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ESSENTIAL CALCULUS CH11 Partial derivatives

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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

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Chapter 11, 11.1, P593 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,.

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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.

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Chapter 11, 11.1, P593

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Chapter 11, 11.1, P594 Domain of

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Chapter 11, 11.1, P594 Domain of

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Chapter 11, 11.1, P594 Domain of

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Chapter 11, 11.1, P594 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 R 3 such that z=f (x, y) and (x, y) is in D.

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Chapter 11, 11.1, P595

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Graph of

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Chapter 11, 11.1, P595 Graph of

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Chapter 11, 11.1, P596

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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).

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Chapter 11, 11.1, P597

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Chapter 11, 11.1, P598

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Contour map of

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Chapter 11, 11.1, P598 Contour map of

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Chapter 11, 11.1, P599 The graph of h (x, y)=4x 2 +y 2 is formed by lifting the level curves.

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Chapter 11, 11.1, P599

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Chapter 11, 11.2, P604 1.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

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Chapter 11, 11.2, P604

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Chapter 11, 11.2, P605 If f( x, y) → L 1 as (x, y) → (a,b) along a path C 1 and f (x, y) → L 2 as (x, y) → (a, b) along a path C 2, where L 1 ≠L 2, then lim (x, y) → (a, b) f (x, y) does not exist.

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Chapter 11, 11.2, P 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.

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Chapter 11, 11.2, P609 5.If f is defined on a subset D of R n, 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

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Chapter 11, 11.3, P611 4, If f is a function of two variables, its partial derivatives are the functions f x and f y defined by

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Chapter 11, 11.3, P612 NOTATIONS FOR PARTIAL DERIVATIVES If Z=f (x, y), we write

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Chapter 11, 11.3, P612 RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) 1.To find f x, regard y as a constant and differentiate f (x, y) with respect to x. 2. To find f y, regard x as a constant and differentiate f (x, y) with respect to y.

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Chapter 11, 11.3, P612 FIGURE 1 The partial derivatives of f at (a, b) are the slopes of the tangents to C 1 and C 2.

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Chapter 11, 11.3, P613

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Chapter 11, 11.3, P614 The second partial derivatives of f. If z=f (x, y), we use the following notation:

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Chapter 11, 11.3, P615 CLAIRAUT ’ S THEOREM Suppose f is defined on a disk D that contains the point (a, b). If the functions f xy and f yx are both continuous on D, then

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Chapter 11, 11.4, P619 FIGURE 1 The tangent plane contains the tangent lines T 1 and T 2

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Chapter 11, 11.4, P Suppose f has continuous partial derivatives. An equation of the tangent plane to the surface z=f (x, y) at the point P (x o,y o,z o ) is

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Chapter 11, 11.4, P621 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)

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Chapter 11, 11.4, P 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).

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Chapter 11, 11.4, P THEOREM If the partial derivatives f x and f y exist near (a, b) and are continuous at (a, b), then f is differentiable at (a, b).

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Chapter 11, 11.4, P623

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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

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Chapter 11, 11.4, P624

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Chapter 11, 11.4, P625 For such functions the linear approximation is and the linearization L (x, y, z) is the right side of this expression.

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Chapter 11, 11.4, P625 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

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Chapter 11, 11.5, P 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

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Chapter 11, 11.5, P628

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Chapter 11, 11.5, P 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

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Chapter 11, 11.5, P630

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4. THE CHAIN RULE (GENERAL VERSION) Suppose that u is a differentiable function of the n variables x 1, x 2, ‧‧‧,x n and each x j is a differentiable function of the m variables t 1, t 2, ‧‧‧,t m Then u is a function of t 1, t 2, ‧‧‧, t m and for each i=1,2, ‧‧‧,m.

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Chapter 11, 11.5, P631

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Chapter 11, 11.5, P632 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

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Chapter 11, 11.5, P632 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.

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Chapter 11, 11.6, P636

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2. DEFINITION The directional derivative of f at (x o,y o ) in the direction of a unit vector u= is if this limit exists.

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Chapter 11, 11.6, P 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= and

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Chapter 11, 11.6, P DEFINITION If f is a function of two variables x and y, then the gradient of f is the vector function ∆f defined by

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Chapter 11, 11.6, P638

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Chapter 11, 11.6, P DEFINITION The directional derivative of f at (x 0, y 0, z 0 ) in the direction of a unit vector u= is if this limit exists.

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Chapter 11, 11.6, P639

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Chapter 11, 11.6, P640

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15. THEOREM Suppose f is a differentiable function of two or three variables. The maximum value of the directional derivative D u f(x) is │ ▽ f (x)│ and it occurs when u has the same direction as the gradient vector ▽ f(x).

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Chapter 11, 11.6, P642 The symmetric equations of the normal line to soot P are The equation of this tangent plane as

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Chapter 11, 11.6, P644

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Chapter 11, 11.7, P647

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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.

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Chapter 11, 11.7, P THEOREM If f has a local maximum or minimum at (a, b) and the first order partial derivatives of f exist there, then f x (a, b)=1 and f y (a, b)=0.

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Chapter 11, 11.7, P647 A point (a, b) is called a critical point (or stationary point) of f if f x (a, b)=0 and f y (a, b)=0, or if one of these partial derivatives does not exist.

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Chapter 11, 11.7, P648

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3. SECOND DERIVATIVES TEST Suppose the second partial derivatives of f are continuous on a disk with center (a, b), and suppose that f x (a, b) and f y (a, b)=0 [that is, (a, b) is a critical point of f]. Let (a)If D>0 and f xx (a, b)>0, then f (a, b) is a local minimum. (b)If D>0 and f xx (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.

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Chapter 11, 11.7, P648 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:

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Chapter 11, 11.7, P649

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Chapter 11, 11.7, P651

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4. EXTREME VALUE THEOREM FOR FUNCTIONS OF TWO VARIABLES If f is continuous on a closed, bounded set D in R 2, then f attains an absolute maximum value f(x 1,y 1 ) and an absolute minimum value f(x 2,y 2 ) at some points (x 1, y 1 ) and (x 2,y 2 ) in D.

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Chapter 11, 11.7, P 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.

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Chapter 11, 11.7, P652

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Chapter 11, 11.8, P654

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Chapter 11, 11.8, P655

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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.

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Chapter 11, 11.8, P657

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