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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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Essential calculus ch11 partial derivatives

ESSENTIAL CALCULUSCH11 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

  • 11.7 Maximum and Minimum Values

  • 11.8 Lagrange Multipliers

    Review


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


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


Chapter 11, 11.1, P593 on by f at the general point (x, y) . The variables x and y are


Domain of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P594


Domain of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P594


Domain of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P594


DEFINITION on by f at the general point (x, y) . The variables x and y are 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


Chapter 11, 11.1, P595 on by f at the general point (x, y) . The variables x and y are


Graph of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P595


Graph of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P595


Chapter 11, 11.1, P596 on by f at the general point (x, y) . The variables x and y are


DEFINITION on by f at the general point (x, y) . The variables x and y are 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


Chapter 11, 11.1, P597 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.1, P597 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.1, P598 on by f at the general point (x, y) . The variables x and y are


Contour map of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P598


Contour map of on by f at the general point (x, y) . The variables x and y are

Chapter 11, 11.1, P598


The graph of h (x, y)=4x on by f at the general point (x, y) . The variables x and y are 2+y2

is formed by lifting the level curves.

Chapter 11, 11.1, P599


Chapter 11, 11.1, P599 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.1, P599 on by f at the general point (x, y) . The variables x and y are


  • DEFINITION on by f at the general point (x, y) . The variables x and y are 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


Chapter 11, 11.2, P604 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.2, P604 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.2, P604 on by f at the general point (x, y) . The variables x and y are


If f( x, y)→L on by f at the general point (x, y) . The variables x and y are 1 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


4. DEFINITION on by f at the general point (x, y) . The variables x and y are 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


5.If f is defined on a subset on by f at the general point (x, y) . The variables x and y are 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


4, If f is a function of two variables, its on by f at the general point (x, y) . The variables x and y are partial derivatives are the functions fx and fy defined by

Chapter 11, 11.3, P611


NOTATIONS FOR PARTIAL DERIVATIVES on by f at the general point (x, y) . The variables x and y are If

Z=f (x, y) , we write

Chapter 11, 11.3, P612


  • RULE FOR FINDING PARTIAL DERIVATIVES OF z=f on by f at the general point (x, y) . The variables x and y are (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


FIGURE 1 on by f at the general point (x, y) . The variables x and y are

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

the slopes of the tangents to C1 and C2.

Chapter 11, 11.3, P612


Chapter 11, 11.3, P613 on by f at the general point (x, y) . The variables x and y are


Chapter 11, 11.3, P613 on by f at the general point (x, y) . The variables x and y are


The on by f at the general point (x, y) . The variables x and y are second partial derivatives of f. If z=f (x, y), we use the following notation:

Chapter 11, 11.3, P614


CLAIRAUT on by f at the general point (x, y) . The variables x and y are ’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


FIGURE 1 on by f at the general point (x, y) . The variables x and y are

The tangent plane contains the

tangent lines T1 and T2

Chapter 11, 11.4, P619


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


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


7. DEFINITION namelyIf 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


8. THEOREM namelyIf 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



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


Chapter 11, 11.4, P624 we define the


For such functions the we define the linear approximation is

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

Chapter 11, 11.4, P625


If w=f (x, y, z), then the we define 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


2. THE CHAIN RULE (CASE 1) we define the 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


Chapter 11, 11.5, P628 we define the


3. THE CHAIN RULE (CASE 2) we define the 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


Chapter 11, 11.5, P630 we define the


Chapter 11, 11.5, P630 we define the


Chapter 11, 11.5, P630 we define the


4. THE CHAIN RULE (GENERAL VERSION) we define the 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


Chapter 11, 11.5, P631 we define the


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

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

Chapter 11, 11.5, P632


F (x, y, z)=0 we define the

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


Chapter 11, 11.6, P636 we define the


2. DEFINITION we define the 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


3. THEOREM we define the 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


8. DEFINITION we define the 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


Chapter 11, 11.6, P638 we define the


10. DEFINITION we define the 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


Chapter 11, 11.6, P639 we define the


Chapter 11, 11.6, P639 we define the


Chapter 11, 11.6, P640 we define the


15. THEOREM we define the 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


The equation of this tangent plane as we define the

The symmetric equations of the normal line to soot P are

Chapter 11, 11.6, P642


Chapter 11, 11.6, P644 we define the


Chapter 11, 11.6, P644 we define the


Chapter 11, 11.7, P647 we define the


1. DEFINITION we define the 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


2. THEOREM we define the 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


A point (a, b) is called a we define the 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


Chapter 11, 11.7, P648 we define the


  • 3. SECOND DERIVATIVES TEST we define the 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


NOTE 1 we define the 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


Chapter 11, 11.7, P649 we define the


Chapter 11, 11.7, P649 we define the


Chapter 11, 11.7, P651 we define the


4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES we define the 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


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


Chapter 11, 11.7, P652 continuous function f on a closed, bounded set D:


Chapter 11, 11.8, P654 continuous function f on a closed, bounded set D:


Chapter 11, 11.8, P655 continuous function f on a closed, bounded set D:


METHOD OF LAGRANGE MULTIPLIERS continuous function f on a closed, bounded set D: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


Chapter 11, 11.8, P657 continuous function f on a closed, bounded set D:


Chapter 11, 11.8, P657 continuous function f on a closed, bounded set D:


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