Mini Exam 12 — Timed Mini Exams

Questions in this quiz (10)

1. Evaluate the limit: $$\lim_{z\to0}\frac{\sqrt{7-z}-\sqrt{7}}{z}$$

   • $$\frac{1}{2\sqrt{7}}$$
   • $$2\sqrt{7}$$
   • $$-\frac{1}{2\sqrt{7}}$$
   • $$0$$

2. Find values for $$a$$ and $$b$$ so that the function is continuous: $$f(x)=\begin{cases}5,&x\le-4\\2ax+b,&-4<x<3\\bx+5,&x\ge3\end{cases}$$

   • $$a=1,b=3$$
   • $$a=-\frac{3}{2},b=-7$$
   • $$a=1,b=-3$$
   • $$a=-1,b=-3$$

3. Find the derivative of the function: $$f(x)=\frac{3\cos x}{4}+2\sqrt{x}$$

   • $$-\frac{3}{4}\sin x+\frac{1}{\sqrt{x}}$$
   • $$\frac{3}{4}\sin x+\frac{1}{\sqrt{x}}$$
   • $$-\frac{3}{4}\cos x+\frac{1}{\sqrt{x}}$$
   • $$-\frac{3}{4}\sin x+\frac{2}{\sqrt{x}}$$

4. Find $$k$$ such that the line $$y=5x-4$$ is tangent to the graph of the function $$f(x)=x^2-kx$$.

   • $$k=3$$
   • $$k=5$$
   • $$k=-1\text{ or }-9$$
   • $$k=9$$

5. Differentiate with respect to $$x$$: $$y=\left(\frac{2x+1}{2x-1}\right)^5$$

   • $$\frac{20(2x+1)^4}{(2x-1)^6}$$
   • $$\frac{10(2x+1)^4}{(2x-1)^6}$$
   • $$-\frac{20(2x+1)^4}{(2x-1)^6}$$
   • $$\frac{5(2x+1)^4}{(2x-1)^5}$$

6. Let $$f'(x)=12x^2-24x$$ and let $$f(x)$$ have critical numbers $$0$$ and $$3$$. Use the Second Derivative Test to determine which of the critical numbers gives a relative maximum.

   • $$0$$
   • $$3$$
   • $$0\text{ and }3$$
   • none

7. Find an equation of the tangent line to the graph of $$y=2\tan x+\cos^2(2x)$$ at $$x=\pi$$.

   • $$y=2x-2\pi+1$$
   • $$y=2x-2\pi-1$$
   • $$y=4x-4\pi+1$$
   • $$y=2x-\pi+1$$

8. A particle is moving along the curve $$y=\sqrt{x}$$. As the particle passes through $$(4,2)$$, its $$x$$-coordinate increases at $$6$$ cm/s. How fast is the distance from the particle to the origin changing?

   • $$15/\sqrt5$$
   • $$12/\sqrt5$$
   • $$9/\sqrt5$$
   • $$18/\sqrt5$$

9. Find the absolute extrema of $$f(x)=\sin x\cos x$$ on $$[0,2\pi]$$.

   • Max $$1/2$$ Min $$-1/2$$
   • Max $$1$$ Min $$-1$$
   • Max $$1/2$$ Min $$0$$
   • Max $$0$$ Min $$-1/2$$

10. An open rectangular box is made from a $$4$$ ft by $$5$$ ft piece of cardboard by cutting congruent squares from the corners and folding up the sides. How long should the sides of the square be to create the box of largest volume?

    • $$\frac{3-\sqrt3}{2}$$
    • $$\frac{9-\sqrt21}{6}$$
    • $$\frac{5-\sqrt5}{2}$$
    • $$\frac{4-\sqrt2}{2}$$