Mini Exam 13 — Timed Mini Exams

Questions in this quiz (10)

1. Apply Newton's Method to approximate the $$x$$-value of the point of intersection of $$f(x)=3x^2-1$$ and $$g(x)=\sin x$$ closest to $$1$$.

   • $$0.52$$
   • $$0.64$$
   • $$0.71$$
   • $$0.83$$

2. If $$g$$ is a differentiable function such that $$g(x)<0$$ for all real numbers $$x$$, and if $$f'(x)=(x^2-4)g(x)$$, which of the following is true?

   • $$f$$ has a relative maximum at $$x=-2$$ and a relative minimum at $$x=2$$
   • $$f$$ has a relative minimum at $$x=-2$$ and a relative maximum at $$x=2$$
   • $$f$$ has relative minima at $$x=-2$$ and $$x=2$$
   • $$f$$ has relative maxima at $$x=-2$$ and $$x=2$$

3. Let $$f(x)=\sqrt{x}$$. If the rate of change of $$f$$ at $$x=c$$ is twice its rate of change at $$x=1$$, then $$c=$$

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

4. Evaluate the integral: $$\int \frac{\cos x}{\sin^2 x}\,dx$$

   • $$-\frac{1}{\sin x}+C$$
   • $$\frac{1}{\sin x}+C$$
   • $$\ln(\sin x)+C$$
   • $$-\ln(\sin x)+C$$

5. Find the sum: $$\sum_{k=1}^{4}(-1)^k(2k)$$

   • $$-4$$
   • $$-2$$
   • $$2$$
   • $$4$$

6. Express the limit as a definite integral on the interval $$[2,3]$$: $$\lim_{|\Delta|\to0}\sum_{i=1}^n(c_i^2-2c_i)\Delta_i$$

   • $$\int_2^3(x^2-2x)dx$$
   • $$\int_2^3(x^2+2x)dx$$
   • $$\int_2^3(2x-x^2)dx$$
   • $$\int_1^3(x^2-2x)dx$$

7. Evaluate the definite integral: $$\int_1^4\left(\sqrt{x}-\frac{1}{x^2}\right)dx$$

   • $$\frac{16}{3}-\frac14$$
   • $$\frac{14}{3}-\frac34$$
   • $$\frac{16}{3}+\frac34$$
   • $$\frac{14}{3}+\frac14$$

8. A store gets 1300 cases of candy every 30 days. $$I(x)=1300-50x$$. Find the average daily inventory and the average daily holding cost if holding a case costs 3 cents per day.

   • 650 cases, $19.50$
   • 600 cases, $18.00$
   • 700 cases, $21.00$
   • 650 cases, $18.00$

9. Find the derivative: $$f(x)=\frac{(x+1)^2(2x^2-3)}{\sqrt{x^2+1}}$$

   • Product + quotient rule expression
   • Chain rule only
   • Quotient rule only
   • Product rule only

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

    • $$\frac{4}{\pi}$$
    • $$\frac{2}{\pi}$$
    • $$\frac{2\sqrt2}{\pi}$$
    • $$\frac{\sqrt2}{\pi}$$