Applications of Differential Calculus — Quiz 1

Questions in this quiz (10)

1. If $$f(x)=(2x-5)^2,\ x\ge\frac52$$, find $$\left(f^{-1}\right)'(x)$$.

   • $$\frac{1}{2\sqrt{x}}$$
   • $$\frac{1}{4\sqrt{x}}$$
   • $$\frac{1}{\sqrt{x}}$$
   • $$\frac{1}{4x}$$

2. If $$g$$ is continuous and differentiable with the table below, evaluate $$\left(g^{-1}\right)'(3)$$. $$\begin{array}{c|ccccc} x&-2&-1&0&1&2\\ \hline g(x)&-3&-1&2&3&5\\ g'(x)&4&-2&1&3&2 \end{array}$$

   • $$3$$
   • $$1$$
   • $$\frac{1}{3}$$
   • $$\frac{1}{2}$$

3. Use a linear approximation centered at $$1$$ to estimate $$h(1.08)$$ if $$h(t)=t^3-\frac{2}{t}+3\ln t$$.

   • $$1.16$$
   • $$1.24$$
   • $$1.32$$
   • $$-0.36$$

4. Evaluate the limit: $$\lim_{x\to4}\frac{\sqrt{x}-2}{x-4}$$

   • $$\frac14$$
   • $$\frac12$$
   • $$2$$
   • $$0$$

5. Evaluate the limit: $$\lim_{x\to0}\sqrt{\frac{x+\sin x}{3x+1}}$$

   • $$1$$
   • $$0$$
   • $$\sqrt2$$
   • Does not exist

6. Evaluate the limit: $$\lim_{x\to1}\frac{e^{x-1}-x+1}{x-1}$$

   • $$0$$
   • $$1$$
   • Does not exist
   • $$2$$

7. Evaluate the limit: $$\lim_{x\to\infty}\left(1+\frac{2}{x}\right)^x$$

   • $$e$$
   • $$2e$$
   • $$2$$
   • $$e^2$$

8. Find the equation of the tangent line to the parametric curve defined by $$x=\ln t,\ y=t^2$$, when $$t=1$$.

   • $$y=2x+1$$
   • $$y=x+1$$
   • $$y=2x$$
   • $$y=x$$

9. What is the derivative of the parametric curve defined by $$x=\sec t,\ y=\tan t$$?

   • $$\sec t$$
   • $$\csc t$$
   • $$\sin t$$
   • $$\tan t$$

10. The position of a particle moving along a line is defined parametrically as $$x=e^t,\ y=t^3-3t+2$$ for $$t\ge0$$. Rank the following values of $$t$$ from least to greatest in terms of speed: $$t=0,\ 0.5,\ 1,\ 2,\ 3$$.

    • $$0,\ 0.5,\ 1,\ 2,\ 3$$
    • $$0.5,\ 0,\ 1,\ 2,\ 3$$
    • $$1,\ 0.5,\ 0,\ 2,\ 3$$
    • $$0,\ 1,\ 0.5,\ 2,\ 3$$