Applications of Differential Calculus — Quiz 3
Practice Applications of Differential Calculus — Quiz 3 on The School of Mathematics (Applications of Differential Calculus) with 10 scored questions. This set covers problems such as: “Find \frac{dy}{dx} at \theta=\frac{5\pi}{6} for the polar equation r=\cos\theta\sin\theta .”; “At what values of t does \frac{dy}{dx}=\frac13 for the parametric function defined by x=1+t^2,\ y=4-2t ?”; “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)}…”. Work carefully, check explanations for misses, then retake to track improvement before test day.
Questions in this quiz (10)
Find $$\frac{dy}{dx}$$ at $$\theta=\frac{5\pi}{6}$$ for the polar equation $$r=\cos\theta\sin\theta$$.
- $$-\frac{5\sqrt3}{3}$$
- $$\sqrt3$$
- $$-1$$
- $$0$$
At what values of $$t$$ does $$\frac{dy}{dx}=\frac13$$ for the parametric function defined by $$x=1+t^2,\ y=4-2t$$?
- $$3$$
- $$-3$$
- $$\pm3$$
- $$\pm1$$
If $$f(x)=3\tan x+x^3$$ and $$g(x)$$ is origin symmetric with the following values estimate $$\lim_{x\to0}\frac{f(x)}{g(x)}$$.
- $$-6.59$$
- $$3$$
- $$4$$
- $$1$$
Find the equation of the vertical tangent line to the curve defined by $$x=e^t\cos t,\ y=3t+2$$.
- $$x=0$$
- $$x=\frac{e^{\pi/4}}{\sqrt2}$$
- $$x=1$$
- $$x=e^\pi$$
If $$k(x)=x^3+2x+1$$ what is the equation of the tangent line to $$k^{-1}(x)$$ when $$x=5$$?
- $$y=\frac15(x-5)+1$$
- $$y=\frac13(x-5)+1$$
- $$y=\frac1{11}(x-5)+1$$
- $$y=\frac17(x-5)+1$$
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?
- $$(0,0)$$
- $$(1,0)$$
- $$(0,1)$$
- $$(1,1)$$
Find an equation of the tangent to $$F(t)=(\sin t,3\cos^2 t)$$ at the point where $$t=\frac{\pi}{4}$$.
- $$y-\frac32=3\left(x-\frac{\sqrt2}{2}\right)$$
- $$y-\frac32=-3\sqrt2\left(x-\frac{\sqrt2}{2}\right)$$
- $$y-1=-3\left(x-\frac{\sqrt2}{2}\right)$$
- $$y-1=3\left(x-\frac{\sqrt2}{2}\right)$$
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.
- $$h=\sqrt2\,r$$
- $$h=r$$
- $$h=2\sqrt2\,r$$
- $$h=3r$$
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?
- $$30$$
- $$40$$
- $$45$$
- $$35$$
Estimate the value of $$\frac{2}{(1-x)^3}$$ at $$x=0.04$$.
- $$2.24$$
- $$2.18$$
- $$2.25$$
- $$2.30$$