Pg 334

Chapter 4 Review
USING THE DERIVATIVE TO ANALYZE
POLYNOMIAL FUNCTION MODELS
CHECK YOUR UNDERSTANDING
1. (a) What does f ′(x) > 0 imply about f (x)?
(b) What does f ′(x) < 0 imply about f (x)?
2. Explain critical numbers and the role they play in identifying maximum and
minimum values.
3. Draw a diagram of a polynomial function on an interval with an absolute
maximum, an absolute minimum, a local maximum, and a local minimum.
4. Describe the first derivative test. Explain how to use it.
5. State the product rule and give an example.
6. What is an inflection point? How do you identify possible points of
inflection?
7. What can you say about the concavity of the graph of f (x) if
(a) f ′′(x) > 0 on an interval? (b) f ′′(x) < 0 on an interval?
8. Describe the second derivative test. Explain how to use it.
9. What is meant by optimization? Outline a method for solving optimization
problems.
10. What is the relation among cost, revenue, and profit? In a business
application, what does marginal revenue measure? When is a company’s
profit maximized?
ADDITIONAL REVIEW QUESTIONS BY SECTION
4.1 Analyzing a Polynomial Function: Intervals of Increase
and Decrease
11. Determine the intervals where each function increases and decreases.
(a) y 3x 2 12x 7 (b) y 4x 3 12x 2 8
(c) y 10x 4 8x 3 (d) y x 4 20x 2 64
(e) y 2x 5 20x 3 50x (f) y x 4 6x 3 5x 2
12. A diver dives from the 3-m springboard. Her height above the water in
metres at t seconds is h(t) 4.9t 2 9.5t 2.2.
(a) When is the height of the diver increasing? decreasing?
(b) When is the upward velocity of the diver increasing? decreasing?
334 CHAPTER 4 USING THE DERIVATIVE TO ANALYZE POLYNOMIAL FUNCTION MODELS

13. The concentration, C, of a drug injected into the bloodstream t hours
t
after injection can be modelled by C(t) 4 2t 2 . Determine when the
concentration of the drug is increasing and when it is decreasing.
4.2 Maximum and Minimum Values of a Polynomial
Function
14. Find the absolute maximum and minimum values.
(a) f (x) x 2 2x 6, 1 ≤ x ≤ 7
(b) f (x) x 3 x 2 , 3 ≤ x ≤ 3
(c) f (x) x 3 12x 2, 5 ≤ x ≤ 5
(d) f (x) 3x 5 5x 3 , 2 ≤ x ≤ 4
(e) f (x) 2x 3 3x 2 12x, 2 ≤ x ≤ 2
(f) f (x) x 4 18x 2 , 4 ≤ x ≤ 4
15. After a football is punted, its height, h, in metres above the ground at
t seconds can be modelled by h(t) 4.9t 2 21t 0.45.
(a) Determine the restricted domain of this model.
(b) When does the ball reach its maximum height?
(c) What is the ball’s maximum height?
16. Determine the equation of the line tangent to f (x) 4x 3 12x 2 96x
with the smallest slope on the interval 4 ≤ x ≤ 2.
4.3 The First Derivative Test
17. Graph y f ′(x) for the given function.
10
8
6
4
2
–4 –3 –2 –1 0
–2
–4
–6
–8
–10
y
1 2 3 4
y = f(x)
x
18. For each function f (x),
i. find the critical numbers
ii. determine where the function increases and decreases
iii. determine whether each critical number is at a maximum, a minimum,
or neither
iv. use all the information to sketch the graph
CHAPTER 4 REVIEW 335

(a) f (x) x 2 7x 18
(b) f (x) 2x 3 9x 2 3
(c) f (x) 2x 4 4x 2 2
(d) f (x) x 5 5x
(e) f (x) x 4 8x 3 18x 2 6
(f) f (x) 3x 5 5x 4
19. For groups of 60 or more, a charter bus company computes total revenue
using R(x) x[10 0.1(x 60)], x ≥ 60. What size of group will
maximize revenue?
4.4 Finding Some Shortcuts — The Product Rule
20. Find the derivative using the product rule.
(a) f (x) (4x 2 9x)(3x 2 5)
(b) f (t) (3t 2 7t 8)(4t 1)
(c) f (x) (5x 2 3x) 2
(d) f (x) 2x 4 (3x 2 3x)(2x 2 3x)
(e) f (x) 4x 3 (4x 2x 2 )
(f) f (z) z(2z 3 5z)
21. Find the equation of the tangent to y (5x 2 9x 2)(x 2 2x 3) at
(1, 48).
22. For f (x) (5x 3)(x 2 x 1), determine the critical numbers, the
intervals of increase or decrease, and the extrema. Use this information to
graph the function.
4.5 Finding Optimal Values for Polynomial Function
Models
23. A park ranger has 600 m of floating rope. She is going to enclose a
rectangular swimming area, using the beach as one border of the area.
Find the maximum area that can be enclosed and the corresponding
dimensions.
24. A rectangular box is to be constructed from two different materials. The box
will have a square base and open top. The material for the bottom costs
$4.25/m 2 . The material for the sides costs $2.50/m 2 . Find the dimensions of
the box with the largest volume if the budget is $500 for the material.
25. The landlord of a 50-unit apartment building is planning to increase the
rent. Currently, residents pay $850/month and all units are occupied.
A real estate agency advises that every $100 increase in rent will result in
10 vacant units. What rent should the landlord charge to maximize revenue?
336 CHAPTER 4 USING THE DERIVATIVE TO ANALYZE POLYNOMIAL FUNCTION MODELS

4.6 Rates of Change in Business and Economics
26. A soft-drink company estimates that the cost, C, in dollars of producing x
litres of product per day is C(x) 0.0005x 2 4x 4000.
(a) Find the marginal cost if 5000 L is produced.
(b) Determine the daily production level to minimize the average cost.
27. A publisher can sell x thousand copies of a monthly sports magazine if it
x
sells for p 5 dollars. The monthly cost of publishing, C, can be
10 0
modelled by C(x) 8000 200x 0.05x 2 .
(a) Determine the revenue function.
(b) Determine the profit function.
(c) Calculate the profit on the sale of 20 000 magazines.
(d) Calculate the marginal profit on the sale of 30 000 magazines.
(e) Calculate the maximum profit and the sales required to realize the
profit.
28. The cost to produce one high-definition TV is $750. The manufacturer
charges $1800 per unit for orders of 50 or less. The price will be reduced
by $15 per unit for larger orders. Determine the size of order that
(a) maximizes revenue
(b) maximizes profit
4.7 Sketching Graphs of Polynomial Functions: Concavity
29. Determine the intervals of concavity and the points of inflection.
(a) y x 3 3x 2 45x (b) y 2x 3 9x 2 108x 200
(c) y x 4 54x 2 250 (d) y x 4 12x 2
30. Determine the intervals where each function increases or decreases.
Also determine where the graph of the function is concave up or
down, the maximum and minimum points, and the points of inflection.
Use the information to graph each function.
(a) y 2x 2 12x 8 (b) y 2x 3 3x 2 12x 3
(c) y 2x 3 3x 2 36x
(d) y x 4 32x
(e) y x 3 x 2 5x 3 (f) y 3x 5 5x 3 60x
31. Use the second derivative test to show that f (x) x 3 2x 2 has a local
maximum at the origin.
REVIEW QUESTIONS BY ACHIEVEMENT
CHART CATEGORIES
Knowledge and Understanding
32. (a) Determine the intervals where f (x) x 4 4x 3 4x 2 1 is increasing
and decreasing.
(b) Identify all maximum and minimum values.
CHAPTER 4 REVIEW 337

33. (a) Find the intervals of concavity and all inflection points for the graph of
f (x) x 3 6x 2 9x 3.
(b) Do you need any more information to graph f ? If so, what would be
this information?
34. Determine the level of production that will maximize profits if the cost,
C, in dollars of producing x units in thousands is C(x) x 2
4x 100
8
and the selling price is p 80 x
.
2
Communication
35. Explain why the following is false: Let c be a critical number of f (x).
Then f (x) has a local maximum or minimum at f (c).
36. f ′(x) < 0 for all x in the interval 3 ≤ x ≤ 10. Explain why f (3) > f (6).
37. For function f (x), f ′(x) is decreasing on an interval. Graph f (x) where
(a) f ′(x) < 0 (b) f ′(x) > 0
Application
38. An object moves horizontally along a line. Its position from a fixed
point can be modelled by s(t) t 3 6t 2 15t 20, where s is the
displacement in centimetres and t is the time in seconds.
Determine the object’s maximum velocity on the interval 0 ≤ t ≤ 5.
39. A cylindrical, open container is to be made to hold 2 L (2000 cm 3 ) of
liquid. The material for the bottom costs $0.05/cm 2 . The material for the
sides costs $0.10/cm 2 . What are the dimensions of the least expensive
container?
40. A toy manufacturer has determined that the total cost, C, of operating a
factory is C(x) 0.5x 2 45x 10 000, where x is the number of units
produced in thousands. Determine the production level that minimizes the
average cost.
Thinking, Inquiry, Problem Solving
41. Graph function f (x) so that
• points (1, 10) and (3, 1) are local extrema on the graph
• (1, 3) is an inflection point
• the graph is concave down only when x < 1
• the x-intercept is 4 and the y-intercept is 8
42. Prove that an nth-degree polynomial, f (x) a 0
x n a 1
x n 1 … a n
,
a 0
≠ 0, has at most n 2 inflection points.
43. Ron is planting a garden all around a rectangular patio. The garden will be
5 m wide, except at the diagonals. The area of the patio must be 150 m 2 .
Find the overall dimensions of the garden and patio in which the area of the
garden is a minimum.
338 CHAPTER 4 USING THE DERIVATIVE TO ANALYZE POLYNOMIAL FUNCTION MODELS