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ESSENTIAL CALCULUS CH11 Partial derivatives. In this Chapter:. 11.1 Functions of Several Variables 11.2 Limits and Continuity 11.3 Partial Derivatives 11.4 Tangent Planes and Linear Approximations 11.5 The Chain Rule 11.6 Directional Derivatives and the Gradient Vector

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slide2

In this Chapter:

  • 11.1 Functions of Several Variables
  • 11.2 Limits and Continuity
  • 11.3 Partial Derivatives
  • 11.4 Tangent Planes and Linear Approximations
  • 11.5 The Chain Rule
  • 11.6 Directional Derivatives and the Gradient Vector
  • 11.7 Maximum and Minimum Values
  • 11.8 Lagrange Multipliers

Review

slide3

DEFINITION A function f 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, .

Chapter 11, 11.1, P593

slide4

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.

Chapter 11, 11.1, P593

slide6

Domain of

Chapter 11, 11.1, P594

slide7

Domain of

Chapter 11, 11.1, P594

slide8

Domain of

Chapter 11, 11.1, P594

slide9

DEFINITION If f is a function of two variables with domain D, then the graph of is the set of all points (x, y, z) in R3 such that z=f (x, y) and (x, y) is in D.

Chapter 11, 11.1, P594

slide11

Graph of

Chapter 11, 11.1, P595

slide12

Graph of

Chapter 11, 11.1, P595

slide14

DEFINITION The level curves of a function f of two variables are the curves with equations f (x, y)=k, where k is a constant (in the range of f).

Chapter 11, 11.1, P596

slide18

Contour map of

Chapter 11, 11.1, P598

slide19

Contour map of

Chapter 11, 11.1, P598

slide20

The graph of h (x, y)=4x2+y2

is formed by lifting the level curves.

Chapter 11, 11.1, P599

slide23

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 ε> 0 there is a corresponding number δ> 0 such that
  • If and then

Chapter 11, 11.2, P604

slide27

If f( x, y)→L1 as (x, y)→ (a ,b) along a path C1 and f (x, y) →L2 as (x, y)→ (a, b) along a path C2, where L1≠L2, then lim (x, y)→ (a, b) f (x, y) does not exist.

Chapter 11, 11.2, P605

slide28

4. 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.

Chapter 11, 11.2, P607

slide29

5.If f is defined on a subset D of Rn, then lim x→a f(x) =L means that for every number ε> 0 there is a corresponding number δ> 0 such that

If and then

Chapter 11, 11.2, P609

slide30

4, If f is a function of two variables, its partial derivatives are the functions fx and fy defined by

Chapter 11, 11.3, P611

slide31

NOTATIONS FOR PARTIAL DERIVATIVES If

Z=f (x, y) , we write

Chapter 11, 11.3, P612

slide32

RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y)

  • To find fx, regard y as a constant and differentiate
  • f (x, y) with respect to x.
  • 2. To find fy, regard x as a constant and differentiate f (x, y) with respect to y.

Chapter 11, 11.3, P612

slide33

FIGURE 1

The partial derivatives of f at (a, b) are

the slopes of the tangents to C1 and C2.

Chapter 11, 11.3, P612

slide36

The second partial derivatives of f. If z=f (x, y), we use the following notation:

Chapter 11, 11.3, P614

slide37

CLAIRAUT’S THEOREM Suppose f is defined on a disk D that contains the point (a, b) . If the functions fxy and fyx are both continuous on D, then

Chapter 11, 11.3, P615

slide38

FIGURE 1

The tangent plane contains the

tangent lines T1 and T2

Chapter 11, 11.4, P619

slide39

2. Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P (xo ,yo ,zo) is

Chapter 11, 11.4, P620

slide40

The linear function whose graph is this tangent plane, namely

3.

is called the linearization of f at (a, b) and the approximation

4.

is called the linear approximation or the tangent plane approximation of f at (a, b)

Chapter 11, 11.4, P621

slide41

7. DEFINITION If z= f (x, y), then f is differentiable at (a, b) if ∆z can be expressed in the form

where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0).

Chapter 11, 11.4, P622

slide42

8. THEOREM If the partial derivatives fx and fy exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b).

Chapter 11, 11.4, P622

slide44

For a differentiable function of two variables, z= f (x ,y), we define the differentials dx and dy to be independent variables; that is, they can be given any values. Then the differential dz, also called the total differential, is defined by

Chapter 11, 11.4, P623

slide46

For such functions the linear approximation is

and the linearization L (x, y, z) is the right side of this expression.

Chapter 11, 11.4, P625

slide47

If w=f (x, y, z), then the increment of w is

The differential dw is defined in terms of the differentials dx, dy, and dz of the independent

variables by

Chapter 11, 11.4, P625

slide48

2. THE CHAIN RULE (CASE 1) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (t) and y=h (t) and are both differentiable functions of t. Then z is a differentiable function of t and

Chapter 11, 11.5, P627

slide50

3. THE CHAIN RULE (CASE 2) Suppose that z=f (x, y) is a differentiable function of x and y, where x=g (s, t) and y=h (s, t) are differentiable functions of s and t. Then

Chapter 11, 11.5, P629

slide54

4. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x1, x2,‧‧‧,xn and each xj is a differentiable function of the m variables t1, t2,‧‧‧,tm Then u is a function of t1, t2,‧‧‧, tm and

for each i=1,2,‧‧‧,m.

Chapter 11, 11.5, P630

slide56

F (x, y)=0. Since both x and y are functions of x, we obtain

But dx /dx=1, so if ∂F/∂y≠0 we solve for dy/dx and obtain

Chapter 11, 11.5, P632

slide57

F (x, y, z)=0

But and

so this equation becomes

If ∂F/∂z≠0 ,we solve for ∂z/∂x and obtain the first formula in Equations 7. The formula for ∂z/∂y is obtained in a similar manner.

Chapter 11, 11.5, P632

slide59

2. DEFINITION The directional derivative of f at (xo,yo) in the direction of a unit vector u=<a, b> is

if this limit exists.

Chapter 11, 11.6, P636

slide60

3. THEOREM If f is a differentiable function of x and y, then f has a directional derivative in the direction of any unit vector u=<a, b> and

Chapter 11, 11.6, P637

slide61

8. DEFINITION If f is a function of two variables x and y , then the gradient of f is the vector function ∆f defined by

Chapter 11, 11.6, P638

slide63

10. DEFINITION The directional derivative of f at (x0, y0, z0) in the direction of a unit vector u=<a, b, c> is

if this limit exists.

Chapter 11, 11.6, P639

slide67

15. THEOREM Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative

Du f(x) is │▽f (x)│ and it occurs when u has the same direction as the gradient vector ▽ f(x) .

Chapter 11, 11.6, P640

slide68

The equation of this tangent plane as

The symmetric equations of the normal line to soot P are

Chapter 11, 11.6, P642

slide72

1. DEFINITION A function of two variables has a local maximum at (a, b) if f (x, y) ≤ f (a, b) when (x, y) is near (a, b). [This means that

f (x, y) ≤ f (a, b) for all points (x, y) in some disk with center (a, b).] The number f (a, b) is

called a local maximum value. If f (x, y) ≥ f (a, b) when (x, y) is near (a, b), then f (a, b) is a local minimum value.

Chapter 11, 11.7, P647

slide73

2. THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then fx(a, b)=1 and fy(a, b)=0.

Chapter 11, 11.7, P647

slide74

A point (a, b) is called a critical point (or stationary point) of f if fx (a, b)=0 and fy (a, b)=0, or if one of these partial derivatives does not exist.

Chapter 11, 11.7, P647

slide76

3. SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b) , and suppose that

  • fx (a, b) and fy (a, b)=0 [that is, (a, b) is a critical point of f]. Let
  • If D>0 and fxx (a, b)>0 , then f (a, b) is a local
  • minimum.
  • (b)If D>0 and fxx (a, b)<0, then f (a, b) is a local maximum.
  • (c) If D<0, then f (a, b) is not a local maximum or minimum.

Chapter 11, 11.7, P648

slide77

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 maximum or local minimum at (a, b), or (a, b) could be a saddle point of f.

NOTE 3 To remember the formula for D it’s helpful to write it as a determinant:

Chapter 11, 11.7, P648

slide81

4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is

continuous on a closed, bounded set D in R2, then f attains an absolute maximum value f(x1,y1) and an absolute minimum value f(x2,y2) at some points (x1,y1) and (x2,y2) in D.

Chapter 11, 11.7, P651

slide82

5. To find the absolute maximum and minimum values of a continuous function f on a closed, bounded set D:

1. Find the values of f at the critical points of 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.

Chapter 11, 11.7, P651

slide86

METHOD OF LAGRANGE MULTIPLIERS To find the maximum and minimum values of f (x, y, z) subject to the constraint g (x, y, z)=k [assuming that these extreme values exist and ▽g≠0 on the surface g (x, y, z)=k]:

(a) Find all values of x, y, z, and such that

and

(b) Evaluate f at all the points (x, y, z) that result from step (a). The largest of these values is the maximum value of f; the smallest is the minimum value of f.

Chapter 11, 11.8, P655

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