Lesson 22: Optimization II (Section 041 handout)

Section 4.5
Optimization II
V63.0121.041, Calculus I
New York University
November 24, 2010
Announcements
No recitation this week
Quiz 4 on §§4.1–4.4 next week in recitation
Happy Thanksgiving!
Announcements
No recitation this week
Quiz 4 on §§4.1–4.4 next
week in recitation
Happy Thanksgiving!
V63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 2 / 25
Objectives
Given a problem requiring
optimization, identify the
objective functions,
variables, and constraints.
Solve optimization problems
with calculus.
Notes
Notes
Notes
1
Section 4.5 : Optimization IIV63.0121.041, Calculus I November 24, 2010

Outline
Recall
More examples
Addition
Distance
Triangles
Economics
The Statue of Liberty
Checklist for optimization problems
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.
Recall: The Closed Interval Method
See Section 4.1
The Closed Interval Method
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/most negative function value are the
global minimum points.
Notes
Notes
Notes
2

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.
Corollary
If f < 0 for all x < c and f (x) > 0 for all x > c, then c is the global
minimum of f on (a, b).
If f < 0 for all x > c and f (x) > 0 for all x < c, then c is the global
maximum of f on (a, b).
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 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).
Corollary
If f (c) = 0 and f (x) > 0 for all x, then c is the global minimum of f
If f (c) = 0 and f (x) < 0 for all x, then c is the global maximum of
fV63.0121.041, Calculus I (NYU) Section 4.5 Optimization II November 24, 2010 8 / 25
Which to use when?
CIM 1DT 2DT
Pro – no need for
inequalities
– gets global extrema
automatically
– works on
non-closed,
non-bounded
intervals
– only one derivative
– works on
non-closed,
non-bounded
intervals
– no need for
inequalities
Con – only for closed
bounded intervals
– Uses inequalities
– More work at
boundary than CIM
– More derivatives
– less conclusive 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.
Notes
Notes
Notes
3

Outline
Recall
More examples
Addition
Distance
Triangles
Economics
Addition with a constraint
Example
Find two positive numbers x and y with xy = 16 and x + y as small as
possible.
Solution
Distance
Example
Find the point P on the parabola y = x2
closest to the point (3, 0).
Notes
Notes
Notes
4

Distance problem
minimization step
Remark
We’ve used each of the methods (CIM, 1DT, 2DT) so far.
Notice how we argued that the critical points were absolute extremes
even though 1DT and 2DT only tell you relative/local extremes.
A problem with a triangle
Example
Find the rectangle of maximal area inscribed in a 3-4-5 right triangle with two sides
on legs of the triangle and one vertex on the hypotenuse.
Solution
3
4
5
y
x
Notes
Notes
Notes
5

Triangle Problem
maximization step
An Economics problem
Example
Let r be the monthly rent per unit in an apartment building with 100
units. A survey reveals that all units can be rented when r = 900 and that
one unit becomes vacant with each 10 increase in rent. Suppose the
average monthly maintenance costs per occupied unit is $100/month.
What rent should be charged to maximize proﬁt?
Solution
Economics Problem
Finishing the model and maximizing
Notes
Notes
Notes
6

Example
The Statue of Liberty stands on top of a pedestal which is on top of on
old fort. The top of the pedestal is 47 m above ground level. The statue
itself measures 46 m from the top of the pedestal to the tip of the torch.
What distance should one stand away from the statue in order to
maximize the view of the statue? That is, what distance will maximize the
portion of the viewer’s vision taken up by the statue?
Seting up the model
Solution
Finding the derivative
Notes
Notes
Notes
7

Finding the critical points
Final answer
Discussion
The length b(a + b) is the geometric mean of the two distances
measured from the ground—to the top of the pedestal (a) and the
top of the statue (a + b).
The geometric mean is of two numbers is always between them and
greater than or equal to their average.
Notes
Notes
Notes
8

Summary
Remember the checklist
Ask yourself: what is the
objective?
Remember your geometry:
similar triangles
right triangles
trigonometric functions
Notes
Notes
Notes
9

Lesson 22: Optimization II (Section 041 handout)

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Lesson 22: Optimization II (Section 041 handout)