • $$\sqrt3$$
    • $$1$$

15. The graph of $$f(x)=(x-2)^3(x+1)^2$$ has a local minimum at $$x=$$

    • $$-1$$
    • $$2$$
    • $$1$$
    • $$\frac15$$

16. Let $$f$$ and $$g$$ be twice differentiable functions such that $$f'(x)0$$, then at $$x=2$$

    • $$g$$ is increasing
    • $$g$$ is decreasing
    • $$f$$ is concave down
    • $$g$$ is concave up

17. What is the area of the region bounded by the curves $$y=x^2+1$$ and $$y=x$$ from $$x=0$$ to $$x=2$$?

    • $$\frac{4}{3}$$
    • $$\frac{8}{3}$$
    • $$2$$
    • $$\frac{10}{3}$$

18. Compute $$\frac{d}{dx}\left(e^{\cos 2x}\right)$$.

    • $$-2\sin2x\,e^{\cos2x}$$
    • $$2\cos2x\,e^{\cos2x}$$
    • $$-2\cos2x\,e^{\cos2x}$$
    • $$2e^{\cos2x}$$

19. An equation of the line tangent to $$y=\cos x+2\sin x$$ at $$(0,1)$$ is

    • $$2x-y=1$$
    • $$y=2x+1$$
    • $$y=-2x+1$$
    • $$x+y=1$$

20. The function $$f(x)=x^4-6x^3+9x^2+1$$. All of these statements are true EXCEPT

    • $$f'(0)=0$$
    • $$f'(1)=0$$
    • $$x=0$$ is critical point
    • $$x=1$$ inflection

21. The function $$f(x)=e^{\cos x}$$ is decreasing over which interval?

    • $$[0,\pi]$$
    • $$[\pi/2,3\pi/2]$$
    • $$[0,\pi/2]$$
    • $$(-\infty,\infty)$$

22. The graph of $$f(x)=\frac{x}{x^2+1}$$ is concave down over which interval(s)?

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