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

D(fg)

where f(x, y, z) = 3x, g(x, y, z) =(

1 2 34 5 6

) xyz

. Do the two methods give the same result?

Bonus Problem (2pts), due Feb 18, write on a separate paper

In R3, prove that if c ⊥ a× b, then there exist s, t ∈ R such that c = sa+ tb.

1

114 Chapter 2 Differentiation in Several Variables

0 < ‖x − a‖ < δi , then | fi (x) − Li | < ε/√

m. Set δ = min(δ1, . . . , δm). Then ifx ∈ X and 0 < ‖x − a‖ < δ, we see that (6) implies

‖f(x) − L‖ <

√ε2

m+ · · · ε2

m=

√m

ε2

m= ε.

Thus, limx→a f(x) = L. ■

Proof of Theorem 2.8 We must show that the composite function g ◦ f is con-tinuous at every point a ∈ X . If a is an isolated point of X , there is nothing toshow. Otherwise, we must show that limx→a(g ◦ f)(x) = (g ◦ f)(a).

Given any ε > 0, continuity of g at f(a) implies that we can find some γ > 0such that if y ∈ range f and 0 < ‖y − f(a)‖ < γ then

‖g(y) − g(f(a))‖ < ε.

Since f is continuous at a, we can find some δ > 0 such that if x ∈ X and 0 <

‖x − a‖ < δ, then

‖f(x) − f(a)‖ < γ.

Therefore, if x ∈ X and 0 < ‖x − a‖ < δ, then

‖g(f(x)) − g(f(a))‖ < ε. ■

2.2 Exercises

In Exercises 1–6, determine whether the given set is open orclosed (or neither).

1. {(x, y) ∈ R2 | 1 < x2 + y2 < 4}2. {(x, y) ∈ R2 | 1 ≤ x2 + y2 ≤ 4}3. {(x, y) ∈ R2 | 1 ≤ x2 + y2 < 4}4. {(x, y, z) ∈ R3 | 1 ≤ x2 + y2 + z2 ≤ 4}5. {(x, y) ∈ R2 | −1 < x < 1} ∪ {(x, y) ∈ R2 | x = 2}6. {(x, y, z) ∈ R3 | 1 < x2 + y2 < 4}

Evaluate the limits in Exercises 7–21, or explain why the limitfails to exist.

7. lim(x,y,z)→(0,0,0)

x2 + 2xy + yz + z3 + 2

8. lim(x,y)→(0,0)

|y|√x2 + y2

9. lim(x,y)→(0,0)

(x + y)2

x2 + y2

10. lim(x,y)→(0,0)

ex ey

x + y + 2

11. lim(x,y)→(0,0)

2x2 + y2

x2 + y2

12. lim(x,y)→(−1,2)

2x2 + y2

x2 + y2

13. lim(x,y)→(0,0)

x2 + 2xy + y2

x + y

14. lim(x,y)→(0,0)

xy

x2 + y2

15. lim(x,y)→(0,0)

x4 − y4

x2 + y2

16. lim(x,y)→(0,0)

x2

x2 + y2

17. lim(x,y)→(0,0),x �=y

x2 − xy√x − √

y

18. lim(x,y)→(2,0)

x2 − y2 − 4x + 4

x2 + y2 − 4x + 4

19. lim(x,y,z)→(0,

√π,1)

exz cos y2 − x

20. lim(x,y,z)→(0,0,0)

2x2 + 3y2 + z2

x2 + y2 + z2

21. lim(x,y,z)→(0,0,0)

xy − xz + yz

x2 + y2 + z2

22. (a) What is limθ→0

sin θ

θ?

(b) What is lim(x,y)→(0,0)

sin (x + y)

x + y?

(c) What is lim(x,y)→(0,0)

sin (xy)

xy?

2.2 Exercises 115

23. Examine the behavior of f (x, y) = x4 y4/(x2 + y4)3

as (x, y) approaches (0, 0) along various straight lines.From your observations, what might you conjecturelim(x,y)→(0,0) f (x, y) to be? Next, consider what hap-pens when (x, y) approaches (0, 0) along the curvex = y2. Does lim(x,y)→(0,0) f (x, y) exist? Why or whynot?

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 discussionas 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) = 4x2 + 2xy + 5y2

3x2 + 5y2

◆T 25. f (x, y) = x2 − y

x2 + y2

◆T 26. f (x, y) = xy5

x2 + y10

◆T 27. f (x, y) =

⎧⎪⎨⎪⎩

x sin1

yif y �= 0

0 if y = 0

Some limits become easier to identify if we switch to a differentcoordinate system. In Exercises 28–33 switch from Cartesian topolar coordinates to evaluate the given limits. In Exercises 34–37 switch to spherical coordinates.

28. lim(x,y)→(0,0)

x2 y

x2 + y2

29. lim(x,y)→(0,0)

x2

x2 + y2

30. lim(x,y)→(0,0)

x2 + xy + y2

x2 + y2

31. lim(x,y)→(0,0)

x5 + y4 − 3x3 y + 2x2 + 2y2

x2 + y2

32. lim(x,y)→(0,0)

x2 − y2√x2 + y2

33. lim(x,y)→(0,0)

x + y√x2 + y2

34. lim(x,y,z)→(0,0,0)

x2 y

x2 + y2 + z2

35. lim(x,y,z)→(0,0,0)

xyz

x2 + y2 + z2

36. lim(x,y,z)→(0,0,0)

x2 + y2√x2 + y2 + z2

37. lim(x,y,z)→(0,0,0)

xz

x2 + y2 + z2

In Exercises 38–45, determine whether the functions are con-tinuous throughout their domains:

38. f (x, y) = x2 + 2xy − y7

39. f (x, y, z) = x2 + 3xyz + yz3 + 2

40. g(x, y) = x2 − y2

x2 + 1

41. h(x, y) = cos

(x2 − y2

x2 + 1

)

42. f (x, y) = cos2 x − 2 sin2 xy

43. f (x, y) =

⎧⎪⎨⎪⎩

x2 − y2

x2 + y2if (x, y) �= (0, 0)

0 if (x, y) = (0, 0)

44. g(x, y) =

⎧⎪⎨⎪⎩

x3 + x2 + xy2 + y2

x2 + y2if (x, y) �= (0, 0)

2 if (x, y) = (0, 0)

45. F(x, y, z) =(

x2 + 3xy,ex ey

2x2 + y4 + 3, sin

(xy

y2 + 1

))

46. Determine the value of the constant c so that

g(x, y) =

⎧⎪⎨⎪⎩

x3 + xy2 + 2x2 + 2y2

x2 + y2if (x, y) �= (0, 0)

c if (x, y) = (0, 0)

is continuous.

47. Show that the function f : R3 → R given by f (x) =(2i − 3j + k) ·x is continuous.

48. Show that the function f: R3 → R3 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 limx→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(x,y)→(x0,y0)

f (x, y) = f (x0, y0) = Ax0 + By0 + C.

2.3 Exercises 131

differentiable.) Now, we consider

‖f(x) − f(a) − Df(a)(x − a)‖‖x − a‖ = ‖(G1, G2, . . . , Gm)‖

‖x − a‖

=(G2

1 + G22 + · · · + G2

m

)1/2

‖x − a‖

≤ |G1| + |G2| + · · · + |Gm |‖x − a‖

= |G1|‖x − a‖ + |G2|

‖x − a‖ + · · · + |Gm |‖x − a‖ .

As x → a, each term |Gi |/‖x − a‖ → 0, by definition of Gi in equation (14) andthe differentiability of the component functions fi of f. Hence, equation (13) holdsand f is differentiable at a. (To see that (G2

1 + · · · + G2m)1/2 ≤ |G1| + · · · + |Gm |,

note that

(|G1| + · · · + |Gm |)2 = |G1|2 + · · · + |Gm |2

+ 2|G1| |G2| + 2|G1| |G3| + · · · + 2|Gm−1| |Gm |≥ |G1|2 + · · · + |Gm |2.

Then, taking square roots provides the inequality.) ■

Proof of Theorem 3.11 In the final paragraph of the proof of Theorem 3.10, weshowed that

‖f(x) − f(a) − Df(a)(x − a)‖‖x − a‖ ≤ |G1|

‖x − a‖ + |G2|‖x − a‖ + · · · + |Gm |

‖x − a‖ ,

where Gi = fi (x) − fi (a) − D fi (a)(x − a) as in equation (14). From this, it fol-lows immediately that differentiability of the component functions f1, . . . , fm ata implies differentiability of f at a. Conversely, for i = 1, . . . , m,

‖f(x) − f(a) − Df(a)(x − a)‖‖x − a‖ = ‖(G1, G2, . . . , Gm)‖

‖x − a‖ ≥ |Gi |‖x − a‖ .

Hence, differentiability of f at a forces differentiability of each componentfunction. ■

2.3 Exercises

In Exercises 1–9, calculate ∂ f/∂x and ∂ f/∂y.

1. f (x, y) = xy2 + x2 y

2. f (x, y) = ex2+y2

3. f (x, y) = sin xy + cos xy

4. f (x, y) = x3 − y2

1 + x2 + 3y4

5. f (x, y) = x2 − y2

x2 + y2

6. f (x, y) = ln (x2 + y2)

7. f (x, y) = cos x3 y

8. f (x, y) = ln

(x

y

)

9. f (x, y) = xey + y sin (x2 + 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) =√

x2 + y2 + z2

14. F(x, y, z) = eax cos by + eaz sin bx

15. F(x, y, z) = x + y + z

(1 + x2 + y2 + z2)3/2

16. F(x, y, z) = sin x2 y3z4

17. F(x, y, z) = x3 + yz

x2 + z2 + 1

Find the gradient ∇ f (a), where f and a are given in Exer-cises 18–25.

18. f (x, y) = x2 y + ey/x , a = (1, 0)

19. f (x, y) = x − y

x2 + y2 + 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) = exy + ln (x − y), a = (2, 1)

23. f (x, y, z) = x + y

ez, a = (3, −1, 0)

24. f (x, y, z) = cos z ln (x + y2), a = (e, 0, π/4)

25. f (x, y, z) = xy2 − x2z

y2 + z2 + 1, a = (−1, 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

y, a = (3, 2)

27. f (x, y, z) = x2 + x ln (yz), a = (−3, e, e)

28. f(x, y, z) = (2x − 3y + 5z, x2 + y, ln (yz)

),

a = (3, −1, −2)

29. f(x, y, z) =(

xyz,√

x2 + y2 + z2)

,

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) = (x2 y, x + y2, cos πxy), a = (2, −1)

33. f(s, t) = (s2, st, t2), a = (−1, 1)

Explain why each of the functions given in Exercises 34–36 isdifferentiable at every point in its domain.

34. f (x, y) = xy − 7x8 y2 + cos x

35. f (x, y, z) = x + y + z

x2 + y2 + z2

36. f(x, y) =(

xy2

x2 + y4,

x

y+ y

x

)

37. (a) Explain why the graph of z = x3 − 7xy + ey hasa 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 ofz = 4 cos xy at the point (π/3, 1, 2).

39. Find an equation for the plane tangent to the graph ofz = ex+y cos xy at the point (0, 1, e).

40. Find equations for the planes tangent to z =x2 − 6x + y3 that are parallel to the plane4x − 12y + z = 7.

41. Use formula (8) to find an equation for the hy-perplane tangent to the 4-dimensional paraboloidx5 = 10 − (x2

1 + 3x22 + 2x2

3 + x24 ) at the point

(2, −1, 1, 3, −8).

42. Suppose that you have the following information con-cerning 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 tangentto 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 Def-inition 3.8 to approximate the indicated value of the givenfunction f . (b) How accurate is the approximation determinedin part (a)?

43. f (x, y) = ex+y , f (0.1, −0.1)

44. f (x, y) = 3 + cos πxy, f (0.98, 0.51)

45. f (x, y, z) = x2 + xyz + y3z, f (1.01, 1.95, 2.2)

46. Calculate the partial derivatives of

f (x1, x2, . . . , xn) = x1 + x2 + · · · + xn√x2

1 + x22 + · · · + x2

n

.

47. Let

f (x, y) =

⎧⎪⎨⎪⎩

xy2 − x2 y + 3x3 − y3

x2 + y2if (x, y) �= (0, 0)

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 systemfor this part.)

(b) Find fx (0, 0) and fy(0, 0).

As mentioned in the text, if a function F(x) of a single variableis 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 itstangent 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 offunctions:

1. f (x, y) = xy + cos x, g(x, y) = sin (xy) + y3

2. f(x, y) = (ex+y, xey), g(x, y) = (ln (xy), yex )

3. f(x, y, z) = (x sin y + z, yez − 3x2), g(x, y, z) =(x3 cos x, xyz)

4. f(x, y, z) = (xyz2, xe−y, y sin xz), g(x, y, z) =(x − y, x2 + y2 + z2, ln (xz + 2))

Verify the product and quotient rules (Proposition 4.2) for thepairs of functions given in Exercises 5–8.

5. f (x, y) = x2 y + y3, g(x, y) = x

y

6. f (x, y) = exy, g(x, y) = x sin 2y

7. f (x, y) = 3xy + y5, g(x, y) = x3 − 2xy2

8. f (x, y, z) = x cos (yz),

g(x, y, z) = x2 + x9 y2 + y2z3 + 2

For the functions given in Exercises 9–17 determine all second-order partial derivatives (including mixed partials).

9. f (x, y) = x3 y7 + 3xy2 − 7xy

10. f (x, y) = cos (xy)

11. f (x, y) = ey/x − ye−x

12. f (x, y) = sin√

x2 + y2

13. f (x, y) = 1

sin2 x + 2ey

14. f (x, y) = ex2+y2

15. f (x, y) = y sin x − x cos y

16. f (x, y) = ln

(x

y

)

17. f (x, y) = x2ey + e2z

18. f (x, y, z) = x − y

y + z

19. f (x, y, z) = x2 yz + xy2z + xyz2

20. f (x, y, z) = exyz

21. f (x, y, z) = eax sin y + ebx cos z

22. Consider the function F(x, y, z) = 2x3 y + xz2 +y3z5 − 7xyz.

(a) Find Fxx , Fyy , and Fzz .

(b) Calculate the mixed second-order partials Fxy ,Fyx , Fxz , Fzx , Fyz , and Fzy , and verify Theorem4.3.

(c) Is Fxyx = Fxxy? Could you have known this with-out resorting to calculation?

(d) Is Fxyz = Fyzx ?

23. Let f (x, y) = ye3x . Give general formulas for∂n f/∂xn and ∂n f/∂yn , where n ≥ 2.

24. Let f (x, y, z) = xe2y + ye3z + ze−x . Give generalformulas for ∂n f/∂xn , ∂n f/∂yn , and ∂n f/∂zn , wheren ≥ 1.

25. Let f (x, y, z) = ln( xy

z

). Give general formulas for

∂n f/∂xn , ∂n f/∂yn , and ∂n f/∂zn , where n ≥ 1. Whatcan you say about the mixed partial derivatives?

26. Let f (x, y, z) = x7 y2z3 − 2x4 yz.

(a) What is ∂4 f/∂x2∂y∂z?

(b) What is ∂5 f/∂x3∂y∂z?

(c) What is ∂15 f/∂x13∂y∂z?

27. Recall from §2.2 that a polynomial in two variables xand y is an expression of the form

p(x, y) =d∑

k,l=0

ckl xk yl ,

where ckl can be any real number for 0 ≤ k, l ≤ d. Thedegree of the term ckl xk yl when ckl �= 0 is k + l andthe degree of the polynomial p is the largest degreeof any nonzero term of the polynomial (i.e., the largestdegree of any term for which ckl �= 0). For example,the polynomial

p(x, y) = 7x6 y9 + 2x2 y3 − 3x4 − 5xy3 + 1

has five terms of degrees 15, 5, 4, 4, and 0. The de-gree of p is therefore 15. (Note: The degree of the zeropolynomial p(x, y) ≡ 0 is undefined.)

(a) If p(x, y) = 8x7 y10 − 9x2 y + 2x , what is the de-gree of ∂p/∂x? ∂p/∂y? ∂2 p/∂x2? ∂2 p/∂y2?∂2 p/∂x∂y?

(b) If p(x, y) = 8x2 y + 2x3 y, what is the degree of∂p/∂x? ∂p/∂y? ∂2 p/∂x2? ∂2 p/∂y2? ∂2 p/∂x∂y?

(c) Try to formulate and prove a conjecture relatingthe degree of a polynomial p to the degree of itspartial derivatives.

28. The partial differential equation

∂2 f

∂x2+ ∂2 f

∂y2+ ∂2 f

∂z2= 0

is known as Laplace’s equation, after Pierre Simonde Laplace (1749–1827). Any function f of class C2