Mini Exam 4 — Timed Mini Exams

Questions in this quiz (10)

1. The graph of $$g$$, shown below, consists of the arcs of two quarter-circles and two straight-line segments. The value of $$\int_0^{12} g(x)\,dx$$ is:

   • $$\pi+2$$
   • $$\frac{7\pi}{4}+\frac{9}{2}$$
   • $$\frac{7\pi}{4}+8$$
   • $$\frac{25\pi}{4}+\frac{21}{2}$$

2. If $$y=\int_2^x \frac{1}{\sqrt{5-3t}}\,dt$$ then:$$\frac{d^2y}{dx^2}=$$?

   • $$-\frac{3}{2(5+3x)^{3/2}}$$
   • $$\frac{1}{(5+3x)^{1/2}}$$
   • $$\frac{3}{2(5+3x)^{1/2}}$$
   • $$\frac{1}{2(5+3x)^{3/2}}$$

3. Suppose $$f'(x)=x(x-2)^2$$. Then $$f''(x)=(x-2)(3x-2)$$.Which of the following statements is true about the graph of $$f(x)$$?

   • The graph has one relative extremum and one inflection point.
   • The graph has one relative extremum and two inflection points.
   • The graph has two relative extrema and one inflection point.
   • The graph has two relative extrema and two inflection points.

4. The $$n$$th derivative of $$\ln(x+2)$$ at $$x=1$$ equals:

   • $$\frac{(-1)^n n!}{3^n}$$
   • $$\frac{(-1)^{n-1}(n-1)!}{3^n}$$
   • $$\frac{(-1)^{n+1}n!}{3^{n+1}}$$
   • $$\frac{(-1)^n(n-1)!}{3^{n+1}}$$

5. Suppose $$f'(x)=x(x-3)^2$$. Then $$f''(x)=(x-3)(3x-3).$$ Over which interval(s) is the graph of $$f$$ both increasing and concave up?$$I.$$ $$x$$ < $$0$$$$II.$$ $$0$$ < $$x$$ < $$1$$$$III.$$ $$1$$ < $$x$$ < $$3$$$$IV.$$ $$x$$ > $$3$$

   • $$I$$ only
   • $$II$$ and $$IV$$ only
   • $$I$$ and $$III$$ only
   • $$IV$$ only

6. If $$f(x)$$ is differentiable at the point where $$x=a$$, which of the following statements must be true?

   • $$\lim_{x\to a}f(x)=f(a)$$
   • $$f(x)$$ is continuous at $$x=a$$
   • $$f'(a)$$ exists
   • All the above

7. The volume, in cubic feet, of a hollow cylinder with inner diameter $$6$$ feet and outer diameter $$10$$ feet and height $$1$$ foot is:

   • $$16\pi$$
   • $$24\pi$$
   • $$32\pi$$
   • $$40\pi$$

8. The graph of function $$h$$ is shown above. Which of these statements is true?$$I.$$ The first derivative is never negative.$$II.$$ The second derivative is constant.$$III.$$ The first and second derivatives equal $$0$$ at the same point.

   • $$I$$ only
   • $$III$$ only
   • $$I$$ and $$II$$ only
   • $$I$$ and $$III$$ only

9. If $$60$$ grams of a radioactive substance decomposes to $$30$$ grams in $$3$$ years, then, to the nearest gram, the amount left after $$5$$ years is:

   • $$19$$
   • $$20$$
   • $$22$$
   • $$24$$

10. The table shows the speed of an object in $$ft/sec$$ at certain times during an $$8$$-second period. Estimate its acceleration in $$ft/sec^2$$ at $$t=4$$ seconds.

    • $$-6$$
    • $$-4$$
    • $$-frac{20}{6}$$
    • $$-2$$