Mini Exam 9 — Timed Mini Exams

Questions in this quiz (10)

1. Estimate $$\int_0^2\sqrt{16-x^2}\,dx$$ using a left rectangular sum and two subintervals of equal width.

   • $$4+2\sqrt{3}$$
   • $$6+2\sqrt{3}$$
   • $$8+2\sqrt{3}$$
   • $$10+2\sqrt{3}$$

2. What is the domain of the particular solution, $$y=f(x)$$, for $$\frac{dy}{dx}=\frac{4x}{x^2-9}$$ containing the point $$(1,0)$$?

   • $$x<3$$
   • $$-3<x<3$$
   • $$x\ne\pm3$$
   • $$x>-3$$

3. If we substitute $$x=\tan\theta$$, which of the following is equivalent to $$\int_0^{\sqrt{3}}\sqrt{1+x^2}\,dx$$?

   • $$\int_0^{\pi/3}\sec\theta\,d\theta$$
   • $$\int_0^{\pi/3}\sec^2\theta\,d\theta$$
   • $$\int_0^{\pi/3}\sec^3\theta\,d\theta$$
   • $$\int_0^{\tan^{-1}\sqrt{3}}\sec^3\theta\,d\theta$$

4. Which of these diverges?

   • $$\sum_{n=1}^{\infty}\frac{1}{2^n}$$
   • $$\sum_{n=1}^{\infty}\frac{1}{2}$$
   • $$\sum_{n=1}^{\infty}\frac{1}{n}$$
   • $$\sum_{n=1}^{\infty}\frac{n}{3^n}$$

5. If $$x=3\cos t$$ and $$y=3\sin t$$, then a single equation in $$x$$ and $$y$$ is

   • $$x^2+y^2=3$$
   • $$x^2+y^2=9$$
   • $$x^2-y^2=9$$
   • $$x^2+3y^2=9$$

6. The $$n$$th derivative of $$\ln(x+2)$$ at $$x=1$$ equals

   • $$\frac{(-1)^n(n-1)!}{3^n}$$
   • $$\frac{(-1)^{n+1}n!}{3^{n+1}}$$
   • $$\frac{(-1)^n n!}{3^n}$$
   • $$\frac{(-1)^{n-1}(n-1)!}{3^{n+1}}$$

7. Suppose $$\int_1^4 f(x+k)\,dx=6$$ where $$k$$ is a constant. Then $$\int_{1+k}^{4+k}f(x)\,dx$$ equals

   • $$3$$
   • $$4$$
   • $$6$$
   • $$6+k$$

8. If $$f(u)=\tan^{-1}(u^3)$$ and $$g(u)=e^{2u}$$, then the derivative of $$f(g(u))$$ is

   • $$\frac{3e^{2u}}{1+e^{6u}}$$
   • $$\frac{6e^{2u}}{1+e^{6u}}$$
   • $$\frac{6e^{6u}}{1+e^{12u}}$$
   • $$\frac{1+e^{12u}}{3e^{4u}}$$

9. If $$\cos(xy)=x$$, then $$\frac{dy}{dx}$$ equals

   • $$\frac{\sin(xy)}{1+y\sin(xy)}$$
   • $$\frac{\sin(xy)}{y\sin(xy)-1}$$
   • $$\frac{-1-y\sin(xy)}{x\sin(xy)}$$
   • $$\frac{1-y\sin(xy)}{x\sin(xy)}$$

10. Find the radius of convergence of $$\sum_{n=1}^{\infty}\frac{n!}{2^n}x^n$$

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