Antidifferentiation — Quiz 3

Questions in this quiz (10)

1. Find the vertical line $$x=c$$ that splits the area bounded by $$y=x^2$$, $$y=0$$, and $$x=3$$ exactly in half.

   • $$\sqrt[3]{\frac{27}{2}}$$
   • $$\sqrt{\frac{9}{2}}$$
   • $$\frac{3}{2}$$
   • $$\sqrt[3]{\frac{9}{2}}$$

2. Evaluate $$\frac{d}{dx}\left(\int_{x^2}^{5}\cos t\,dt\right)$$.

   • $$2x\cos(x^2)$$
   • $$-2x\cos(x^2)$$
   • $$-\cos(x^2)$$
   • $$2x\sin(x^2)$$

3. If $$f(x)=\int_0^{x/2}\sin t\,dt$$, evaluate $$f(\pi)$$, $$f'(\pi)$$, and $$f''(\pi)$$.

   • $$1,\frac12,0$$
   • $$2,0,\frac12$$
   • $$1,0,-\frac12$$
   • $$1,\frac12,0$$

4. Find the average value of the function $$f(x)=\sqrt{x}+2x$$ on $$[1,4]$$.

   • $$\frac{19}{6}$$
   • $$\frac{17}{6}$$
   • $$\frac{59}{9}$$
   • $$\frac{11}{6}$$

5. Find the average value of $$f(x)=\sec^2 x$$ on the interval $$\left[\frac{\pi}{6},\frac{\pi}{3}\right]$$.

   • $$\frac{4\sqrt3}{\pi}$$
   • $$\frac{3}{\pi/6}$$
   • $$\frac{2}{\pi/4}$$
   • $$\frac{4}{\pi}$$

6. If $$\int_3^8 f(x)\,dx=12$$ and $$\int_3^{10} f(x)\,dx=18$$, what is the average value of $$f(x)$$ on the interval $$[8,10]$$?

   • $$2$$
   • $$3$$
   • $$4$$
   • $$6$$

7. Find the value $$c$$ guaranteed by the Mean Value Theorem for Integrals for $$f(x)=x^2+\frac{1}{x}$$ on $$[1,3]$$.

   • $$\sqrt2$$
   • $$2$$
   • $$\sqrt3$$
   • $$2$$

8. A particle travels along the x-axis according to the position function $$s(t)=\sin t+\cos t$$. Find the average velocity on the interval $$[0,\pi]$$.

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

9. Evaluate $$\int \frac{\ln(2x)}{x}dx$$.

   • $$\frac{(\ln(2x))^2}{2}+C$$
   • $$(\ln(2x))^2+C$$
   • $$\frac{\ln(2x)}{2}+C$$
   • $$\frac{(\ln x)^2}{2}+C$$

10. Evaluate $$\int_1^2 6x^2\sqrt{2x^3+1}dx$$.

    • $$\frac{(\ln(2x))^2}{2}+C$$
    • $$(\ln(2x))^2+C$$
    • $$\frac{\ln(2x)}{2}+C$$
    • $$\frac{(\ln x)^2}{2}+C$$