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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 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, f ( x, y) ( x, y) D . 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 x y 1 Domain of f ( x, y ) x 1 Chapter 11, 11.1, P594 Domain of f ( x, y ) x ln( y 2 x) Chapter 11, 11.1, P594 Domain of g ( x, y) 9 x y 2 2 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 R3 such that z=f (x, y) and (x, y) is in D. Chapter 11, 11.1, P594 Chapter 11, 11.1, P595 Graph of g ( x, y) 9 x 2 y 2 Chapter 11, 11.1, P595 h ( x, y ) 4 x 2 y 2 Graph of Chapter 11, 11.1, P595 x2 y 2 (a) f ( x, y) ( x 3y )e 2 2 x2 y 2 (b) f ( x, y) ( x 3y )e 2 2 Chapter 11, 11.1, P596 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 Chapter 11, 11.1, P597 Chapter 11, 11.1, P597 Chapter 11, 11.1, P598 Contour map of f ( x, y) 6 3x 2 y Chapter 11, 11.1, P598 Contour map of g ( x, y) 9 x 2 y 2 Chapter 11, 11.1, P598 The graph of h (x, y)=4x2+y2 is formed by lifting the level curves. Chapter 11, 11.1, P599 Chapter 11, 11.1, P599 Chapter 11, 11.1, P599 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 lim f ( x, y ) L ( x , y ) ( a ,b ) if for every number ε> 0 there is a corresponding number δ> 0 such that If ( x, y) Dand 0 ( x a)2 ( y b)2 then f ( x, y) L Chapter 11, 11.2, P604 Chapter 11, 11.2, P604 Chapter 11, 11.2, P604 Chapter 11, 11.2, P604 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 4. DEFINITION A function f of two variables is called continuous at (a, b) if lim f ( x, y ) f (a, b) ( x , y ) ( a ,b ) 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 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 x D and 0 x a then f ( x) L Chapter 11, 11.2, P609 4, If f is a function of two variables, its partial derivatives are the functions fx and fy defined by f ( x h, y ) f ( x, y ) f x ( x, y ) lim h 0 h f ( x y, h) f ( x, y ) f y ( x, y ) lim h 0 h Chapter 11, 11.3, P611 NOTATIONS FOR PARTIAL DERIVATIVES If Z=f (x, y) , we write f z f x ( x, y ) f x f ( x, y ) f1 D1 f Dx f x x x f z f y ( x, y ) f y f ( x, y ) f 2 D2 f D y f y y y Chapter 11, 11.3, P612 RULE FOR FINDING PARTIAL DERIVATIVES OF z=f (x, y) 1.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 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 Chapter 11, 11.3, P613 The second partial derivatives of f. If z=f (x, y), we use the following notation: f 2 f 2z ( f x ) x f xx f11 x x x 2 x 2 f 2 f 2z ( f x ) y f xy f12 y x yx yx f 2 f 2 z ( f y ) x f yx f 21 y xy xy x f 2 f 2 z ( f y ) y f yy f 22 y y 2 y 2 y Chapter 11, 11.3, P614 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 f xy ( a, b) f yx ( a, b) Chapter 11, 11.3, P615 FIGURE 1 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 z z0 f x ( x0 , y0 )( x x0 ) f y ( x0 , y0 )( y y0 ) Chapter 11, 11.4, P620 The linear function whose graph is this tangent plane, namely 3. L( x, y ) f (a, b) f x (a, b)( x a) f y (a, b)( y b) is called the linearization of f at (a, b) and the approximation 4. f ( x, y ) f (a, b) f x (a, b)( x a) f y (a, b)( y b) is called the linear approximation or the tangent plane approximation of f at (a, b) Chapter 11, 11.4, P621 7. DEFINITION If z= f (x, y), then f is differentiable at (a, b) if ∆z can be expressed in the form z f x (a, b)x f y (a, b)y 1x 2 y where ε1 and ε2→ 0 as (∆x, ∆y)→(0,0). Chapter 11, 11.4, P622 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 Chapter 11, 11.4, P623 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 z z dx f x ( x, y )dx f y ( x, y )dy dx dy x y Chapter 11, 11.4, P623 Chapter 11, 11.4, P624 For such functions the linear approximation is f ( x, y, z) f (a, b, c) f x (a, b, c)(x a) f y (a, b, c)( y b) f z (a, b, c)(z c) 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 increment of w is w f ( x x, y y, z z) f ( x, y, z ) The differential dw is defined in terms of the differentials dx, dy, and dz of the independent variables by w w w dw dx dy dz x y a Chapter 11, 11.4, P625 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 dz f dx f dy dt x dt y dt Chapter 11, 11.5, P627 dz z dx z dy dt x dt y dt Chapter 11, 11.5, P628 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 dz z dx z dy dz z dx z dy dx x ds y ds dt x dt y dt Chapter 11, 11.5, P629 Chapter 11, 11.5, P630 Chapter 11, 11.5, P630 Chapter 11, 11.5, P630 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 u u dx1 u x2 u xn ‧‧‧ ti x1 dti x2 dti xn ti for each i=1,2,‧‧‧,m. Chapter 11, 11.5, P630 Chapter 11, 11.5, P631 F (x, y)=0. Since both x and y are functions of x, we obtain F dx F dy 0 x dx y dx But dx /dx=1, so if ∂F/∂y≠0 we solve for dy/dx and obtain F dy x Fx dx F Fy y Chapter 11, 11.5, P632 F (x, y, z)=0 F dx F dy F z 0 x dx y dx z x But x ( x) 1 and ( y) 1 x so this equation becomes F F z 0 x z x 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. F F dz x dz y F dx dy F Chapter 11, 11.5, P632 z z Chapter 11, 11.6, P636 2. DEFINITION The directional derivative of f at (xo,yo) in the direction of a unit vector u=<a, b> is f ( x0 ha, y0 hb) f ( x0 , y0 ) Du f ( x0 , y0 ) lim h0 h if this limit exists. Chapter 11, 11.6, P636 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 Du f ( x, y ) f x ( x, y )a f y ( x, y )b Chapter 11, 11.6, P637 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 f f f ( x, y ) f x ( x, y ), f y ( x, y ) i j x y Chapter 11, 11.6, P638 Du f ( x, y ) f ( x, y ) u Chapter 11, 11.6, P638 10. DEFINITION The directional derivative of f at (x0, y0, z0) in the direction of a unit vector u=<a, b, c> is f ( x0 ha, y0 hb, z0 hc) f ( x0 , y0 , z0 ) Du f ( x0 , y0 , z0 ) lim h 0 h if this limit exists. Chapter 11, 11.6, P639 f ( x0 hu) f ( x0 ) Du f ( x0 ) lim h 0 h Chapter 11, 11.6, P639 f f f f f x , f y , f z i j k x y z Chapter 11, 11.6, P639 Du f ( x, y, z ) f ( x, y, z ) u Chapter 11, 11.6, P640 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 The equation of this tangent plane as F0 ( x0 , y0 , z0 )( x x0 ) Fy ( x0 , y0 , z0 )( y y0 ) Fz ( x0 , y0 , z0 )( z z0 ) The symmetric equations of the normal line to soot P are x x0 y y0 z z0 Fx ( x0 , y0 , z0 ) Fy ( x0 , y0 , z0 ) Fz ( x0 , y0 , z0 ) Chapter 11, 11.6, P642 Chapter 11, 11.6, P644 Chapter 11, 11.6, P644 Chapter 11, 11.7, P647 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 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 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 Chapter 11, 11.7, P648 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 D D(a, b) f xx (a, b) f yy (a, b) [ f xy (a, b)]2 (a)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 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: f xy f xy D f xx f yy ( f xy ) 2 f yx f yy Chapter 11, 11.7, P648 z x y 4 xy 1 4 4 Chapter 11, 11.7, P649 Chapter 11, 11.7, P649 Chapter 11, 11.7, P651 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 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 Chapter 11, 11.8, P654 f ( x0 , y0 , z0 ) g ( x0 , y0 , z0 ) Chapter 11, 11.8, P655 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 f ( x, y, z) g ( x, y, z ) and g ( x, y, z) k (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 Chapter 11, 11.8, P657