Applications of Differential Calculus — Quiz 2

Questions in this quiz (10)

1. Find the slope of the tangent line to $$r=1+2\sin\theta$$ if $$\theta=\frac{\pi}{4}$$.

   • $$-1$$
   • $$-(2\sqrt2+1)$$
   • $$1$$
   • $$\frac{3+2\sqrt2}{3}$$

2. At what values of $$\theta$$, $$0\le\theta\le\pi$$, does $$r=3\sin(2\theta)$$ have vertical or horizontal tangent lines?

   • Horizontal: $$\frac{\pi}{8},\frac{3\pi}{8},\frac{5\pi}{8},\frac{7\pi}{8}$$
   • Horizontal: $$0,\frac{\pi}{2},\pi$$
   • Horizontal: $$0,\arcsin\sqrt{\frac23},\pi-\arcsin\sqrt{\frac23},\pi$$; Vertical: $$\arccos\sqrt{\frac23},\frac{\pi}{2},\pi-\arccos\sqrt{\frac23}$$
   • Horizontal: $$\frac{\pi}{4},\frac{3\pi}{4}$$

3. Find the slopes of the tangent lines to $$r=a\sin\left(2\theta\right)$$, $$a$$>$$0$$ at the four points furthest from the origin.

   • $$\pm1$$
   • $$\pm2$$
   • $$\pm \frac{1}{2}$$
   • $$\pm \frac{3}{2}$$

4. Find $$\frac{dy}{dx}$$ at $$\theta=\frac{5\pi}{6}$$ for the polar equation $$r=\cos\theta\sin\theta$$.

   • $$-\sqrt3$$
   • $$\sqrt3$$
   • $$-1$$
   • $$0$$

5. A function $$f$$ has the derivative shown. Which of the following statements must be false?

   • $$f$$ is continuous at $$x=a$$
   • $$f$$ has a vertical asymptote at $$x=a$$
   • $$-\sqrt3$$
   • $$f$$ has a jump discontinuity at $$x=a$$
   • $$\sqrt3$$
   • $$f$$ has a removable discontinuity at $$x=a$$
   • $$-1$$
   • $$0$$

6. The rate of change of $$f(x)$$ is least at:

   • $$-3$$
   • $$-1.3$$
   • $$\pm1$$
   • $$0.7$$
   • $$\pm3$$
   • $$2.7$$
   • $$-\!3$$
   • $$\pm\frac12$$

7. At what values of $$t$$ does $$\frac{dy}{dx}=\frac13$$ for the parametric function $$x=1+t^2,\ y=4-2t$$?

   • $$\pm1$$
   • $$\pm3$$
   • $$-\!3$$
   • $$\pm\frac12$$

8. If $$f(x)=3\tan x+x^3$$ and $$g(x)$$ is origin symmetric with the following values, estimate $$\lim_{x\to0}\frac{f(x)}{g(x)}$$.

   • $$1$$
   • $$3$$
   • $$4$$
   • $$0$$

9. Find the equation of the vertical tangent line to $$x=e^t\cos t,\ y=3t+2$$.

   • $$x=0$$
   • $$x=e^{\pi/2}$$
   • $$x=\frac{e^{\pi/4}}{\sqrt2}$$
   • $$x=e^\pi$$

10. If $$k(x)=x^3+2x+1$$, what is the equation of the tangent line to $$k^{-1}(x)$$ when $$x=5$$?

    • $$y=\frac15(x-5)+1$$
    • $$y=\frac13(x-5)+1$$
    • $$y=\frac1{11}(x-5)+1$$
    • $$y=\frac17(x-5)+1$$