Calculus of Optimization Review for AAE 320
Dr. Paul D. Mitchell (608) 2656514,
[email protected]
This document provides a quick review of calculusbased optimization for AAE 320.
It
assumes a basic understanding of calculus, but that you need a quick “refresher.”
It goes
through the general rules, then works through an example, then gives sample problems
(Think Breaks).
A useful review of calculus based optimization is available at
http://www.swlearning.com/economics/mcguigan/mcguigan9e/web_chapter_a.pdf
.
This
pdf is a web chapter supplement for the text
Managerial Economics
(9
th
ed.) by
McGuigan, Moyer, and Harris.
Note that I would like students to know univariate and
multivariate unconstrained optimization, and so for AAE 320 you do not need to know
the constrained optimization methods described in the web chapter.
However, you may
want to know them for other AAE courses.
Part 1: Review of Derivatives
Before explaining optimization, first let’s review derivatives.
A derivative is the slope of
a function.
For some functions the slope is constant, say 1.5, for other functions, the
slope changes depending where on the function you are, and so the slope if a function as
well, say 1.5 + 3x.
Notation for Derivatives
Three different notations are used for 1
st
and 2
nd
derivatives of y = f(x)
1)
dy
dx
and
2
2
d y
dx
Newton
2)
f
′
(x) and f
′′
(x)
Leibniz
3)
f
x
(x) and f
xx
(x)
Leibniz
They all mean the same thing, but sometimes one is easier to use than the other.
Finally, not that the second derivative is just the derivative of the derivative
Rules of Differentiation (Web Chapter A, Table A.1, page 12)
Constant Function
If y= f(x) = a
f
′
(x) = 0
Power Function
If f(x) =
ax
b
f
′
(x) =
bax
b
– 1
If f(x) = 7x
f
′
(x) = 7
If f(x) = 7
x
0.34
f
′
(x) = 7(0.34)x
0.34 – 1
= 2.38x
–0.66
Sum of Functions
If y = f(x) + g(x), then
dy
dx
= f
′
(x) + g
′
(x)
If h(x) = 3 + 4x – 7x
2
, then h
′
(x) = 4 – 2(7)x
21
= 4 – 12x
If k(x) = 56 + 44x
0.5
– 17x, then h
′
(x) = 0.5(44)x
0.51
– 17 = 22x
0.5
– 17
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Review Rules of Differentiation
Find the first and second derivatives of the following functions.
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 Spring '08
 MITCHELL
 Derivative, FOC

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