Applications of Differential Calculus — Quiz 3

Questions in this quiz (10)

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

   • $$-\frac{5\sqrt3}{3}$$
   • $$\sqrt3$$
   • $$-1$$
   • $$0$$

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

   • $$3$$
   • $$-3$$
   • $$\pm3$$
   • $$\pm1$$

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

   • $$-6.59$$
   • $$3$$
   • $$4$$
   • $$1$$

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

   • $$x=0$$
   • $$x=\frac{e^{\pi/4}}{\sqrt2}$$
   • $$x=1$$
   • $$x=e^\pi$$

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

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

6. At what rectangular coordinates do the tangent lines to $$r=\cos^2\theta$$ at $$\theta=\frac{\pi}{4}$$ and $$\theta=\frac{3\pi}{4}$$ intersect?

   • $$(0,0)$$
   • $$(1,0)$$
   • $$(0,1)$$
   • $$(1,1)$$

7. Find an equation of the tangent to $$F(t)=(\sin t,3\cos^2 t)$$ at the point where $$t=\frac{\pi}{4}$$.

   • $$y-\frac32=3\left(x-\frac{\sqrt2}{2}\right)$$
   • $$y-\frac32=-3\sqrt2\left(x-\frac{\sqrt2}{2}\right)$$
   • $$y-1=-3\left(x-\frac{\sqrt2}{2}\right)$$
   • $$y-1=3\left(x-\frac{\sqrt2}{2}\right)$$

8. The volume of a cone equals $$V$$ cubic units, where $$V$$ is a constant. Find the proportions of the cone that minimize total surface area.

   • $$h=\sqrt2\,r$$
   • $$h=r$$
   • $$h=2\sqrt2\,r$$
   • $$h=3r$$

9. A concert organizer sells tickets for a show as follows: At least $$30$$ people must attend. The cost when $$30$$ participate is $$\$60$$ per person. The price will drop by $$\$1.50$$ per ticket for each additional person beyond $$30$$. If the venue can accommodate $$50$$ people, how many participants maximize revenue?

   • $$30$$
   • $$40$$
   • $$45$$
   • $$35$$

10. Estimate the value of $$\frac{2}{(1-x)^3}$$ at $$x=0.04$$.

    • $$2.24$$
    • $$2.18$$
    • $$2.25$$
    • $$2.30$$