Download Calculus of Several Variables

Chapter 8
Calculus of Several Variables
8.1
Function of Two Variables
Many functions have several variables.
Ex. There are 3 types of football tickets. Type A costs $50, type B costs $30, and type C costs
$20. If in a match, x tickets of type A, y tickets of type B, and z tickets of type C are sold,
the total income of ticket sale is f (x, y, z) = 50x + 30y + 20z.
Def. A real-valued function of two variables f consists of
1. A domain A consisting of ordered pairs of some real numbers (x, y).
2. A rule that associates with each ordered pair in A with one real number, denoted by
z = f (x, y).
Ex. (Ex 1, p.532) f (x, y) = x + xy + y w + 2. Compute f (0, 0), f (1, 2), and f (2, 1).
p
2
Ex. Find the domain of a. f (x, y) = x2 +y 2 , b. g(x, y) = x−y
, c. h(x, y) = 1 − x2 − y 2 .
√
Ex. Find the domain of g(r, s) = rs.
The graph of a function z = f (x, y) is the collection of all points {(x, y, f (x, y)) : x, y ∈ A}
in R3 (Fig 5, p.534; Fig 6, p.535).
The graph of z = f (x, y) is 3 dimensional and it is difficult to draw. So we use level curves.
A level curve is the graph of
c = f (x, y)
on xy-plane for a constant c. By drawing the level curves corresponding to several admissible
values of c, we obtain a contour map. (Fig 7, p.535; Fig 8, p.536)
Ex. Ex 5, p.536.
HW. C8.1:
SC1, 2, 3,
Ex 31, 33
23
24
8.2
CHAPTER 8. CALCULUS OF SEVERAL VARIABLES
Partial Derivatives
I. First Order Partial Derivatives
For a function z = f (x, y), we hope to know how z changes with respect to a small change of
x or y.
Let (a, b) be a point in the domain of f (x, y). The first derivative of f with respect
to x at (a, b) is
∂f
f (a + h, b) − f (a, b)
(a, b) = lim
.
h→0
∂x
h
∂z
It can be written as ∂f
∂x |(a,b) = ∂x (a, b) = fx (a, b).
Similarly, the first derivative of f with respect to y at (a, b) is
∂f
f (a, b + h) − f (a, b)
(a, b) = lim
.
h→0
∂y
h
It can be written as
∂f
∂y |(a,b)
Ex. (Ex 1, p.542) Find
∂f
∂x
=
∂z
∂y (a, b)
and
∂f
∂y
= fy (a, b).
for f (x, y) = x2 − xy 2 + y 3 .
Ex. (HW24, p.550) Find the first partial derivatives of f (x, y, z) = xey/z .
p
Ex. Find the first partial derivatives of f (x, y) = x3 + 2y.
II. Second Order Partial Derivatives
(Fig 15, p.548) Differentiate the first partial derivatives. We obtain four second partial
derivatives:
∂2f
∂
fxx =
=
(fx )
2
∂x
∂x
∂2f
∂
fxy =
=
(fx )
∂y∂x
∂y
∂2f
∂
fyx =
=
(fy )
∂x∂y
∂x
∂2f
∂
fyy =
=
(fy )
2
∂y
∂y
When fxy and fyx are continuous, fxy = fyx .
Ex. (HW36, p.550) Find the second-order partial derivatives of f (x, y) = x3 + x2 y + x + 4.
p
Ex. Find the second partial derivatives of f (x, y) = x3 + 2y.
2
Ex. (Ex 7, p.549) Find the second-order partial derivatives of f (x, y) = exy .
HW. C8.2:
SC 1, 2, 3,
Ex 13, 15, 35, 37
8.3. MAXIMA AND MINIMA OF FUNCTIONS OF SEVERAL VARIABLES
8.3
25
Maxima and Minima of Functions of Several Variables
Finding the extreme values of a function are very useful in economic applications.
Def. Let f be a function defined on a region containing (a, b). Then (a, b) is a
• relative maximum if f (x, y) ≤ f (a, b) for all (x, y) near (a, b);
the number f (a, b) is a relative maximum value.
• absolute maximum if f (x, y) ≤ f (a, b) for all (x, y) in the domain of f .
• relative minimum if f (x, y) ≥ f (a, b) for all (x, y) near (a, b);
the number f (a, b) is a relative minimum value.
• absolute minimum if f (x, y) ≥ f (a, b) for all (x, y) in the domain of f .
Ex. Fig 16, p.554
The extrema can be determined by the First and the Second Derivative Tests.
Thm 8.1. If f (x, y) has a relative maximum/minimum at (a, b) and the partial derivatives of
f at (a, b) exist, then
∂f
∂f
(a, b) = 0,
(a, b) = 0.
∂x
∂y
(Explain by differential. See Fig 17 of p.555.)
∂f
The converse is not true. ∂f
∂x (a, b) = 0 and ∂y (a, b) = 0 don’t imply that (a, b) is a relative
maximum/minimum of f .
By the above theorem, a (relative/absolute) (maximum/minimum) can exist ONLY on
those points (a, b) that either
•
∂f
(a, b) = 0
∂x
and
∂f
(a, b) = 0,
∂y
or
• at least one of the partial derivatives dose not exist.
Any point in the domain that meets one of the above requirements is called a critical point.
Thm 8.2. The following steps are used to determine relative extrema:
1. Find critical points by solving the system of equations
(
fx = 0
fy = 0.
26
CHAPTER 8. CALCULUS OF SEVERAL VARIABLES
2. The second derivative test: Let
2
D(x, y) = fxx fyy − fxy
.
Then for any solution (a, b),
(a) D(a, b) > 0 and fxx (a, b) < 0 imply that f has a relative maximum at (a, b).
(b) D(a, b) > 0 and fxx (a, b) > 0 imply that f has a relative minimum at (a, b).
(c) D(a, b) < 0 implies that f has a saddle point at (a, b) (neither maximum nor
minimum).
(d) D(a, b) = 0 implies that the test is inconclusive. Other techniques must be used.
Ex. Ex 1, p.557.
Ex. (HW 26, p.562.) An open rectangular box having a volume of 108 in.3 is to be constructed
from a tin sheet. Find the dimensions of such a box of the amount of material used in its
construction is to be minimal.
(Hint: Let the dimensions of the box be x” by y” by z”. Then xyz = 108, i.e. z = 108/xy.
216
The amount of material used is S = xy + 2yz + 2xz = xy + 216
x + y . Minimize f (x, y) =
216
xy + 216
x + y .)
Ex. HW14, p.561. (saddle point at (0, −1)).
Ex. (Applied Ex 4, p.559, Maximizing Profits)
HW. C8.3:
8.4
SC 1, 2,
Ex 11, 13, 15
The Method of Least Squares (skip)
8.5. CONSTRAINED MAXIMA AND MINIMA AND THE METHOD OF LAGRANGE MULTIPLIERS27
8.5
Constrained Maxima and Minima and the Method of Lagrange Multipliers
We shall find the constrained relative extrema of f (x, y) subject to the constrain g(x, y) = 0.
Ex. (Ex 1, p.576) Find the relative minimum of f (x, y) = 2x2 + y 2 subject to the constrain
g(x, y) = x + y − 1 = 0. (Hint: Substitute y = −x + 1. Fig 27, p.577)
Thm 8.3 (Method of Lagrange Multipliers). To find the relative extrema of f (x, y)
subject to the constraint g(x, y) = 0,
1. Introduce a new variable λ (called the Lagrange multiplier), and form an auxiliary function
F (x, y, λ) = f (x, y) + λg(x, y).
2. Solve the system of equations


Fx = 0
Fy = 0


Fλ = 0
3. The solutions found in step 2 are candidates for the relative extrema of f .
Ex. Redo (Ex 1) to find the relative minimum of f (x, y) = 2x2 + y 2 subject to g(x, y) =
x + y − 1 = 0.
Ex. (HW 12, p.583) Find the maximum and minimum values of f (x, y) = exy subject to the
constraint x2 + y 2 = 8.
The Method of Lagrange Multipliers can be similarly applied to functions of more variables.
Ex. (HW 14, p.583) Maximize f (x, y, z) = xyz subject to the constraint 2x + 2y + z = 84.
Below are some applications.
Ex. (optional, HW 18, p.584, maximizing profit) The profit function in publishing and selling
Weston Publishing’s dictionaries is
P (x, y) = −0.005x2 − 0.003y 2 − 0.002xy + 14x + 12y − 200,
where x is the number of deluxe editions and y is the number of standard editions sold daily.
Constraint x + y = 400. How many deluxe copies and how many standard copies should be
published daily to maximize the profit?
Ex. (HW 22, p. 584,minimizing construction costs) An open rectangular box is to be constructed from materials that costs $3/ft2 for the bottom and $1/ft2 for its sides. Find the
dimensions of the box of greatest volume that can be constructed for $36.
HW. C8.5:
SC 1, 2,
Ex 5, 11, 19