Mini Exam 5 — Timed Mini Exams

Questions in this quiz (10)

1. At what point in the interval $$[0,2]$$ is the rate of change of $$f(x)=\cos x$$ equal to its average rate of change on the interval?

   • $$0.785$$
   • $$1.047$$
   • $$1.215$$
   • $$1.571$$

2. Suppose $$\int_1^4 (f(x)+k)\,dx=9$$ where $$k$$ is a constant. Then $$\int_1^4 f(x)\,dx$$ equals:

   • $$9-3k$$
   • $$9-k$$
   • $$9$$
   • $$9+k$$

3. If $$\int_{-2}^5 f(x)\,dx=7$$ then $$\int_{-3}^4 f(x+1)\,dx=$$?

   • $$-7$$
   • $$6$$
   • $$7$$
   • $$8$$

4. If $$f(u)=\tan^{-1}(3u^2)$$ and $$g(u)=e^{2u}$$ then the derivative of $$f(g(u))$$ is:

   • $$\frac{6ue^{2u}}{1+9u^4}$$
   • $$\frac{6e^{2u}}{1+9e^{4u}}$$
   • $$\frac{6e^{4u}}{1+9e^{4u}}$$
   • $$\frac{12e^{2u}}{1+9e^{4u}}$$

5. The region bounded by $$y=e^{2x}$$, $$y=1$$, and $$x=1$$ is rotated about the $$x$$-axis. The volume of the solid generated is given by the integral:

   • $$\pi\int_0^1 e^{4x}\,dx$$
   • $$2\pi\int_1^e (1-\ln y)(y-1)\,dy$$
   • $$\pi\int_0^1 (e^{4x}-1)\,dx$$
   • $$\pi\int_0^1 (e^{2x}-1)^2\,dx$$

6. Let $$f(x)=\int_{0}^{x}(2-\sin^{2}t)dt$$ for $$0\leq x\leq 2\pi$$. On which interval is $$f$$ increasing?

   • $$0$$<$$x$$<$$\pi$$
   • $$0.785$$<$$x$$<$$5.498$$
   • $$\pi$$<$$x$$<$$2\pi$$
   • $$0$$<$$x$$<$$2\pi$$

7. An ellipse has major axis $$16$$ and minor axis $$8$$. Rounded to the nearest integer, the maximum area of an inscribed rectangle is:

   • $$32$$
   • $$48$$
   • $$64$$
   • $$80$$

8. The average value of $$y=x\ln x$$ on the interval $$1\le x\le 3$$ is:

   • $$1.10$$
   • $$1.47$$
   • $$1.65$$
   • $$2.00$$

9. Shown is the graph of $$f(x)=\frac{4}{x^{2}+1}$$.Let $$H(x)=\int_{0}^{x}f(t)dt$$. The local linearization of $$H$$ at $$x=1$$ is

   • $$y=2x$$
   • $$y=-2x-4$$
   • $$y=2x+\pi-2$$
   • $$y=-2x+\pi+2$$

10. In a protected lake, the fish population increases at a rate$$\frac{dP}{dt}=k(800-P)$$where $$P(t)$$ represents the population at $$t$$ years. If $$200$$ fish were initially placed in the lake and the population grew to $$400$$ in $$4$$ years, how many fish will there be after $$8$$ years?

    • $$533$$
    • $$560$$
    • $$600$$
    • $$640$$