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4.4 Concavity and the Second Derivative Test
240
____ 24.
Find the cubic function of the form
where
and the
coefficients are real numbers, which satisfies the conditions given below.
Relative maximum:
Relative minimum:
Inflection point:
a.
b.
c.
d.
e.
____ 25. Suppose the deflection D of a beam of length L is where x is the distance from one end of
the beam. Find the value of x that yields the maximum deflection.
a.
b.
c.
d.
e.
____ 26. Suppose a manufacturer has determined that the total cost C of operating a factory is
where x is the number of units produced. At what level of production will
the average cost per unit be minimized? (The average cost per unit is C/x.)
a.
b.
c.
d.
e.
241 Chapter 4: Applications of Differentiation
4.4 Concavity and the Second Derivative Test
Answer Section
4.4 Concavity and the Second Derivative Test 242
243 Chapter 4: Applications of Differentiation
4.5 Limits at Infinity
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. Match the function with one of the following graphs.
a. d.
b. e.
c.
4.5 Limits at Infinity
244
____ 2.
Match the function
with one of the following graphs.
a.
d.
b. e.
c.
245
Chapter 4: Applications of Differentiation
____
3.
Match the function
with one of the following graphs.
a.
d.
b. e.
c.
4.5 Limits at Infinity
246
____ 4.
For the function
, use a graphing utility to complete the table and
estimate the limit as x approaches infinity.
0.8000
1.8000
1.2500
2.2500
0.2500
____ 5. For the function , use a graphing utility to complete the table and
estimate the limit as x approaches infinity.
–0.2500
1.0000
2.1800
–0.9000
does not exist
247 Chapter 4: Applications of Differentiation
____ 6. For the function , use a graphing utility to complete the table
and estimate the limit as x approaches infinity.
–6
2
6
0
None of the above
____ 7. Find the limit.
3
–3
5
____ 8. Find the limit.
2
0
4.5 Limits at Infinity
248
____ 9. Find the limit.
1
0
does not exist
____ 10. Find the limit.
1
0
____ 11. Find the limit.
1
–6
249 Chapter 4: Applications of Differentiation
____ 12. Find the limit.
a.
b.
c.
1
d.
7
e.
____ 13.
Sketch the graph of the function
using any extrema, intercepts,
symmetry, and asymptotes.
a.
d.
b. e.
4.5 Limits at Infinity
250
c.
____ 14. Sketch the graph of the function using any extrema, intercepts,
symmetry, and asymptotes.
a. d.
b. e.
251 Chapter 4: Applications of Differentiation
c.
____ 15. Sketch the graph of the relation using any extrema, intercepts, symmetry,
and asymptotes.
a. d.
b. e.
4.5 Limits at Infinity
252
c.
____ 16. Sketch the graph of the relation using any extrema, intercepts, symmetry,
and asymptotes.
a. d.
b. e.
4.5 Limits at Infinity
254
d.
e.
____ 19. A heat probe is attached to the heat exchanger of a heating system. The temperature T (in
degrees Celsius) is recorded t seconds after the furnace is started. A model for the data recorded
for the first two minutes is given by . Find .
a.
b.
c.
d.
e.
____ 20. A container holds 3 liters of a 25% brine solution. A model for the concentration C of the
mixture after adding x liters of a 0.67 % brine solution to the container and then draining x liters
of the well-mixed solution is given as . Find . Round your answer to two decimal
places.
1.00 %
0.17 %
0.70 %
0.67 %
0.25 %
____ 21. Find the point on the graph of the function that is closest to the point
. Round all numerical values in your answer to four decimal places.
a.
b.
c.
d.
e.
255 Chapter 4: Applications of Differentiation
____ 22. Find the point on the graph of the function that is closest to the point
.
a.
b.
c.
d.
e.
____ 23. A rectangular page is to contain square inches of print. The margins on each side are 1
inch. Find the dimensions of the page such that the least amount of paper is used.
a.
b.
c.
d.
e.
4.5 Limits at Infinity
256
4.5 Limits at Infinity
Answer Section
257 Chapter 4: Applications of Differentiation
4.6 A Summary of Curve Sketching
258
4.6 A Summary of Curve Sketching
Multiple Choice
Identify the choice that best completes the statement or answers the question.
____ 1. The graph of a function f is is shown below.
Sketch the graph of the derivative .
a. d.
b. e.
259 Chapter 4: Applications of Differentiation
c.
____ 2. The graph of a function f is is shown below. Sketch the graph of the derivative .
a. d.
4.6 A Summary of Curve Sketching
260
b. e.
c.
