Module 4 Assessment MC Q
1. Given $$f(x)=9-\left(\frac{14}{x}\right)$$ , find all $$c$$ in the interval
$$(2,7)$$ such that $$f^{\prime}(c)=\frac{f(7)-f(2)}{7-2}$$ .
$$\frac{9}{2}$$
$$\pm \sqrt{14}$$
*$$\sqrt{14}$$
$$\frac{14}{9}$$
2. Let $$f(x)=x^3-x^2+3$$ . Determine all critical numbers.
*$$0 \text{ , } \frac{2}{3}$$
$$1 \text{ , } 3$$
$$0 \text{ , } 3$$
$$\frac{2}{3} \text{ , } 3$$
3. Which of the following statements is true of $$f(x)=-x^3+9x^2-24x+18$$ ?
$$f$$ is decreasing on $$(2,4)$$
*$$f$$ is increasing on $$(2,4)$$
$$f$$ is increasing on $$(3,\infty)$$
$$f$$ is increasing on $$(-\infty,2)$$
4. Given $$f(x)=2x^4(x^2-3)$$ . For what value(s) is the graph of $$f$$ concave
upwards?
$$\left(0,\sqrt{\frac{6}{5}}\right)$$
$$\left(-\sqrt{\frac{6}{5}},0\right)$$
*$$\left(-\infty,\sqrt{\frac{6}{5}}\right)\cup\left(\sqrt{\frac{6}{5}},\infty\right)$$
$$\left(-\sqrt{\frac{6}{5}},\sqrt{\frac{6}{5}}\right)$$
5. Given that $$f(x)=\frac{4}{x}$$ , choose the correct statement.
$$f$$ is concave up on the interval $$(-\infty,0)$$
*$$f$$ is concave down on the interval $$(-\infty,0)$$
$$f$$ is concave up on the interval $$(-\infty,\infty)$$
$$f$$ is concave down on the interval $$(0,\infty)$$
6. Given that $$f(x)=-x^2+12x-34$$ has a relative maximum at $$x=6$$ , choose
the correct statement.
$$f^{\prime}$$ is positive on the interval $$(6,\infty)$$
*$$f^{\prime}$$ is negative on the interval $$(6,\infty)$$
$$f^{\prime}$$ is negative on the interval $$(-\infty,6)$$
$$f^{\prime}$$ is positive on the interval $$(-\infty,\infty)$$
7. Find all points of inflection of the function $$f(x)=x^4+x^3$$ .
*$$(0,0) \text{ and } \left(-\frac{1}{2},-\frac{1}{16}\right)$$
$$(0,0)$$
$$\left(-\frac{1}{2},-\frac{1}{16}\right)$$
$$\left(-\frac{3}{4},-\frac{27}{256}\right) \text{ and } \left(-\frac{1}{2},\frac{1}{16}\right)$$
8. The figure shows the graph of $$f^{\prime}$$ , the derivative of the function
$$f$$. The domain of $$f$$ is $$[-10,10]$$ . For what value(s) of $$x$$ does the
function have a relative maximum?
3 and 5
2
*6
4
9. The figure shows the graph of $$f^{\prime}$$ , the derivative of the function
$$f$$. The domain of $$f$$ is $$[1,6]$$ . For what value(s) of $$x$$ is the graph
of $$f$$ concave upwards?
$$(-\infty,6)$$
*$$(1,6)$$
$$(-\infty,1)$$
$$(4,6)$$
10. the graph of the derivative of $$f(x)$$ is shown.
Which of the following could be $$f(x)$$ ?
*
*
11. A point moves along the curve $$y=\sqrt{x}$$ in such a way that the y-value
is increasing at the rate of 2 units per second. At what rate is $$x$$ changing
when$$ x=\frac{1}{2}$$ ?
*$$2\sqrt{2}$$
$$\frac{\sqrt{5}}{5}$$
$$1$$
$$\frac{\sqrt{2}}{2}$$
12. Two vehicles are approaching an intersection. One truck from the west at 15
m/s and one van from the north at 20 m/s. How fast is the distance between the
vehicles changing at the instant the truck is 60 m west and the van 80 m north of
the intersection?
$$5\sqrt{10} \text{ m/s}$$
$$10 \text{ m/s}$$
$$20 \text{ m/s}$$
*$$25 \text{ m/s}$$
13. A searchlight is located at a perpendicular distance of 315 yards from a fixed
point F on a straight shoreline. This light revolves at 1 revolution per minute. How
fast does its beam sweep along the shoreline at a point P, 425 yards downshore
from F?
$$1024\pi \text{ yd/min}$$
$$1259\pi \text{ yd/min}$$
$$1681\pi \text{ yd/min}$$
*$$1777\pi \text{ yd/min}$$
14. The product of two positive numbers is 588. The sum of the first and three
times the second is minimized. Find the two numbers.
*42 and 14
28 and 21
49 and 12
Both numbers are $$14\sqrt{3}$$
15. Given $$f(x)=\sqrt{1-x^2}$$ intersects the x-axis at points A and B. Find the
abscissa (x-coordinate) of point C on the semicircle so that the area of triangle ABC
is a maximum.
*$$0$$
$$\sqrt{3}$$
$$\frac{1}{\sqrt{2}}$$
$$\frac{1}{\sqrt{3}}$$
16. A farmer has 160 meters of fence to enclose two adjacent pig pens. What
dimensions should be used for each pig pen so that the enclosed area will be a
maximum?
*$$20 \text{ m by } \frac{80}{3} \text{ m}$$
$$40 \text{ m by } \frac{80}{3} \text{ m}$$
$$40 \text{ m by } 40 \text{ m}$$
$$\frac{40}{3} \text{ m by } 60 \text{ m}$$
17. A right circular cylinder is inscribed in a sphere of radius a cm. Find the
maximum volume of the cylinder.
*$$\frac{4\pi a^3}{3\sqrt{3}}$$
$$\frac{4\pi a^2}{3}$$
$$\frac{2\pi a^3}{3}$$
$$\frac{4a^3}{\sqrt{3}}$$
18. Use Newton’s method to find a root of $$x^3=5x-1$$ .
0.0021
*0.2016
0.2164
0.2166
19. $$\lim_{x \to 0}\frac{e^{x^2}-1}{2x^2}$$
$$0$$
*$$\frac{1}{2}$$
$$\infty$$
$$-1$$
20. Use differentials to approximate $$\sqrt{4.9}$$ .
*2.225
2.250
2.214
2.450