1. Chapter 14 Partial Derivatives
14.1 Functions of Several Variables
14.2 Limits and Continuity
14.3 Partial Derivatives
14.4 Tangent Planes and Linear Approximations
14.5 The Chain Rule
*14.6 Directional Derivatives and the Gradient Vector
14.7 Maximum and Minimum Values
14.8 Lagrange Multipliers

2. So far we have dealt with the calculus of functions of a single variable. But, in the real world, physical quantities often depend on two or more variables, so in this chapter we turn our attention to functions of several variables and extend the basic ideas of differential calculus to such functions.

3. 14.1 Functions of Several Variables
In this section we study functions of two or
more variables from four points of view:
Verbally (by a description in words)
Numerically (by a table of values)
Algebraically (by an explicit formula)
Visually (by a graph or level curves)

4. Functions of Two Variables
Definition A function of two variables is a rule that assigns to each
ordered pair of real numbers (x,y) in a set D a unique real number
denoted by f(x,y). The set D is the domain of f and its range is the set of
values that f takes on, that is , {f(x,y)| (x,y) D}.
We often write z= f(x,y) to make explicit the value taken on by f at the
general point (x,y) . The variables x and y are independent variables
and z is the dependent variable. y f x z
f(x,y)
（x,y）

5. Example 1 Find the domains of the following functions and evaluate f(3,2).
(a) (b)
Solution
(a)
(b)

6. Graphs Another way of visualizing the behavior of a function of two variables is to consider its graph.
Definition If f is a function of two variables with domain D,
then the graph of f is the set of all points (x,y,z) in such
that z=f(x,y) and (x,y) is in D. z S y D x

7. Functions of Three or More Variables
A function of three variables, f, is a rule that assigns to each ordered triple (x,y,z) in a domain D a unique real number denoted by f(x,y,z).
A function of n variables, f, is a rule that assigns a number to an n-tuple of real
number. We denoted by the set of all such n-tuples.

8. 14.2 Limits and Continuity
Definition Let f be a function of two variables whose
domain D includes points arbitrarily close to (a,b).
Then we say that the limit of f(x,y) as (x,y) approaches (a,b) is L and we write
if for every number there is a corresponding number such that whenever
and

9. If as along a path and
as along a path , where ,
then does not exist.
Example does not exist.
Solution
as along the x-axis
as along the y-axis

10. Continuity Definition A function f of two variables is called
continuous at (a,b) if
We say f is continuous on D if f is continuous at
every point (a,b) in D.
Example

11. 14.3 Partial Derivatives
If f is a function of two variables , its partial derivatives
are the functions and defined by
Notations for partial derivatives If , we write

12. Rule for finding partial derivatives of
1.To find regard as a constant and differentiate with respect to
2.To find regard as a constant and differentiate with respect to
Example If , find and
Solution

13. Interpretations of partial derivatives
Stand for differentiate of tangent through
of curve
Stand for differentiate of tangent through of curve
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14. Higher derivatives
If then

15. Example
Clairaut’s Theorem Suppose f is defined on a disk D that contains the point (a,b). If the functions and are both continuous on D, then

16. 14.4 Tangent Planes and Linear Approximations
Definition If , then f is differentiable at
(a,b) if can be expressed in the form
where and as
Theorem If the partial derivatives and exist
near (a,b) and are continuous at (a,b), then f is
differentiable at (a,b).

17. Differentials For a differentiable function of two variables,
, we define the differentials and
to be independent variables; that is, they can
be given any values. Then the differential ,
also called the total differential , is defined by

18. 14.5 The Chain Rule The Chain Rule (Case 1) Suppose that
is a differentiable function of and , where
and are both differentiable functions of .
Then is a differentiable function of and or

19. Example Let z = excosy, x = sint and y = t2. Find
Solution We find that
The Chain Rule (Case 2) Suppose that is
a differentiable function of and , where
and are differentiable functions of and .Then

20. Example Let z = xy, x = 3u2 + v2, and y = 4u +2v. Find and Solution

21. Implicit Differentiation
We suppose that an equation of the form
defines implicitly as a differentiable function of ,
that is, , where for all in the
domain of f. If F is differentiable. Then

22. We suppose that an equation of the form
defines implicitly as a differentiable function of
and , that is, , where for all
in the domain of f. If F is differentiable. Then

23. Example 1 Find if
Solution Let
then
Example 2 Find and if

24. 14.7 Maximum and Minimum Values
Definition A function of two variables has a local
maximum at (a,b) if when (x,y) is near
(a,b). [ This means that for all points (x,y)
in some disk with center (a,b).] The number f(a,b)
is called a local maximum value. If
when (x,y) is near (a,b), then f(a,b) is called a local
minimum value. If the inequalities hold for all points
(x,y) in the domain of f, then f has an absolute maxi-
mum (or absolute minimum) at (a,b).

25. Theorem If f has a local maximum or minimum
at (a,b) and the first-order partial derivatives of f
exist there, then and
A point (a,b) is called a critical point (or stationary
point ) of f if and , or one of partial
Derivatives does not exist.
Theorm says that if f has a local maximum or mini-
mum at (a,b), then (a,b) is a critical point of f .
However, not all critical points give rise to maximum
or minimum.

26. Second derivatives test Suppose the second
partial derivatives of f are continuous on a disk with
center (a,b), and suppose that and
[that is, (a,b) is a critical point of f ]. Let
(a) If and , then f(a,b) is a local
minimum.
(b) If and , then f(a,b) is a local
maximum.
(c) If ,then f(a,b) is not a local minimum or local

27. Note 1 In case (c) the point (a,b) is called a saddle
point of f and the graph of f crosses its tangent
plane at (a,b).
Note 2 If D=0, the test gives no information: f could
have a local minimum or local maximum at (a,b), or
(a,b) could be a saddle point of f .
Note 3 We often use the following symbols:

28. Example Let f(x, y) = x2 – y2. Show that the origin is the only critical point but that f has no extreme value at the origin. Solution The partial derivatives of f exist at every point in the domain of f, and we have that f’x(x, y) = 2x, f’y(x, y) = -2y, so that (0, 0) is the only critical point of f. However, f(0, 0) = 0 is no a extreme value of f, because f(x, 0) = x2 > 0 for all x≠0 and f(0, y) = -y2 < 0 for all y ≠0.

29. Example Given f(x, y) = x3 +y3 –3xy, find the extreme values of function f.
Solution f has continuous second partial derivatives so the critical points of f are those at which f’x and f’y are 0. Since
f’x(x, y) = 3x2 –3y and f’y(x, y) = 3y2 –3x,
Solving for x, y from x2 –3y = 0,
3y2 –3x = 0,
we obtain the critical points of f, which are (1, 1) and (0, 0).
For (1, 1) we have A = 6, B = -3, C = 6, and D=27> 0.
Since A>0, we know that f(1, 1) = -1 is a local minimum.
For (0, 0) we have A = 0, B = -3, C = 0, and D= -9< 0,
hence f has no extreme value at (0, 0).

30. Absolute Maximum and Minimum Values
Extreme value theorem for functions of two variables
If f is continuous on a closed, bounded set D in , then f
attains an absolute maximum value and an absolute
minimum value at some points and in D.
To find the absolute maximum and minimum values of a
continuous function of f on a closed, bounded set D :
1. Find the values of f at the critical points of f in D.
2. Find the extreme values of f on the boundary of D.
3. The largest of the values from steps 1 and 2 is the absolute
maximum value; the smallest of these values is the absolute
minimum value.

31. Example Find the absolute maximum and minimum values
of the function on
Solution First find the critical points. These occur when
So the only critical point is (1,1), and the value is f(1,1)=1.
In step 2 we look at the values of f on the boundary of D, which
consists of the four line segments of D. We attain that f(0,0)=0
and f(3,2)=1 are minimum values of f on the boundary of D ,
we also attain that f(3,0)=9 and f(0,2)=4 are maximum values
of f on the boundary of D .
In step 3 we compare these values and conclude that the
absolute maximum value of f on D is f(3,0)=9 and the absolute
minimum value is f(0,0)=0.

32. 14.8 Lagrange Multipliers
Conditional extreme values and Lagrange multipliers
In previous discussion on extreme values of functions of two
variables x and y, the variables x and y are independent to
each other, i.e., they are not constrained by any side
condition on the values of x and y. However, we may
encounter problems of finding the extreme values of function
f(x, y) subject to a side condition, also called a constraint,
of the form g(x, y) = 0. Such extreme values are called
conditional extreme values.

33. We will describe a general method of finding the extreme values of a function subject to a constraint, which is called the Lagrange multipliers method, and the number is called a Lagrange multiplier. The method of determining extreme values of function f(x, y) subject to constraint g(x, y) = 0 by means of Lagrange multipliers proceed as follows.

34. 1. Form a function F by multiplying g(x, y) with the Lagrange
multiplier and then adding to f(x, y), that is,
2. Find the partial derivatives of F, F’x and F’y , and then
solve the equations
3. Calculate the value of f at each point (x, y) that is obtained
in step 2.

35. Then, subject to the constraint g(x, y) = 0, if f(x, y) has a
(conditional) maximum value, it will be the largest of the
values computed; if f(x, y) has a (conditional) minimum
value, it will be the smallest of the values computed.
Example 1 Find the extreme values of f(x, y) = x2 + 4y3
subject to the constraint x2 + 2y2 =1.
Solution Let
Then
Solving

36. we obtain as the solutions
we obtain as the solutions. Thus the only possible extreme values of f occur at points in the xy plane:

37. Since we conclude that the maximum value of f(x, y) on the ellipse is , which occurs at point , and the minimum value of f(x,y) on the ellipse is , which occurs at point .

38. Example 2 A rectangular box without a lid is to be made from 12 m2 of cardboard. Find the maximum volume of such a box.
Solution Let x, y, and z be the length, width, and height, respectively, of the box in meters. We wish to maximize V= xyz subject to the constraint
g(x,y,z)=2xz+2yz+xy-12=0
Let
Solving the equations

39. we have x=y=2 and z=1.
Then V=xyz=4, so the maximum volume of the box is 4 m3.