Differentiation — Quiz 4

Questions in this quiz (10)

1. Determine the value of $$\frac{dy}{dx}$$ if $$x=1$$ for $$e^{xy}+\sin y=2$$.

   • $$1$$
   • $$0$$
   • $$-1$$
   • $$2$$

2. Write the equation of the tangent line to $$y=\frac{x^2+3}{2x+1}$$ at $$x=1$$.

   • $$y=\frac{4}{3}-\frac{2}{9}(x-1)$$
   • $$y-\frac{4}{3}=\frac{2}{9}(x-1)$$
   • $$y-\frac{4}{3}=-\frac{2}{9}(x-1)$$
   • $$y=-\frac{2}{9}x+\frac{4}{3}$$

3. At what $$x$$-values does $$(2x^2-x)(x+2)^{3/5}$$ have horizontal tangent lines?

   • $$x=1$$
   • $$x=-2$$
   • $$x=2$$
   • $$x=0,\frac12$$

4. If $$x=t^3+2$$ and $$y=3t^2$$, then $$\frac{dy}{dx}=$$

   • $$2t$$
   • $$\frac{2}{t}$$
   • $$6t$$
   • $$\frac{6t}{3t^2+2}$$

5. If $$x=t^2+2t$$ and $$y=t^3-t$$, then when $$t=1$$, $$\frac{d^2y}{dx^2}$$ is

   • $$0$$
   • $$1$$
   • $$2$$
   • $$-1$$

6. If $$x=e^\theta\sin\theta$$ and $$y=e^\theta\cos\theta$$, then when $$\theta=\frac{\pi}{2}$$, $$\frac{dy}{dx}$$ is

   • $$0$$
   • $$1$$
   • $$-1$$
   • $$e^{\pi/2}$$

7. The graph in the xy-plane represented by $$x=2+3\sin t$$ and $$y=4\cos t+1$$, for $$-\pi\le t\le\pi$$, is

   • circle
   • ellipse
   • parabola
   • half ellipse

8. $$\lim_{x\to0} x\sin\left(\frac{1}{x}\right)$$

   • $$0$$
   • $$1$$
   • $$\infty$$
   • oscillates

9. Choose the alternative that is the derivative $$\frac{dy}{dx}$$. If $$x=\frac12 t$$ and $$y=\ln(2-t)+3\ (t<2)$$.

   • $$\frac12 t$$
   • $$2-t$$
   • $$\frac{1}{(2-t)^2}$$
   • $$(2-t)^2$$

10. Choose the alternative that is the derivative $$\frac{dy}{dx}$$. If $$x=\frac13 t$$ and $$y=2-\ln(3-t)\ (t<3)$$.

    • $$\frac13 t$$
    • $$3-t$$
    • $$\frac{1}{(3-t)^2}$$
    • $$(3-t)^2$$