Lesson 20: Derivatives and the Shapes of Curves (handout)
Lesson 20: Optimization (slides)
1. Section 4.5
Optimization Problems
V63.0121.006/016, Calculus I
New York University
April 6, 2010
Announcements
Thank you for the evaluations
Quiz 4 April 16 on §§4.1–4.4
. . . . . .
2. Announcements
Thank you for the evaluations
Quiz 4 April 16 on §§4.1–4.4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 2 / 36
3. Evaluations: The good
“Very knowledgeable”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
4. Evaluations: The good
“Very knowledgeable”
“Knows how to teach”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
5. Evaluations: The good
“Very knowledgeable”
“Knows how to teach”
“Very good at projecting voice”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
6. Evaluations: The good
“Very knowledgeable”
“Knows how to teach”
“Very good at projecting voice”
“Office hours are accessible”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
7. Evaluations: The good
“Very knowledgeable”
“Knows how to teach”
“Very good at projecting voice”
“Office hours are accessible”
“Clean”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
8. Evaluations: The good
“Very knowledgeable”
“Knows how to teach”
“Very good at projecting voice”
“Office hours are accessible”
“Clean”
“Great syllabus”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 3 / 36
9. Evaluations: The bad
Too fast, not enough examples
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 4 / 36
10. Evaluations: The bad
Too fast, not enough examples
Not enough time to do everything
Lecture is not the only learning time (recitation and independent
study)
I try to balance concept and procedure
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 4 / 36
11. Evaluations: The bad
Too fast, not enough examples
Not enough time to do everything
Lecture is not the only learning time (recitation and independent
study)
I try to balance concept and procedure
Too many proofs
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 4 / 36
12. Evaluations: The bad
Too fast, not enough examples
Not enough time to do everything
Lecture is not the only learning time (recitation and independent
study)
I try to balance concept and procedure
Too many proofs
In this course we care about concepts
There will be conceptual problems on the exam
Concepts are the keys to overcoming templated problems
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 4 / 36
13. Evaluations: The ugly
“The projector blows.”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 5 / 36
14. Evaluations: The ugly
“The projector blows.”
“Sometimes condescending/rude.”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 5 / 36
15. Evaluations: The ugly
“The projector blows.”
“Sometimes condescending/rude.”
“Can’t pick his nose without checking his notes, and he still gets it
wrong the first time.”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 5 / 36
16. Evaluations: The ugly
“The projector blows.”
“Sometimes condescending/rude.”
“Can’t pick his nose without checking his notes, and he still gets it
wrong the first time.”
“If I were chained to a desk and forced to see this guy teach, I
would chew my arm off in order to get free.”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 5 / 36
17. A slide on slides
Pro
“Excellent slides and examples”
“clear and well-rehearsed”
“Slides are easy to follow and posted”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 6 / 36
18. A slide on slides
Pro
“Excellent slides and examples”
“clear and well-rehearsed”
“Slides are easy to follow and posted”
Con
“I wish he would actually use the chalkboard occasionally”
“Sometimes the slides skip steps”
“too fast”
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 6 / 36
19. A slide on slides
Pro
“Excellent slides and examples”
“clear and well-rehearsed”
“Slides are easy to follow and posted”
Con
“I wish he would actually use the chalkboard occasionally”
“Sometimes the slides skip steps”
“too fast”
Why I like them
Board handwriting not an issue
Easy to put online; notetaking is more than transcription
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 6 / 36
20. My handwriting
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 7 / 36
21. A slide on slides
Pro
“Excellent slides and examples”
“clear and well-rehearsed”
“Slides are easy to follow and posted”
Con
“I wish he would actually use the chalkboard occasionally”
“Sometimes the slides skip steps”
“too fast”
Why I like them
Board handwriting not an issue
Easy to put online; notetaking is more than transcription
What we can do
if you have suggestions for details to put in, I’m listening
Feel free to ask me to fill in something on the board
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 8 / 36
22. Outline
Leading by Example
The Text in the Box
More Examples
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 9 / 36
23. Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 10 / 36
24. Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
.
. . . . . .
25. Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
.
.
ℓ
. . . . . .
26. Leading by Example
Example
What is the rectangle of fixed perimeter with maximum area?
Solution
Draw a rectangle.
w
.
.
.
ℓ
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 10 / 36
27. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
28. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
29. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
p − 2w
Since p = 2ℓ + 2w, we have ℓ = ,
2
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
30. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
p − 2w
Since p = 2ℓ + 2w, we have ℓ = , so
2
p − 2w 1 1
A = ℓw = · w = (p − 2w)(w) = pw − w2
2 2 2
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
31. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
p − 2w
Since p = 2ℓ + 2w, we have ℓ = , so
2
p − 2w 1 1
A = ℓw = · w = (p − 2w)(w) = pw − w2
2 2 2
Now we have A as a function of w alone (p is constant).
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
32. Solution Continued
Let its length be ℓ and its width be w. The objective function is
area A = ℓw.
This is a function of two variables, not one. But the perimeter is
fixed.
p − 2w
Since p = 2ℓ + 2w, we have ℓ = , so
2
p − 2w 1 1
A = ℓw = · w = (p − 2w)(w) = pw − w2
2 2 2
Now we have A as a function of w alone (p is constant).
The natural domain of this function is [0, p/2] (we want to make
sure A(w) ≥ 0).
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 11 / 36
33. Solution Concluded
1
We use the Closed Interval Method for A(w) = pw − w2 on [0, p/2].
2
At the endpoints, A(0) = A(p/2) = 0.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 12 / 36
34. Solution Concluded
1
We use the Closed Interval Method for A(w) = pw − w2 on [0, p/2].
2
At the endpoints, A(0) = A(p/2) = 0.
dA 1
To find the critical points, we find = p − 2w.
dw 2
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 12 / 36
35. Solution Concluded
1
We use the Closed Interval Method for A(w) = pw − w2 on [0, p/2].
2
At the endpoints, A(0) = A(p/2) = 0.
dA 1
To find the critical points, we find = p − 2w.
dw 2
The critical points are when
1 p
0= p − 2w =⇒ w =
2 4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 12 / 36
36. Solution Concluded
1
We use the Closed Interval Method for A(w) = pw − w2 on [0, p/2].
2
At the endpoints, A(0) = A(p/2) = 0.
dA 1
To find the critical points, we find = p − 2w.
dw 2
The critical points are when
1 p
0= p − 2w =⇒ w =
2 4
Since this is the only critical point, it must be the maximum. In this
p
case ℓ = as well.
4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 12 / 36
37. Solution Concluded
1
We use the Closed Interval Method for A(w) = pw − w2 on [0, p/2].
2
At the endpoints, A(0) = A(p/2) = 0.
dA 1
To find the critical points, we find = p − 2w.
dw 2
The critical points are when
1 p
0= p − 2w =⇒ w =
2 4
Since this is the only critical point, it must be the maximum. In this
p
case ℓ = as well.
4
We have a square! The maximal area is A(p/4) = p2 /16.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 12 / 36
38. Outline
Leading by Example
The Text in the Box
More Examples
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 13 / 36
39. Strategies for Problem Solving
1. Understand the problem
2. Devise a plan
3. Carry out the plan
4. Review and extend
György Pólya
(Hungarian, 1887–1985)
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 14 / 36
40. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
41. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
42. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
43. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
44. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
5. If Q is a function of more than one “decision variable”, use the
given information to eliminate all but one of them.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
45. The Text in the Box
1. Understand the Problem. What is known? What is unknown?
What are the conditions?
2. Draw a diagram.
3. Introduce Notation.
4. Express the “objective function” Q in terms of the other symbols
5. If Q is a function of more than one “decision variable”, use the
given information to eliminate all but one of them.
6. Find the absolute maximum (or minimum, depending on the
problem) of the function on its domain.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 15 / 36
46. Name [_
Problem Solving Strategy
Draw a Picture
Kathy had a box of 8 crayons.
She gave some crayons away.
She has 5 left.
How many crayons did Kathy give away?
UNDERSTAND
•
What do you want to find out?
Draw a line under the question.
You can draw a picture
to solve the problem.
What number do I
add to 5 to get 8?
8 - = 5
crayons 5 + 3 = 8
CHECK
Does your answer make sense?
Explain.
What number
Draw a picture to solve the problem. do I add to 3
Write how many were given away. to make 10?
I. I had 10 pencils. ft ft ft A
I gave some away. 13 ill
i :i
I
'•' I I
I have 3 left. How many i? «
11 I
pencils did I give away? I
H 11
M i l
~7 U U U U> U U
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 16 / 36
47. Recall: The Closed Interval Method
See Section 4.1
To find the extreme values of a function f on [a, b], we need to:
Evaluate f at the endpoints a and b
Evaluate f at the critical points x where either f′ (x) = 0 or f is not
differentiable at x.
The points with the largest function value are the global maximum
points
The points with the smallest or most negative function value are
the global minimum points.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 17 / 36
48. Recall: The First Derivative Test
See Section 4.3
Theorem (The First Derivative Test)
Let f be continuous on [a, b] and c a critical point of f in (a, b).
If f′ changes from negative to positive at c, then c is a local
minimum.
If f′ changes from positive to negative at c, then c is a local
maximum.
If f′ does not change sign at c, then c is not a local extremum.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 18 / 36
49. Recall: The Second Derivative Test
See Section 4.3
Theorem (The Second Derivative Test)
Let f, f′ , and f′′ be continuous on [a, b]. Let c be be a point in (a, b) with
f′ (c) = 0.
If f′′ (c) < 0, then f(c) is a local maximum.
If f′′ (c) > 0, then f(c) is a local minimum.
Warning
If f′′ (c) = 0, the second derivative test is inconclusive (this does not
mean c is neither; we just don’t know yet).
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 19 / 36
50. Which to use when?
CIM 1DT 2DT
Pro – no need for – works on – works on
inequalities non-closed, non-closed,
– gets global non-bounded non-bounded
extrema intervals intervals
automatically – only one derivative – no need for
inequalities
Con – only for closed – Uses inequalities – More derivatives
bounded intervals – More work at – less conclusive
boundary than CIM than 1DT
– more work at
boundary than CIM
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 20 / 36
51. Which to use when?
CIM 1DT 2DT
Pro – no need for – works on – works on
inequalities non-closed, non-closed,
– gets global non-bounded non-bounded
extrema intervals intervals
automatically – only one derivative – no need for
inequalities
Con – only for closed – Uses inequalities – More derivatives
bounded intervals – More work at – less conclusive
boundary than CIM than 1DT
– more work at
boundary than CIM
Use CIM if it applies: the domain is a closed, bounded interval
If domain is not closed or not bounded, use 2DT if you like to take
derivatives, or 1DT if you like to compare signs.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 20 / 36
52. Outline
Leading by Example
The Text in the Box
More Examples
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 21 / 36
53. Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 22 / 36
55. Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 24 / 36
56. Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
Known: amount of fence used
Unknown: area enclosed
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 24 / 36
57. Another Example
Example (The Best Fencing Plan)
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With
800m of wire at your disposal, what is the largest area you can
enclose, and what are its dimensions?
Known: amount of fence used
Unknown: area enclosed
Objective: maximize area
Constraint: fixed fence length
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 24 / 36
59. Solution
1. Everybody understand?
2. Draw a diagram.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 25 / 36
60. Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
. .
.
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 26 / 36
61. Solution
1. Everybody understand?
2. Draw a diagram.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 27 / 36
62. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 27 / 36
63. Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
. .
.
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 28 / 36
64. Diagram
A rectangular plot of farmland will be bounded on one side by a river
and on the other three sides by a single-strand electric fence. With 800
m of wire at your disposal, what is the largest area you can enclose,
and what are its dimensions?
.
ℓ
w
.
. .
.
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 28 / 36
65. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
66. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
67. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
68. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
dQ p
6. = p − 4w, which is zero when w = .
dw 4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
69. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
dQ p
6. = p − 4w, which is zero when w = .
dw 4
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
70. Solution
1. Everybody understand?
2. Draw a diagram.
3. Length and width are ℓ and w. Length of wire used is p.
4. Q = area = ℓw.
5. Since p = ℓ + 2w, we have ℓ = p − 2w and so
Q(w) = (p − 2w)(w) = pw − 2w2
The domain of Q is [0, p/2]
dQ p
6. = p − 4w, which is zero when w = . Q(0) = Q(p/2) = 0, but
dw 4
(p) p p2 p2
Q =p· −2· = = 80, 000m2
4 4 16 8
so the critical point is the absolute maximum.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 29 / 36
71. Your turn
Example (The shortest fence)
A 216m2 rectangular pea patch is to be enclosed by a fence and
divided into two equal parts by another fence parallel to one of its
sides. What dimensions for the outer rectangle will require the smallest
total length of fence? How much fence will be needed?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 30 / 36
72. Your turn
Example (The shortest fence)
A 216m2 rectangular pea patch is to be enclosed by a fence and
divided into two equal parts by another fence parallel to one of its
sides. What dimensions for the outer rectangle will require the smallest
total length of fence? How much fence will be needed?
Solution
Let the length and width of the pea patch be ℓ and w. The amount of
fence needed is f = 2ℓ + 3w. Since ℓw = A, a constant, we have
A
f(w) = 2 + 3w.
w
The domain is all positive numbers.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 30 / 36
73. Diagram
. .
w
.
.
.
ℓ
f = 2ℓ + 3w A = ℓw ≡ 216
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 31 / 36
74. Solution (Continued)
2A
We need to find the minimum value of f(w) = + 3w on (0, ∞).
w
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 32 / 36
75. Solution (Continued)
2A
We need to find the minimum value of f(w) = + 3w on (0, ∞).
w
We have
df 2A
=− 2 +3
dw w
√
2A
which is zero when w = .
3
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 32 / 36
76. Solution (Continued)
2A
We need to find the minimum value of f(w) = + 3w on (0, ∞).
w
We have
df 2A
=− 2 +3
dw w
√
2A
which is zero when w = .
3
Since f′′ (w) = 4Aw−3 , which is positive for all positive w, the
critical point is a minimum, in fact the global minimum.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 32 / 36
77. Solution (Continued)
2A
We need to find the minimum value of f(w) = + 3w on (0, ∞).
w
We have
df 2A
=− 2 +3
dw w
√
2A
which is zero when w = .
3
Since f′′ (w) = 4Aw−3 , which is positive for all positive w, the
critical point is a minimum, in fact the global minimum.
√
2A
So the area is minimized when w = = 12 and
√ 3
A 3A
ℓ= = = 18. The amount of fence needed is
w 2
(√ ) √ √
2A 3A 2A √ √
f =2· +3 = 2 6A = 2 6 · 216 = 72m
3 2 3
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 32 / 36
78. Try this one
Example
An advertisement consists of a rectangular printed region plus 1 in
margins on the sides and 1.5 in margins on the top and bottom. If the
total area of the advertisement is to be 120 in2 , what dimensions should
the advertisement be to maximize the area of the printed region?
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 33 / 36
79. Try this one
Example
An advertisement consists of a rectangular printed region plus 1 in
margins on the sides and 1.5 in margins on the top and bottom. If the
total area of the advertisement is to be 120 in2 , what dimensions should
the advertisement be to maximize the area of the printed region?
Answer
√ √
The optimal paper dimensions are 4 5 in by 6 5 in.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 33 / 36
80. Solution
Let the dimensions of the
printed region be x and y, P 1
. .5 cm
the printed area, and A the .
Lorem ipsum dolor sit amet,
consectetur adipiscing elit. Nam
paper area. We wish to dapibus vehicula mollis. Proin nec
tristique mi. Pellentesque quis
maximize P = xy subject to placerat dolor. Praesent a nisl diam.
the constraint that Phasellus ut elit eu ligula accumsan
euismod. Nunc condimentum
lacinia risus a sodales. Morbi nunc
risus, tincidunt in tristique sit amet,
A = (x + 2)(y + 3) ≡ 120
. cm
. cm
y
. ultrices eu eros. Proin pellentesque
aliquam nibh ut lobortis. Ut et
1
1
sollicitudin ipsum. Proin gravida
Isolating y in A ≡ 120 gives ligula eget odio molestie rhoncus
sed nec massa. In ante lorem,
120 imperdiet eget tincidunt at, pharetra
y= − 3 which yields sit amet felis. Nunc nisi velit,
x+2 tempus ac suscipit quis, blandit
vitae mauris. Vestibulum ante ipsum
( ) primis in faucibus orci luctus et
120 120x . ultrices posuere cubilia Curae;
P=x −3 = −3x
x+2 x+2 1
. .5 cm
The domain of P is (0, ∞) x
.
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 34 / 36
81. Solution (Concluded)
We want to find the absolute maximum value of P. Taking derivatives,
dP (x + 2)(120) − (120x)(1) 240 − 3(x + 2)2
= −3=
dx (x + 2)2 (x + 2)2
There is a single critical point when
√
(x + 2)2 = 80 =⇒ x = 4 5 − 2
(the negative critical point doesn’t count). The second derivative is
d2 P −480
2
=
dx (x + 2)3
which is negative all along the domain of P. Hence the unique critical
( √ )
point x = 4 5 − 2 cm is the absolute maximum of P. This means
√ 120 √
the paper width is 4 5 cm, and the paper length is √ = 6 5 cm.
4 5 . . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 35 / 36
82. Summary
Remember the checklist
Ask yourself: what is the objective?
Remember your geometry:
similar triangles
right triangles
trigonometric functions
. . . . . .
V63.0121, Calculus I (NYU) Section 4.5 Optimization Problems April 6, 2010 36 / 36