Mini Exam 1 — Timed Mini Exams

Questions in this quiz (9)

1. The function $$f$$ is given by $$f(x)=2x^4-4x^3+3x^2-1$$. On which of the following intervals is $$f'$$ decreasing?

   • $$(-\infty,\infty)$$
   • $$(-\infty,0)$$
   • $$(1,\infty)$$
   • None

2. What is the area under the curve described by the parametric equations $$x=\cos t,\ y=\sin^2 t$$, for $$0\le t\le\frac{\pi}{2}$$?

   • $$\frac{1}{4}$$
   • $$\frac{2}{3}$$
   • $$\frac{1}{2}$$
   • $$1$$

3. The function $$f$$ is given by $$f(x)=x^3+6x^2+12x+8$$. All of these statements are true EXCEPT

   • $$-2$$ is a zero of $$f$$
   • $$-2$$ is a point of inflection
   • $$-2$$ is a local extremum
   • $$-2$$ is a zero of $$f'$$

4. Evaluate $$\int x e^x\,dx$$

   • $$x e^x+e^x+C$$
   • $$x e^x-e^x+C$$
   • $$e^x+x+C$$
   • $$x^2 e^x+C$$

5. Let $$h(x)=\ln(f(x))$$. If $$f$$ is increasing for all $$x$$ in its domain, then

   • $$h$$ is strictly increasing
   • $$h$$ is strictly decreasing
   • $$h$$ has a relative maximum
   • none of the above

6. Evaluate $$\int_0^1\left(\frac{x^2+2}{x+1}\right)\,dx$$

   • $$1$$
   • $$\frac{3}{2}$$
   • $$1.58$$
   • $$\frac{5}{2}$$

7. The function $$f$$ given by $$f(x)=x^5-5x^3+4x$$ is increasing and concave down over which of these intervals?

   • $$(-\infty,-1)$$
   • $$(-1,0)$$
   • $$(0,1)$$
   • $$(1,\infty)$$

8. The length of the curve described by the parametric equations $$x=t^2,\ y=t^3$$ where $$0\le t\le1$$ is

   • $$\frac{1}{2}$$
   • $$1$$
   • $$1.44$$
   • $$2$$

9. What is the average value of $$f(x)=2\sin^2x+\cos^2x$$ over $$\left[0,\frac{\pi}{2}\right]$$?

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