Mini Exam 3 — Timed Mini Exams

Questions in this quiz (10)

1. Evaluate $$\int_1^{\infty}\frac{x}{e^x}\,dx$$.

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

2. The population $$P(t)$$ satisfies the logistic differential equation $$\frac{dP}{dt}=\frac{1}{2}P\left(4-\frac{P}{200}\right)$$ What is $$\lim_{t\to\infty}P(t)$$

   • $$200$$
   • $$400$$
   • $$600$$
   • $$800$$

3. If $$\sum_{n=0}^{\infty}a_n(x-c)^n$$ is a Taylor series that converges to $$f(x)$$, then $$f'(c)=$$

   • $$a_0$$
   • $$a_1$$
   • $$2a_2$$
   • $$3a_3$$

4. The graph of the function represented by the Taylor series centered at $$x=0$$, $$1+x+\frac{x^2}{2!}+\frac{x^3}{3!}+\cdots$$ intersects the graph of $$y=2$$ at $$x=$$

   • $$0.567$$
   • $$0.693$$
   • $$1.000$$
   • $$1.386$$

5. If $$f$$ is a vector-valued function defined by $$f(t)=\langle\sin t,\ e^t\rangle$$ then $$f'(t)=$$

   • $$\langle\cos t,\ e^t\rangle$$
   • $$\langle-\cos t,\ e^t\rangle$$
   • $$\langle\cos t,\ t e^t\rangle$$
   • $$\langle\sin t,\ e^t\rangle$$

6. If $$f$$ is continuous over the set of real numbers and $$f(x)=\frac{x^2-4}{x-2},\quad x\ne2$$ then $$f(2)=$$

   • $$0$$
   • $$2$$
   • $$4$$
   • None of the above

7. If $$0\le k\le3$$ and the area between the curves $$y=x^2\quad\text{and}\quad y=x$$ from $$x=0$$ to $$x=k$$ is $$1$$, then $$k=$$

   • $$1.145$$
   • $$1.442$$
   • $$1.732$$
   • $$2.000$$

8. Determine $$\frac{dy}{dx}$$ for the curve defined by $$x\cos y=1$$

   • $$\frac{\sin y}{x}$$
   • $$\frac{\sin y}{x}$$
   • $$\frac{\cos y}{x\sin y}$$
   • $$\frac{\cos y}{x}$$

9. If $$f(x)=p(x)+q(x)$$ for $$0\le x\le5$$, then $$\int_0^5 (f(x)-p(x)+2)\,dx=$$

   • $$\int_0^5 (q(x)+2)\,dx$$
   • $$\int_0^5 (q(x)-p(x))\,dx+10$$
   • $$\int_0^5 q(x)\,dx+10$$
   • $$\int_0^5 (q(x)+2)\,dx+5$$

10. Use a fourth-degree Taylor polynomial centered at $$x=0$$ to estimate $$e^{1.5}$$

    • $$4.375$$
    • $$4.398$$
    • $$4.462$$
    • $$4.500$$