Download Week 11 & 12

Week 11 & 12: Optimization and
Modeling, Trigonometric Functions
Goals:
• Discuss optimizations problems in economics
• Review trigonometric functions
• Study the derivatives of trigonometric functions.
Suggested Readings: Chapter 13: §13.6, and Trigonometric Functions (on-line notes)
• Review: 7, 11, 17, 23, 27, 33, 35*, 49, 55, 61, 63, 67, 69
• Practice questions in the on-line notes - Trigonometric Functions.
Note: the function given in question 35 is different in two editions.
35. (12th) f (x) = (x2 + 1)e−x .
35. (13th) f (x) = x ln x.
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2
Applied Maxima and Minima (continued)
To prepare for this topic, please read section §13.6 in the textbook.
Example 1: (The discussion at the end of §13.6) Recall that the total profit function P is defined as P (x) = R(x) − C(x), where R is the total revenue and C is the total
cost when x units of product produced and sold.
(a) Show that at the level of production x0 that yields the maximum profit for the company, the following two conditions are satisfied:
R0 (x0 ) = C 0 (x0 )
and
R00 (x0 ) < C 00 (x0 )
(b) Interpret the two conditions in part (a) in economic terms and explain why they
make sense.
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Example 2: The marginal revenue and marginal cost for a certain item are graphed
below. Do the following quantities maximize profit for the company?
$/unit
MC
6
MR
-
a
b
q
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Next we look at some examples in which a function is given graphically and the optimum
values are read from a graph.
Optimization without formulae
Example 3: (December Exam 2004) You are starting your own business doing tutoring for high school kids. If you spend x dollars on advertising, the number of hours
H of business you receive is given by the graph below.
H 6
100 r
80
r
60
r
40
r
20
r
r
r
r
r
r
r
r
-
x
100 200 300 400 500 600
Suppose that you change $12 per hour.
(a) What is your profit function? How much should you spend on advertising to maximize your profit?
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(b) Suppose a tax of 10% is applied on your revenue. How much should you spend on
advertising to maximize your profit?
H
6
100 r
80
r
60
r
40
r
20
r
r
r
r
r
r
r
r
-
x
100 200 300 400 500 600
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6
Example 4: (December Exam 2004) Your scheme to raise money for the field trip
is to make scarves in the school colours and sell them to students. You intend to spend
the long weekend producing scarves of a fixed width. Your only cost is the wool at $4 per
meter of scarf length. The question is, how long should you make each scarf to maximize
your profit? The total number of meters you can produce is a fixed number L, so the
longer you make each scarf, the fewer scarves you will get, but the more you can sell
each scarf for. The following graph indicates the relationship between the length x of a
scarf (in meters) and its selling price S (in dollars). Note that no one will buy a scarf
less than 50 cm in length, and scarves that are too long are unwieldy.
v 6
25 q
20 q
15 q
10 q
5
q
q
q
q
q
q
q -
x
10 20 30 40 50
Assume you have no problem to sell all your scarves.
(a) Write an expression for your total profit function.
(b) Find a construction on the graph above which maximizes your profit.
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(c) Suppose you have to pay your school $5 per scarf sold for the use of the school colors,
and you decide to keep your price the same as before. What is your new profit function?
If you wish to maximize your profit, how long should you make each scarf ?
v 6
25 q
20 q
15 q
10 q
5
q
q
q
q
q
q
q -
x
10 20 30 40 50
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Unit Circle and Angles
The unit circle is the circle of radius 1 centered at the origin on the xy-plane. The
equation of the unit circle is x2 + y 2 = 1. An angle is formed when one ray in the
plane is rotated into the other about their common endpoint. The angle generated is a
positive angle if the ray is rotated counterclockwise, and a negative angle if the ray is
rotated clockwise. The size of an angle can be represented in either degrees or radians.
Relationship between Degrees and Radians
◦
180
◦
180 = π radians, 1 radian =
,
π
1◦ =
π
radians
180
π
. To convert radians to degrees, multiply
To convert degrees to radians, multiply by
180
180
.
by
π
Example 5:
Radians
Degrees
45◦
π
6
90◦
2π
3
150◦
Length of a Circular Arc
In a circle of radius r, the length s of an arc that subtends a central angle of θ radians
is
s = rθ
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Trigonometric Functions
Right Triangle Definition
a
sin θ = ,
c
c a
c
1
csc θ = =
a
sin θ
θ
b
cos θ = ,
c
sec θ =
c
1
=
b
cos θ
a
tan θ = ,
b
cot θ =
b
1
=
a
tan θ
b
π π
Example 6: Find trigonometric ratios of θ = 0, , .
6 2
Unit Circle Definition
Let t be any real number. Let’s mark off a distance θ along the unit circle, starting
at the point (1, 0) and moving in a counterclockwise direction if θ is positive or in a
clockwise direction if θ is negative. In this way we arrive at a point P (x, y) on the unit
circle. The point P (x, y) is called the terminal point determined by the number θ. We
define
sin θ = y, cos θ = x, tan θ =
y
1
1
x
, csc θ = , sec θ = , cot θ =
x
y
x
y
Example 7: Evaluate the trigonometric functions y = sin t, y = cos t and y = tan t at
π
t=
4
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Special Values of the Trigonometric Functions
θ in radians
0
sin θ
0
cos θ
1
tan θ
0
π
6
1
2
√
3
2
1
√
3
π
4
√
2
2
√
2
2
1
π
3
√
3
2
1
2
√
3
π
2
1
0
∞
2π
3
√
3
2
−
1
2
√
− 3
3π
4
√
2
2
√
2
−
2
−1
5π
6
1
2
√
−
3
2
1
−√
3
π
0
−1
0
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The Graphs of sine, cosine and tangent Functions
The graphs of sine, cosine and tangent functions are as follows.
y
6
1 r
sin(x)
cos(x)
r
r
− π2
π
2
r
r
r
3π
2
π
r
2π
r -
5π
2
x
−1 r
y
6
y = tan x
r
r
r
r
− π2
− 3π
−π
2
r
π
2
r
π
r
-
3π
2
x
Transformations of the Graphs of Trigonometric Functions
sin(x +
π
) = cos x
2
sin(x + π) = − sin x
y
6
1 r
y = sin(x)
y = sin(x + π2 )
r
− π2
r
−1 r
π
2
r
r
π
r
3π
2
r
2π
r -
5π
2
x
y = sin(x + π)
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Similarly we have
cos(t +
π
) = − sin t
2
cos(t + π) = − cos t
Inverse Trigonometric Functions
By their nature of being periodic functions, trigonometric functions are not one-to-one.
However, we can restrict the domain so that the trigonometric function is one-to-one
and then we can find its inverse function.
Trigonometric Inverse Functions
Range
h π πi
− ,
2 2
Domain
y = arcsin x
[−1, 1]
y = arccos x
[−1, 1]
y = arctan x
[−∞, ∞]
[0, π]
h π πi
− ,
2 2
Fundamental Trigonometric Identities
sin2 t + cos2 t = 1,
sin(arcsin x) = x
cos(arccos x) = x
tan2 t + 1 = sec2 t,
for − 1 ≤ x ≤ 1,
for − 1 ≤ x ≤ 1,
tan t =
arcsin(sin x) = x
arccos(cos x) = x
sin t
cos t
π
π
≤x≤ ,
2
2
for 0 ≤ x ≤ π,
for −
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Derivatives of Trigonometric Functions
In this section we develop rules for differentiating trigonometric functions. To derive the
rule for differentiating the sine function, we need the following two important limits.
sin h
=1
h→0 h
lim
and
cos(h) − 1
=0
h→0
h
lim
Example 8: Find the derivative of the function y = sin x at x = 0,
π
, and π
2
Example 9: Draw the graph of the derivative function of y = sin t.
y
6
1 r
r
r
− π2
y = sin(t)
π
2
r
r
π
r
r
3π
2
2π
r
r
r -
5π
2
x
−1 r
y
6
r
r
r
r
r
r -
x
r
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From our graphical intuition, we are lead to the following elegant result:
The derivative of the sine function y = sin x is y 0 = cos x, i.e.,
d
sin x = cos x
dx
Example 10: Differentiate.
(a) y = sin 3x
(b) y = cos x
(c) y = cos(x2 )
Example 11: Prove that (tan x)0 =
1
cos2 x
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The derivative of the cosine and tangent functions
d
cos x = − sin x
dx
1
d
= sec2 x
tan x =
dx
cos2 x
Example 12: The revenue of a fish farm is approximately
π R(t) = 2 5 − 4 cos t
0 ≤ t ≤ 12
6
during the t-th month, where R is measured in thousands of dollars.
(a) Find the marginal revenue function M R.
(b) Find the values of M R at t = 3 and t = 9.
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Now we derive the derivative of the inverse trigonometric functions.
Example 13: Prove that
d
1
where |x| < 1.
arcsin x = √
dx
1 − x2
The derivative of the inverse trigonometric functions
d
1
arccos x = − √
, |x| < 1
dx
1 − x2
d
1
arctan x =
dx
1 + x2
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