Mini Exam 14 — Timed Mini Exams

Questions in this quiz (10)

1. Find any points of inflection, if any, of $$f(x)=xe^x$$.

   • $$(-2,-2e^{-2})$$
   • $$(-1,-e^{-1})$$
   • $$(0,0)$$
   • $$(1,e)$$

2. Suppose the downward velocity of a sky diver is given by $$v(t)=30(1-e^{-t})$$ ft/s for the first $$5$$ seconds of a jump. Compute the distance fallen.

   • $$150+30e^{-5}$$ ft
   • $$120+30e^{-5}$$ ft
   • $$150-30e^{-5}$$ ft
   • $$120-30e^{-5}$$ ft

3. Find the derivative of $$f(x)=\ln\left(\frac{e^x+e^{-x}}{2}\right)$$.

   • $$\frac{e^x-e^{-x}}{e^x+e^{-x}}$$
   • $$\frac{e^x+e^{-x}}{e^x-e^{-x}}$$
   • $$\frac{e^x}{e^x+e^{-x}}$$
   • $$\frac{e^{-x}}{e^x+e^{-x}}$$

4. The half-life of carbon-14 is about $$5730$$ years. Find the fraction $$\left(\frac{A}{A_0}\right)$$ of carbon-14 as a function of time and determine that fraction at $$8{,}585$$ years.

   • $$0.35$$
   • $$0.50$$
   • $$0.25$$
   • $$0.75$$

5. Estimate the maximum population for Dallas for the year $$2006$$ given $$P(t)=\frac{1{,}301{,}642}{1+21.602e^{-0.05054t}}$$.

   • $$1{,}301{,}642$$
   • $$1{,}150{,}000$$
   • $$980{,}000$$
   • $$1{,}450{,}000$$

6. Let $$R$$ be the region in the first quadrant enclosed by $$y=\sqrt{x+2}$$, the line $$x=8$$, the $$x$$-axis, and the $$y$$-axis. The volume of the solid generated when $$R$$ is revolved about the $$y$$-axis is:

   • $$\pi\int_0^8 (x+2)dx$$
   • $$2\pi\int_0^8 x\sqrt{x+2}dx$$
   • $$\pi\int_0^{\sqrt{10}}(y^2-2)^2dy$$
   • $$2\pi\int_0^{\sqrt{10}} y(y^2-2)dy$$

7. Find the area of the region bounded by $$f(x)=\frac{1}{\sqrt{4x-x^2}}$$, the $$x$$-axis, and the lines $$x=1$$ and $$x=3$$.

   • $$\frac{\pi}{6}$$
   • $$\frac{\pi}{4}$$
   • $$\frac{\pi}{3}$$
   • $$\frac{\pi}{2}$$

8. Find the particular solution of $$y'=\frac{x^2+5x+2}{y}$$, $$y(0)=3$$.

   • $$y^2=\frac23x^3+5x^2+4x+9$$
   • $$y^2=\frac23x^3+5x^2+4x+3$$
   • $$y=\frac13x^3+\frac52x^2+2x+3$$
   • $$y^2=\frac13x^3+5x^2+4x+9$$

9. Find the inverse function of $$f(x)=\sqrt[3]{\frac{2x+1}{x-4}}$$.

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

10. Evaluate $$\lim_{x\to1}\frac{\ln(x+3)-\ln4}{x-1}$$.

    • $$\frac12$$
    • $$\frac13$$
    • $$\frac14$$
    • $$1$$