Hw 4 Due Feb 22 2.2 Exercise 7,8,10,12,15,18,28,35,36,46 2.3 Exercise 3,11,39,40,47(b) 2.4 Exercise 6,7 Use both the direct method and product rule to calculate where f(x, y, z) = 3x, g(x, y, z) = ( 1 2 3 4 5 6 D(fg) ) Bonus Problem (2pts), due Feb 18, write on a separate paper In R 3, prove that if c a b, then there exist s, t R such that c = sa + tb. x y z. Do the two methods give the same result? 1
114 Chapter 2 Differentiation in Several Variables 2.2 Exercises 0 < <δ i, then f i (x) L i <ɛ/ m. Set δ = min(δ 1,...,δ m ). Then if x X and 0 < <δ, we see that (6) implies ɛ 2 f(x) L < + ɛ2 m m = m ɛ2 m = ɛ. Thus, lim x a f(x) = L. Proof of Theorem 2.8 We must show that the composite function g f is continuous at every point a X. Ifa is an isolated point of X, there is nothing to show. Otherwise, we must show that lim x a (g f)(x) = (g f)(a). Given any ɛ>0, continuity of g at f(a) implies that we can find some γ>0 such that if y range f and 0 < y f(a) <γ then g(y) g(f(a)) <ɛ. Since f is continuous at a, we can find some δ>0such that if x X and 0 < <δ, then f(x) f(a) <γ. Therefore, if x X and 0 < <δ, then g(f(x)) g(f(a)) <ɛ. In Exercises 1 6, determine whether the given set is open or closed (or neither). 1. {(x, y) R 2 1 < x 2 + y 2 < 4} 2. {(x, y) R 2 1 x 2 + y 2 4} 3. {(x, y) R 2 1 x 2 + y 2 < 4} 4. {(x, y, z) R 3 1 x 2 + y 2 + z 2 4} 5. {(x, y) R 2 1 < x < 1} {(x, y) R 2 x = 2} 6. {(x, y, z) R 3 1 < x 2 + y 2 < 4} Evaluate the limits in Exercises 7 21, or explain why the limit fails to exist. 7. lim (x,y,z) (0,0,0) x 2 + 2xy + yz + z 3 + 2 y 8. lim (x,y) (0,0) x 2 + y 2 (x + y) 2 9. lim e x e y 10. lim (x,y) (0,0) x + y + 2 2x 2 + y 2 11. lim 2x 2 + y 2 12. lim (x,y) ( 1,2) x 2 + y 2 x 2 + 2xy + y 2 13. lim (x,y) (0,0) x + y xy 14. lim x 4 y 4 15. lim x 2 16. lim x 2 xy 17. lim (x,y) (0,0),x y x y x 2 y 2 4x + 4 18. lim (x,y) (2,0) x 2 + y 2 4x + 4 19. lim (x,y,z) (0, e xz cos y 2 x π,1) 2x 2 + 3y 2 + z 2 20. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 xy xz + yz 21. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 sin θ 22. (a) What is lim? θ 0 θ sin (x + y) (b) What is lim? (x,y) (0,0) x + y sin (xy) (c) What is lim? (x,y) (0,0) xy
2.2 Exercises 115 23. Examine the behavior of f (x, y) = x 4 y 4 /(x 2 + y 4 ) 3 as (x, y) approaches (0, 0) along various straight lines. From your observations, what might you conjecture lim (x,y) (0,0) f (x, y) to be? Next, consider what happens when (x, y) approaches (0, 0) along the curve x = y 2. Does lim (x,y) (0,0) f (x, y) exist? Why or why not? In Exercises 24 27, (a) use a computer to graph z = f (x, y); (b) use your graph in part (a) to give a geometric discussion as to whether lim (x,y) (0,0) f (x, y) exists; (c) give an analytic (i.e., nongraphical) argument for your answer in part (b). T 24. f (x, y) = 4x 2 + 2xy + 5y 2 3x 2 + 5y 2 T 25. f (x, y) = x 2 y x 2 + y 2 T 26. f (x, y) = xy5 x 2 + y 10 x sin 1 if y 0 T 27. f (x, y) = y 0 if y = 0 Some limits become easier to identify if we switch to a different coordinate system. In Exercises 28 33 switch from Cartesian to polar coordinates to evaluate the given limits. In Exercises 34 37 switch to spherical coordinates. x 2 y 28. lim x 2 29. lim x 2 + xy + y 2 30. lim x 5 + y 4 3x 3 y + 2x 2 + 2y 2 31. lim x 2 y 2 32. lim (x,y) (0,0) x 2 + y 2 x + y 33. lim (x,y) (0,0) x 2 + y 2 x 2 y 34. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 xyz 35. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 x 2 + y 2 36. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 xz 37. lim (x,y,z) (0,0,0) x 2 + y 2 + z 2 In Exercises 38 45, determine whether the functions are continuous throughout their domains: 38. f (x, y) = x 2 + 2xy y 7 39. f (x, y, z) = x 2 + 3xyz + yz 3 + 2 40. g(x, y) = x 2 y 2 x 2 + 1 ( x 2 y 2 ) 41. h(x, y) = cos x 2 + 1 42. f (x, y) = cos 2 x 2 sin 2 xy x 2 y 2 if (x, y) (0, 0) 43. f (x, y) = x 2 + y 2 0 if (x, y) = (0, 0) x 3 + x 2 + xy 2 + y 2 if (x, y) (0, 0) 44. g(x, y) = x 2 + y 2 2 if(x, y) = (0, 0) ( 45. F(x, y, z) = x 2 e x e y ( )) xy + 3xy, 2x 2 + y 4 + 3, sin y 2 + 1 46. Determine the value of the constant c so that x 3 + xy 2 + 2x 2 + 2y 2 if (x, y) (0, 0) g(x, y) = x 2 + y 2 c if (x, y) = (0, 0) is continuous. 47. Show that the function f : R 3 R given by f (x) = (2i 3j + k) x is continuous. 48. Show that the function f: R 3 R 3 given by f(x) = (6i 5k) x is continuous. Exercises 49 53 involve Definition 2.2 of the limit. 49. Consider the function f (x) = 2x 3. (a) Show that if x 5 <δ, then f (x) 7 < 2δ. (b) Use part (a) to prove that lim x 5 f (x) = 7. 50. Consider the function f (x, y) = 2x 10y + 3. (a) Show that if (x, y) (5, 1) <δ, then x 5 < δ and y 1 <δ. (b) Use part (a) to show that if (x, y) (5, 1) <δ, then f (x, y) 3 < 12δ. (c) Show that lim (x,y) (5,1) f (x, y) = 3. 51. If A, B, and C are constants and f (x, y) = Ax + By + C, show that lim f (x, y) = f (x 0, y 0 ) = Ax 0 + By 0 + C. (x,y) (x 0,y 0 )
2.3 Exercises 131 differentiable.) Now, we consider f(x) f(a) Df(a)(x a) = (G 1, G 2,...,G m ) ( G 2 = 1 + G 2 1/2 2 m) + +G2 G 1 + G 2 + + G m = G 1 + G 2 + + G m. As x a, each term G i / 0, by definition of G i in equation (14) and the differentiability of the component functions f i of f. Hence, equation (13) holds and f is differentiable at a. (To see that (G 2 1 + +G2 m )1/2 G 1 + + G m, note that ( G 1 + + G m ) 2 = G 1 2 + + G m 2 + 2 G 1 G 2 + 2 G 1 G 3 + +2 G m 1 G m G 1 2 + + G m 2. Then, taking square roots provides the inequality.) Proof of Theorem 3.11 In the final paragraph of the proof of Theorem 3.10, we showed that f(x) f(a) Df(a)(x a) G 1 + G 2 + + G m, where G i = f i (x) f i (a) Df i (a)(x a) as in equation (14). From this, it follows immediately that differentiability of the component functions f 1,..., f m at a implies differentiability of f at a. Conversely, for i = 1,...,m, f(x) f(a) Df(a)(x a) = (G 1, G 2,...,G m ) G i. Hence, differentiability of f at a forces differentiability of each component function. 2.3 Exercises In Exercises 1 9, calculate f/ x and f/ y. 1. f (x, y) = xy 2 + x 2 y 2. f (x, y) = e x2 +y 2 3. f (x, y) = sin xy + cos xy 4. f (x, y) = x 3 y 2 1 + x 2 + 3y 4 5. f (x, y) = x 2 y 2 x 2 + y 2 6. f (x, y) = ln (x 2 + y 2 ) 7. f (x, y) = cos x 3 y 8. ( ) x f (x, y) = ln y 9. f (x, y) = xe y + y sin (x 2 + y) In Exercises 10 17, evaluate the partial derivatives F/ x, F/ y, and F/ z for the given functions F. 10. F(x, y, z) = x + 3y 2z
132 Chapter 2 Differentiation in Several Variables 11. F(x, y, z) = x y y + z 12. F(x, y, z) = xyz 13. F(x, y, z) = x 2 + y 2 + z 2 14. F(x, y, z) = e ax cos by + e az sin bx x + y + z 15. F(x, y, z) = (1 + x 2 + y 2 + z 2 ) 3/2 16. F(x, y, z) = sin x 2 y 3 z 4 17. F(x, y, z) = x 3 + yz x 2 + z 2 + 1 Find the gradient f (a), where f and a are given in Exercises 18 25. 18. f (x, y) = x 2 y + e y/x, a = (1, 0) 19. x y f (x, y) =, x 2 + y 2 + 1 a = (2, 1) 20. f (x, y, z) = sin xyz, a = (π, 0,π/2) 21. f (x, y, z) = xy+ y cos z x sin yz, a = (2, 1,π) 22. f (x, y) = e xy + ln (x y), a = (2, 1) 23. f (x, y, z) = x + y e z, a = (3, 1, 0) 24. f (x, y, z) = cos z ln (x + y 2 ), a = (e, 0,π/4) 25. f (x, y, z) = xy2 x 2 z, a = ( 1, 2, 1) y 2 + z 2 + 1 In Exercises 26 33, find the matrix Df(a) of partial derivatives, where f and a are as indicated. 26. f (x, y) = x, a = (3, 2) y 27. f (x, y, z) = x 2 + x ln (yz), a = ( 3, e, e) 28. f(x, y, z) = ( 2x 3y + 5z, x 2 + y, ln (yz) ), a = (3, 1, 2) 29. f(x, y, z) = (xyz, ) x 2 + y 2 + z 2, a = (1, 0, 2) 30. f(t) = (t, cos 2t, sin 5t), a = 0 31. f(x, y, z,w) = (3x 7y + z, 5x + 2z 8w, y 17z + 3w), a = (1, 2, 3, 4) 32. f(x, y) = (x 2 y, x + y 2, cos π xy), a = (2, 1) 33. f(s, t) = (s 2, st, t 2 ), a = ( 1, 1) Explain why each of the functions given in Exercises 34 36 is differentiable at every point in its domain. 34. f (x, y) = xy 7x 8 y 2 + cos x 35. f (x, y, z) = x + y + z x 2 + y 2 + z 2 ( xy 2 36. f(x, y) = x 2 + y, x 4 y + y ) x 37. (a) Explain why the graph of z = x 3 7xy + e y has a tangent plane at ( 1, 0, 0). (b) Give an equation for this tangent plane. 38. Find an equation for the plane tangent to the graph of z = 4 cos xy at the point (π/3, 1, 2). 39. Find an equation for the plane tangent to the graph of z = e x+y cos xy at the point (0, 1, e). 40. Find equations for the planes tangent to z = x 2 6x + y 3 that are parallel to the plane 4x 12y + z = 7. 41. Use formula (8) to find an equation for the hyperplane tangent to the 4-dimensional paraboloid x 5 = 10 (x1 2 + 3x 2 2 + 2x 3 2 + x 4 2 ) at the point (2, 1, 1, 3, 8). 42. Suppose that you have the following information concerning a differentiable function f : f (2, 3) = 12, f (1.98, 3) = 12.1, f (2, 3.01) = 12.2. (a) Give an approximate equation for the plane tangent to the graph of f at (2, 3, 12). (b) Use the result of part (a) to estimate f (1.98, 2.98). In Exercises 43 45, (a) use the linear function h(x) in Definition 3.8 to approximate the indicated value of the given function f. (b) How accurate is the approximation determined in part (a)? 43. f (x, y) = e x+y, f (0.1, 0.1) 44. f (x, y) = 3 + cos π xy, f (0.98, 0.51) 45. f (x, y, z) = x 2 + xyz + y 3 z, f (1.01, 1.95, 2.2) 46. Calculate the partial derivatives of f (x 1, x 2,...,x n ) = x 1 + x 2 + +x n. x1 2 + x 2 2 + +x n 2 47. Let xy 2 x 2 y + 3x 3 y 3 if (x, y) (0, 0) f (x, y) = x 2 + y 2. 0 if(x, y) = (0, 0) (a) Calculate f/ x and f/ y for (x, y) (0, 0). (You may wish to use a computer algebra system for this part.) (b) Find f x (0, 0) and f y (0, 0). As mentioned in the text, if a function F(x) of a single variable is differentiable at a, then, as we zoom in on the point (a, F(a)), the graph of y = F(x) will straighten out and look like its tangent line at (a, F(a)). For the differentiable functions given
2.4 Exercises 141 2.4 Exercises In Exercises 1 4, verify the sum rule for derivative matrices (i.e., part 1 of Proposition 4.1) for each of the given pairs of functions: 1. f (x, y) = xy + cos x, g(x, y) = sin (xy) + y 3 2. f(x, y) = (e x+y, xe y ), g(x, y) = (ln (xy), ye x ) 3. f(x, y, z) = (x sin y + z, ye z 3x 2 ), g(x, y, z) = (x 3 cos x, xyz) 4. f(x, y, z) = (xyz 2, xe y, y sin xz), g(x, y, z) = (x y, x 2 + y 2 + z 2, ln (xz + 2)) Verify the product and quotient rules (Proposition 4.2) for the pairs of functions given in Exercises 5 8. 5. f (x, y) = x 2 y + y 3, g(x, y) = x y 6. f (x, y) = e xy, g(x, y) = x sin 2y 7. f (x, y) = 3xy + y 5, g(x, y) = x 3 2xy 2 8. f (x, y, z) = x cos (yz), g(x, y, z) = x 2 + x 9 y 2 + y 2 z 3 + 2 For the functions given in Exercises 9 17 determine all secondorder partial derivatives (including mixed partials). 9. f (x, y) = x 3 y 7 + 3xy 2 7xy 10. f (x, y) = cos (xy) 11. f (x, y) = e y/x ye x 12. f (x, y) = sin x 2 + y 2 1 13. f (x, y) = sin 2 x + 2e y 14. f (x, y) = e x2 +y 2 15. f (x, y) = y sin x x cos y ( ) x 16. f (x, y) = ln y 17. f (x, y) = x 2 e y + e 2z 18. f (x, y, z) = x y y + z 19. f (x, y, z) = x 2 yz + xy 2 z + xyz 2 20. f (x, y, z) = e xyz 21. f (x, y, z) = e ax sin y + e bx cos z 22. Consider the function F(x, y, z) = 2x 3 y + xz 2 + y 3 z 5 7xyz. (a) Find F xx, F yy, and F zz. (b) Calculate the mixed second-order partials F xy, F yx, F xz, F zx, F yz, and F zy, and verify Theorem 4.3. (c) Is F xyx = F xxy? Could you have known this without resorting to calculation? (d) Is F xyz = F yzx? 23. Let f (x, y) = ye 3x. Give general formulas for n f/ x n and n f/ y n, where n 2. 24. Let f (x, y, z) = xe 2y + ye 3z + ze x. Give general formulas for n f/ x n, n f/ y n, and n f/ z n, where n 1. ( xy ) 25. Let f (x, y, z) = ln. Give general formulas for z n f/ x n, n f/ y n, and n f/ z n, where n 1. What can you say about the mixed partial derivatives? 26. Let f (x, y, z) = x 7 y 2 z 3 2x 4 yz. (a) What is 4 f/ x 2 y z? (b) What is 5 f/ x 3 y z? (c) What is 15 f/ x 13 y z? 27. Recall from 2.2 that a polynomial in two variables x and y is an expression of the form p(x, y) = d c kl x k y l, k,l=0 where c kl can be any real number for 0 k, l d. The degree of the term c kl x k y l when c kl 0isk + l and the degree of the polynomial p is the largest degree of any nonzero term of the polynomial (i.e., the largest degree of any term for which c kl 0). For example, the polynomial p(x, y) = 7x 6 y 9 + 2x 2 y 3 3x 4 5xy 3 + 1 has five terms of degrees 15, 5, 4, 4, and 0. The degree of p is therefore 15. (Note: The degree of the zero polynomial p(x, y) 0 is undefined.) (a) If p(x, y) = 8x 7 y 10 9x 2 y + 2x, what is the degree of p/ x? p/ y? 2 p/ x 2? 2 p/ y 2? 2 p/ x y? (b) If p(x, y) = 8x 2 y + 2x 3 y, what is the degree of p/ x? p/ y? 2 p/ x 2? 2 p/ y 2? 2 p/ x y? (c) Try to formulate and prove a conjecture relating the degree of a polynomial p to the degree of its partial derivatives. 28. The partial differential equation 2 f x + 2 f 2 y + 2 f 2 z = 0 2 is known as Laplace s equation, after Pierre Simon de Laplace (1749 1827). Any function f of class C 2