Differentiation — Quiz 3

Questions in this quiz (10)

1. Given $$f(u)=3u^2+4u$$ and $$u=g(x)=2x^3-1$$ and $$H(x)=f(g(x)),$$ evaluate $$H'(1).$$

   • $$10$$
   • $$60$$
   • $$45$$
   • $$30$$

2. For the function $$f(x)=x^3-2x,$$ approximate $$f'(2)$$ using the symmetric difference quotient with $$h=0.01.$$

   • $$9.94$$
   • $$9.98$$
   • $$10.04$$
   • $$10.0001$$

3. If $$x=3\cos\theta$$ and $$y=\sin(2\theta),$$ find $$\frac{dy}{dx}$$ at $$\theta=\frac{\pi}{6}.$$

   • $$-\frac{2}{3}$$
   • $$-\frac{4}{3}$$
   • $$\frac{4}{3}$$
   • $$\frac{2}{3}$$

4. For the parametric equations $$x=2\sin\theta$$ and $$y=3\cos\theta,$$ find the equation of the tangent line at $$\theta=\frac{\pi}{4}.$$

   • $$y=-\frac{3}{2}x+3\sqrt{2}$$
   • $$y=\frac{3}{2}x-3\sqrt{2}$$
   • $$y=-\frac{3}{2}x+\frac{3\sqrt{2}}{2}$$
   • $$y=\frac{3}{2}x+\frac{3\sqrt{2}}{2}$$

5. Suppose $$2$$ objects are moving in a plane during the time interval $$0\le t\le4$$. Their positions at time $$t$$ are described by the parametric equations$$x_1=2t$$$$y_1=4t-t^2$$and$$x_2=t+1$$$$y_2=4-t$$Find all collision points.

   • $$(1,3)$$
   • $$(2,3)$$
   • $$(2,2)$$
   • $$(1,2)$$

6. If $$x^2-2xy+3y^2=2$$, find $$\frac{dy}{dx}$$.

   • $$2x-2y+3y^2$$
   • $$\frac{x-y}{x-3y}$$
   • $$2x-2+6y$$
   • $$\frac{x+y}{x+2y}$$

7. If $$x\cos y=\sin(x-y)$$, find $$\frac{dy}{dx}$$.

   • $$\frac{\cos(x-y)-\cos y}{x\sin y+\cos(x-y)}$$
   • $$\frac{\cos(x-y)+\cos y}{x\sin y-\cos(x-y)}$$
   • $$\frac{\cos y-\cos(x-y)}{x\sin y+\cos(x-y)}$$
   • $$\frac{\cos(x-y)-\cos y}{x\sin y-\cos(x-y)}$$

8. Find $$\frac{dy}{dx}$$ and $$\frac{d^2y}{dx^2}$$ using implicit differentiation on the equation$$x^2+4y^2=16$$.

   • $$\frac{dy}{dx}=-\frac{x}{4y},\frac{d^2y}{dx^2}=-\frac{4y^2+x^2}{16y^3}$$
   • $$\frac{dy}{dx}=-\frac{x}{2y},\frac{d^2y}{dx^2}=-\frac{4y^2+x^2}{8y^3}$$
   • $$\frac{dy}{dx}=-\frac{x}{4y},\frac{d^2y}{dx^2}=-\frac{4y^2-x^2}{16y^3}$$
   • $$\frac{dy}{dx}=-\frac{2x}{4y},\frac{d^2y}{dx^2}=-\frac{4y^2+x^2}{16y^2}$$

9. You leave your house and drive $$240$$ miles to a nearby city, arriving $$4$$ hours later. What does the Mean Value Theorem guarantee about your speed during the trip?

   • You traveled exactly $$60$$ miles per hour for the entire trip.
   • There was at least one moment during the trip when your instantaneous speed was exactly $$60$$ miles per hour.
   • Your speed was never greater than $$60$$ miles per hour.
   • Your speed increased at a constant rate throughout the trip.

10. Using implicit differentiation, verify the formula for the derivative of the inverse cosine function, $$y=\cos^{-1}x=\arccos x$$, with domain $$[-1,1]$$ and range $$[0,\pi]$$. What is $$\frac{dy}{dx}$$?

    • $$\frac{1}{\sqrt{1-x^2}}$$
    • $$-\frac{1}{\sqrt{1-x^2}}$$
    • $$-\frac{x}{\sqrt{1-x^2}}$$
    • $$\frac{x}{\sqrt{1-x^2}}$$