Mini Exam 6 — Timed Mini Exams

Questions in this quiz (10)

1. Which of the following sequences converge? I. $$\left\{\frac{5n}{2n^2+1}\right\}$$ II. $$\left\{\frac{4}{n}\right\}$$ III. $$\left\{\frac{2n^3}{n^2+1}\right\}$$

   • I only
   • II only
   • I and II
   • I, II, and III

2. Evaluate $$\lim_{x\to2}\frac{\int_1^x t^2\,dt}{x-2}$$

   • $$0$$
   • $$2$$
   • $$4$$
   • It does not exist

3. Evaluate $$\int\frac{x^2+1}{x}\,dx$$

   • $$\frac{x^3}{3}+\ln|x|+C$$
   • $$\frac{x^2}{2}+\ln|x|+C$$
   • $$\frac{x^3}{3}+x+C$$
   • $$x^2+\ln|x|+C$$

4. What is the instantaneous rate of change of the derivative of the function $$f(x)=\ln(x^3)$$ when $$x=1$$?

   • $$-3$$
   • $$-1$$
   • $$1$$
   • $$3$$

5. Evaluate $$\lim_{x\to\infty}\frac{3x^2+5x-1}{(x+1)(2x-1)}$$

   • $$0$$
   • $$\frac{1}{2}$$
   • $$1$$
   • $$\frac{3}{2}$$

6. If $$f(x)=\tan^2x$$, then $$f'\left(\frac{\pi}{4}\right)=$$

   • $$1$$
   • $$2$$
   • $$4$$
   • $$8$$

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

   • $$2x\cot(x^2)$$
   • $$2x\tan(x^2)$$
   • $$\cot(x^2)$$
   • $$\tan(x^2)$$

8. What is the approximation of the area under $$y=x^2+1$$ for $$0\le x\le2$$ using the trapezoidal rule with $$2$$ subintervals?

   • $$\frac{14}{3}$$
   • $$5$$
   • $$6$$
   • $$7$$

9. Let $$f$$ be the function given by the first four nonzero terms of the Maclaurin polynomial used to approximate $$\sin x$$ Determine the area bounded by the graph and the $$x$$-axis for $$0\le x\le1$$.

   • $$\frac{1}{2}$$
   • $$\frac{5}{12}$$
   • $$\frac{1}{3}$$
   • $$\frac{18511}{40320}$$

10. Evaluate $$\int_0^4\frac{e^{\sqrt{x}}}{\sqrt{x}}\,dx$$

    • $$2(e^2-1)$$
    • $$e^2-1$$
    • $$4(e^2-1)$$
    • $$2(e^4-1)$$