Chapter 16 Partial Differentiation

More documents

Recommendations

Info

384 Chapter 16 Partial DifferentiationExercises 16.6.1. Let f = xy/(x 2 +y 2 ); compute f xx , f yx , and f yy . ⇒2. Find all first and second partial derivatives of x 3 y 2 +y 5 . ⇒3. Find all first and second partial derivatives of 4x 3 +xy 2 +10. ⇒4. Find all first and second partial derivatives of xsiny. ⇒5. Find all first and second partial derivatives of sin(3x)cos(2y). ⇒6. Find all first and second partial derivatives of e x+y2 . ⇒7. Find all first and second partial derivatives of ln √ x 3 +y 4 . ⇒8. Findallfirstandsecond partialderivativesofz withrespecttoxandy ifx 2 +4y 2 +16z 2 −64 =0. ⇒9. Find all first and second partial derivatives of z with respect to x and y if xy+yz+xz = 1.⇒10. Let α and k be constants. Prove that the function u(x,t) = e −α2 k 2t sin(kx) is a solution tothe heat equation u t = α 2 u xx11. Let a be a constant. Prove that u = sin(x−at)+ln(x+at) is a solution to the wave equationu tt = a 2 u xx .12. How many third-order derivatives does a function of 2 variables have? How many of theseare distinct?13. How many nth order derivatives does a function of 2 variables have? How many of these aredistinct?½ºÅÜÑ Ò ÑÒÑSuppose a surface given by f(x,y) has a local maximum at (x 0 ,y 0 ,z 0 ); geometrically, thispoint on the surface looks like the top of a hill. If we look at the cross-section in theplane y = y 0 , we will see a local maximum on the curve at (x 0 ,z 0 ), and we know fromsingle-variable calculus that ∂z∂x = 0 at this point. Likewise, in the plane x = x 0, ∂z∂y = 0.So if there is a local maximum at (x 0 ,y 0 ,z 0 ), both partial derivatives at the point mustbe zero, and likewise for a local minimum. Thus, to find local maximum and minimumpoints, we need only consider those points at which both partial derivatives are 0. As inthe single-variable case, it is possible for the derivatives to be 0 at a point that is neithera maximum or a minimum, so we need to test these points further.You will recall that in the single variable case, we examined three methods to identifymaximum and minimum points; the most useful is the second derivative test, though itdoes not always work. For functions of two variables there is also a second derivative test;again it is by far the most useful test, though it doesn’t always work.THEOREM 16.7.1 Suppose that the second partial derivatives of f(x,y) are continuousnear (x 0 ,y 0 ), and f x (x 0 ,y 0 ) = f y (x 0 ,y 0 ) = 0. We denote by D the discriminant
16.7 Maxima and minima 385D(x 0 ,y 0 ) = f xx (x 0 ,y 0 )f yy (x 0 ,y 0 ) − f xy (x 0 ,y 0 ) 2 . If D > 0 and f xx (x 0 ,y 0 ) < 0 there isa local maximum at (x 0 ,y 0 ); if D > 0 and f xx (x 0 ,y 0 ) > 0 there is a local minimum at(x 0 ,y 0 ); if D < 0 there is neither a maximum nor a minimum at (x 0 ,y 0 ); if D = 0, thetest fails.EXAMPLE 16.7.2 Verify that f(x,y) = x 2 +y 2 has a minimum at (0,0).First, we compute all the needed derivatives:f x = 2x f y = 2y f xx = 2 f yy = 2 f xy = 0.The derivatives f x and f y are zero only at (0,0). Applying the second derivative test there:D(0,0) = f xx (0,0)f yy (0,0)−f xy (0,0) 2 = 2·2−0 = 4 > 0,so there is a local minimum at (0,0), and there are no other possibilities.EXAMPLE 16.7.3 Find all local maxima and minima for f(x,y) = x 2 −y 2 .The derivatives:f x = 2x f y = −2y f xx = 2 f yy = −2 f xy = 0.Again there is a single critical point, at (0,0), andD(0,0) = f xx (0,0)f yy (0,0)−f xy (0,0) 2 = 2·−2−0 = −4 < 0,so there is neither a maximum nor minimum there, and so there are no local maxima orminima. The surface is shown in figure 16.7.1.EXAMPLE 16.7.4 Find all local maxima and minima for f(x,y) = x 4 +y 4 .The derivatives:f x = 4x 3 f y = 4y 3 f xx = 12x 2 f yy = 12y 2 f xy = 0.Again there is a single critical point, at (0,0), andD(0,0) = f xx (0,0)f yy (0,0)−f xy (0,0) 2 = 0·0−0 = 0,so we get no information. However, in this case it is easy to see that there is a minimumat (0,0), because f(0,0) = 0 and at all other points f(x,y) > 0.EXAMPLE 16.7.5 Find all local maxima and minima for f(x,y) = x 3 +y 3 .
Page 5 and 6: ½º¾ÄÑØ× Ò ÓÒØÒÙØÝ16.
Page 7 and 8: 16.2 Limits and Continuity 367We wa
Page 9 and 10: ½º¿ÈÖØÐ ÖÒØØÓÒ16.3 Par
Page 11 and 12: 16.3 Partial Differentiation 371sur
Page 13 and 14: 16.3 Partial Differentiation 373In
Page 15 and 16: 16.4 The Chain Rule 375Exercises 16
Page 17 and 18: 16.4 The Chain Rule 377We can write
Page 19 and 20: 16.5 Directional Derivatives 379So
Page 21 and 22: 16.5 Directional Derivatives 381ori
Page 23: 16.6 Higher order derivatives 38320
Page 27 and 28: 16.7 Maxima and minima 387ThenA l =
Page 29 and 30: 16.8 Lagrange Multipliers 3896. Fin
Page 31 and 32: 16.8 Lagrange Multipliers 391points
Page 33 and 34: 16.8 Lagrange Multipliers 393Anothe

Chapter 16 Partial Differentiation

Create successful ePaper yourself

Delete template?

Save as template?