Mini Exam 5 — Timed Mini Exams

Questions in this quiz (9)

1. If $$x=\ln t$$ and $$y=e^{3t}$$, then $$\frac{dy}{dx}=$$

   • $$3e^{3t}$$
   • $$\frac{3e^{3t}}{t}$$
   • $$3te^{3t}$$
   • $$\frac{te^{3t}}{3}$$

2. The function $$y=\frac{(x-1)^2}{x^2-4x+3}$$ has a local minimum at $$x=$$

   • $$0$$
   • $$1$$
   • $$2$$
   • The function has no local minimum.

3. Evaluate $$\frac{d}{dx}\left(e^{2x}\ln(\sin e^x)\right)$$

   • $$2e^{2x}\ln(\sin e^x)+e^{3x}\cot(e^x)$$
   • $$e^{2x}\ln(\sin e^x)+e^{3x}\cot(e^x)$$
   • $$2e^{2x}\ln(\sin e^x)+e^{2x}\cot(e^x)$$
   • $$e^{3x}\cot(e^x)$$

4. If $$f(x)=\frac{\cos x}{x^2}$$ then $$f'(\pi)=$$

   • $$-\frac{1}{\pi^2}$$
   • $$\frac{1}{\pi^2}$$
   • $$-\frac{2}{\pi^3}$$
   • $$\frac{2}{\pi^3}$$

5. Evaluate $$\int_2^{\infty}\frac{\ln x}{x^3}\,dx$$

   • $$\frac{\ln2+1}{4}$$
   • $$\frac{\ln2}{4}$$
   • $$\frac{2\ln2+1}{16}$$
   • It diverges

6. Evaluate $$\int x\tan^2x\,dx$$

   • $$x\tan x+\ln|\cos x|+C$$
   • $$x\tan x+\ln|\cos x|-\frac{x^2}{2}+C$$
   • $$x\tan x-\ln|\sec x+\tan x|+C$$
   • $$x\tan x+\ln|\sec x+\tan x|+C$$

7. Evaluate $$\lim_{x\to1}\frac{(\ln x)^3}{x^2-1}$$

   • $$0$$
   • $$1$$
   • $$\frac{1}{2}$$
   • It does not exist

8. What is the approximation of the value of $$\sin(1)$$ obtained by using the fifth-degree Taylor polynomial about $$x=0$$ for $$\sin x$$?

   • $$1-\frac{1}{6}+\frac{1}{120}$$
   • $$1-\frac{1}{2}-\frac{1}{24}$$
   • $$1-\frac{1}{6}-\frac{1}{120}$$
   • $$1-\frac{1}{2}+\frac{1}{120}$$

9. A particle moves on a plane curve so that at any time $$t>0$$ its position is defined by $$x(t)=t^2+1,\quad y(t)=\frac{t^3+2}{t}$$ The acceleration vector of the particle at $$t=1$$ is

   • $$\langle2,1\rangle$$
   • $$\langle2,2\rangle$$
   • $$\langle2,6\rangle$$
   • $$\langle3,1\rangle$$