220
Chapter 4: Applications of Differentiation
____
11.
Find the open interval(s) on which the function
is increasing in the
interval
Round numerical values in your answer to three decimal places.
increasing on:
increasing on:
increasing on:
increasing on:
increasing on:
____ 12. Find the relative maxima of on the interval by applying the
First Derivative Test. Round numerical values in your answer to three decimal places.
relative maxima:
relative maxima:
relative maxima:
relative maxima:
relative maxima:
____ 13. Find the relative minima of on the interval by applying the
First Derivative Test. Round numerical values in your answer to three decimal places.
relative minima:
relative minima:
relative minima:
relative minima:
relative minima:
4.3 Increasing and Decreasing Functions and the First Derivative Test
221
____ 14.
The graph of f is shown in the figure. Sketch a graph of the derivative of f.
a. d.
b. e. The derivative of f does not exist.
222 Chapter 4: Applications of Differentiation c.
____ 15. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
a. d.
223
b. e.
c.
____ 16. The graph of f is shown in the figure. Sketch a graph of the derivative of f.
224 Chapter 4: Applications of Differentiation
a. d.
b. e.
c.
4.3 Increasing and Decreasing Functions and the First Derivative Test
225
____ 17.
The graph of f is shown in the figure. Sketch a graph of the derivative of f.
a. d.
b. e.
226 Chapter 4: Applications of Differentiation
c.
____ 18.
A ball bearing is placed on an inclined plane and begins to roll. The angle of
elevation of the plane is
radians. The distance (in meters) the ball bearing rolls in t
seconds is
.
Determine the speed of the ball bearing after t seconds.
a.
speed:
meters per second
b.
speed:
meters per second
c.
speed:
meters per second
d.
speed:
meters per second
e.
speed:
meters per second
____ 19.
A ball bearing is placed on an inclined plane and begins to roll. The angle of
elevation of the plane is
radians. The distance (in meters) the ball bearing rolls in t
seconds is
. Determine the value of
after one second. Round numerical
values in your answer to one decimal place.
a.
b.
c.
d.
e.
____ 20. The resistance R of a certain type of resistor is where R is measured in ohms and the
temperature T is measured in degrees Celsius. Use a computer algebra system to find the critical
number of the function. Round numerical values in your answer to the nearest whole number.
a.
b.
c.
d.
e.
4.3 Increasing and Decreasing Functions and the First Derivative Test
227
____ 21.
The resistance R of a certain type of resistor is
where R is
measured in ohms and the temperature T is measured in degrees Celsius. Use a computer
algebra system to find
a.
b.
c.
d.
e.
228 Chapter 4: Applications of Differentiation
4.3 Increasing and Decreasing Functions and the First Derivative Test
Answer Section
4.3 Increasing and Decreasing Functions and the First Derivative Test
229
4.4 Concavity and the Second Derivative Test
230
4.4 Concavity and the Second Derivative Test
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____
1.
Determine the open intervals on which the graph of
is concave
downward or concave upward.
a.
concave downward on
b.
concave upward on
; concave downward on
c.
concave upward on
; concave downward on
d.
concave upward on
e.
concave downward on
; concave upward on
____
2.
Determine the open intervals on which the graph of
is
concave downward or concave upward.
a.
concave downward on
b.
concave upward on
; concave downward on
c.
concave upward on
; concave downward on
d.
concave downward on
; concave upward on
e.
concave downward on
; concave upward on
____
3.
Determine the open intervals on which the graph of the function
is
concave upward or concave downward.
a.
concave upward:
; concave downward:
b.
concave upward:
; concave downward:
c.
concave upward:
; concave downward:
d.
concave upward:
; concave downward:
e.
concave upward:
; concave downward:
231
Chapter 4: Applications of Differentiation
____ 4.
Determine the open intervals on which the graph of the function
is concave upward or concave downward.
a.
concave
upward:
; concave downward:
b.
concave
upward:
; concave downward:
c.
concave
upward:
; concave downward:
d.
concave
upward:
; concave downward:
e.
concave
upward:
; concave downward:
____ 5.
Determine the open intervals on which the graph of
is concave
downward or concave upward.
a.
concave downward on
; concave upward on
b.
concave downward on
; concave upward on
c. concave upward on ; concave downward on
d. concave downward on ; concave upward on
e. concave upward on ; concave downward on
4.4 Concavity and the Second Derivative Test
232
____
6.
Find all points of inflection on the graph of the function
.
a.
b.
c.
d.
e.
____
7.
Find the points of inflection and discuss the concavity of the function.
inflection point at ; concave upward on ; concave downward on
inflection point at ; concare downward on ; concave upward on
inflection point at ; concave downward on ; concave upward on
inflection point at ; concave downward on ; concave upward on
inflection point at ; concave upward on ; concave downward on
233
Chapter 4: Applications of Differentiation
____
8.
Find the points of inflection and discuss the concavity of the function
.
no inflection points; concave up on
no inflection points; concave down on
inflection point at x = 16; concave up on
d.
inflection point at x = 0; concave up on
; concave down on
e.
inflection point at x = 16; concave down on
____ 9.
Find all points of inflection, if any exist, of the graph of the function
. Round your answers to two decimal places.
a.
b.
c.
d.
e.
no points of inflection
____ 10. Find the point of inflection of the graph of the function on the interval
.
a.
b.
c.
d.
e.
4.4 Concavity and the Second Derivative Test
234
____ 11.
Find the points of inflection and discuss the concavity of the function
on the interval
.
no inflection points. concave up on
concave upward on ; concave downward on ; inflection point at
no inflection points. concave down on
concave downward on ; concave upward on ; inflection point at
none of the above
____
12.
Find all points of inflection of the graph of the function
on
the interval
. Round your answer to three decimal places wherever applicable.
a.
b.
c.
d.
e.
____
13.
Find the points of inflection and discuss the concavity of the function
on the interval
.
a.
concave down on
; no points of inflection
b.
concave downward on
; concave upward on
;
inflection points at
and
c.
concave upward on
;
concave downward on
;
inflection points at
and
d.
concave up on
; no points of inflection
e.
none of the above
235
Chapter 4: Applications of Differentiation
____
14.
Find all points of inflection on the graph of the function
.
a.
b.
c.
d.
e.
____ 15. Find all relative extrema of the function . Use the Second
Derivative Test where applicable.
relative max: ; no relative min
no relative max; no relative min
c.
relative min:
; relative max:
d.
relative min:
; no relative max
e.
relative min:
; relative max:
____ 16.
Find all relative extrema of the function
. Use the Second
Derivative Test where applicable.
a.
relative max:
; no relative min
b.
relative min:
; no relative max
c.
relative min:
; no relative max
d.
relative max:
; no relative min
e. no relative max or min
____ 17. Find all relative extrema of the function . Use the Second
Derivative Test where applicable.
relative minimum:
relative minimum:
relative maximum:
relative minimum:
relative maximum:
4.4 Concavity and the Second Derivative Test
236
____ 18.
Find all relative extrema of the function
. Use the Second Derivative
Test where applicable.
relative max: (1, –2); no relative min
no relative max or min
relative min: (0, –3); no relative max
relative max: (1, –2); relative min: (0, –3)
relative max: (0, –3); no relative min
____
19.
Locate any relative extrema and inflection points of the function
. Use
a graphing utility to confirm your results.
a.
relative minimum value:
; no inflection points
b.
relative minimum value:
inflection point:
c.
relative maximum value:
inflection point:
d.
relative minimum value:
no inflection points
e.
relative maximum value:
; no inflection points
____
20.
Determine the x-coordinate(s) of any relative extrema and inflection points of the
function
.
a.
b.
relative minimum:
; inflection point:
c.
no relative extrema; inflection point:
relative minimum:
; no inflection points
d.
e.
no relative extrema; inflection point:
relative maximum:
inflection point:
____
21.
Determine the x-coordinate(s) of any relative extrema and inflection points of the
function
.
a.
relative minimum:
; no inflection points
237 Chapter 4: Applications of Differentiation
b.
relative maximum: ; inflection point:
c.
no relative maximum or minimum; inflection point:
d. no relative extrema or inflection points.
e.
relative minimum: ; inflection point:
____ 22. The graph of f is shown. Graph f, f’ and f” on the same set of coordinate axes.
a. d.
4.4 Concavity and the Second Derivative Test
238
b.
e. none of the above
c.
____ 23. The graph of f is shown. Graph f, f’ and f” on the same set of coordinate axes.
239 Chapter 4: Applications of Differentiation
a. d.
b. e. none of the above
c.