12.3   
1. Suppose that the function f (x, y) is a sum of terms where each term contains x or y but not both. Explain why f xy = 0.
2. In Definition 3.1, explain how to remember which partial derivative involves the term f (x+h, y) and which involves the term f (x, y+h).
3. In section 2.8, we computed derivatives implicitly, by using the chain rule and differentiating both sides of an equation with respect to x. In the process of doing so, we made calculations such as (x2y2)' = 2xy2+2x2yy'. Explain why this derivative is computed differently than the partial derivatives of this section.
4. For f (x, y, z) = x3e4xsin y+y2sin xy+4xyz, you could compute f xyz in a variety of orders. Discuss how many different orders are possible and which order(s) would be the easiest.
In exercises 5 – 16, find both first-order partial derivatives.
5. f (x, y) = x3-4xy2+y4 6. f (x, y) = x2y3-3x
7. f (x, y) = x2ey-4y 8. f (x, y) = ysin x2+x3
9. f (x, y) = x2sin xy-3y3 10.
 11.
12.
13. f (x, y, z) = 3xsin y+4x3y2z
14. f (x, y, z) = 4ysin z-3x2z
15.
16.
In exercises 17 – 26, find the indicated partial derivatives.
17. f (x, y) = x3-4xy2+3y
18. f (x, y) = x2y-4x+3sin y
19. f (x, y) = x4-3x2y3+5y f xx f xy f xyy
20. f xx f xy f yyx
21. f (x, y, z) = x3y2-sin yz f xx f yz, f xyz
22. f (x, y, z) = xy2z3-exyz f yy f zz f xyy
23. f xx f yy f yyzz
24. f (x, y, z) = ln(xyz2); f xx, f yyz, f xxyzz
25. f (w, x, y, z) = w2xy-ewz; f ww, f wxy, f wwxyz
26. ; f xx, f yy, f wxyz
In exercises 27 – 32, (a) sketch the graph of z = f (x, y) and ( b) on this graph, highlight the appropriate two-dimensional trace and interpret the partial derivative as a slope.
27. f (x, y) = 4-x2-y2,
28. ,
29. f (x, y) = 4-x2-y2,
30. ,
31. f (x, y) = 4-x2-y2,
32. ,
33. Compute and interpret for van der Waals' equation (see example 3.3).
34. For van der Waals' equation, show that . If you misunderstood the chain rule, why might you expect this product to equal to 1?
35. In example 3.7, show that .
36. In example 3.7, show that .
37. If the sag in the beam of example 3.7 were given by , determine which variable would have the greatest proportional effect.
38. Based on example 3.7 and your result in exercise 37, state a simple rule for determining which variable has the greatest proportional effect.
In exercises 39 – 42, find all points at which and interpret the significance of the points graphically.
39. f (x, y) = x2+y2 40. f (x, y) = x2+y2-x4
41. f (x, y) = sin xsin y 42.
In exercises 43 – 46, use the contour plot to estimate and at the origin.
43.


44.


45.


46.


47. The table shows wind chill (how cold it “feels'' outside) as a function of temperature (degrees Fahrenheit) and wind speed (mph). We can think of this as a function C(t, s). Estimate the partial derivatives and . Interpret each partial derivative and explain why it is surprising that .
Speed vs. Temp 30 20 10 0 -10
0 30 20 10 0 -10
5 27 16 6 -5 -15
10 16 4 -9 -24 -33
15 9 -5 -18 -32 -45
20 4 -10 -25 -39 -53
25 0 -15 -29 -44 -59
30 -2 -18 -33 -48 -63
48. Rework exercise 47 using the point (10, 20). Explain the significance of the inequality .
49. Using the baseball data in example 3.8, estimate and interpret and .
50. According to the data in example 3.8, a baseball with initial velocity 170 ft/s and backspin 3000 rpm flies 395 ft. Suppose that the ball must go 400 ft to clear the fence for a home run. Based on your answers to exercise 49, how much extra backspin is needed for a home run?
51. Carefully write down a definition for the three first-order partial derivatives of a function of three variables f (x, y, z).
52. Determine how many second-order partial derivatives there are of f (x, y, z). Assuming a result analogous to Theorem 3.1, how many of these second-order partial derivatives are actually different?
53. Show that the functions f n(x, t ) = sin n xcos n ct satisfy the wave equation for any positive integer n and any constant c.
54. Show that if f (x) is a function with a continuous second derivative, then f (x-ct) is a solution of the wave equation of exercise 53. If x represents position and t  represents time, explain why c can be interpreted as the velocity of the wave.
55. The value of an investment of $1000 invested at a constant 10% rate for 5 years is where T is the tax rate and I is the inflation rate. Compute and and discuss whether the tax rate or the inflation rate has a greater influence on the value of the investment.
56. The value of an investment of $1000 invested at a rate r for 5 years with a tax rate of 28% is where I is the inflation rate. Compute and and discuss whether the investment rate or the inflation rate has a greater influence on the value of the investment.
57. Suppose that the position of a guitar string of length L varies according to p(x, t ) = sin xcos t , where x represents the distance along the string, 0 x L, and t  represents time. Compute and interpret and .
58. Suppose that the concentration of some pollutant in a river as a function of position x and time t  is given by p(x, t ) = for constants p0, c and . Show that . Interpret both and and explain how this equation relates the change in pollution at a specific location to the current of the river and the rate at which the pollutant decays.
59. In a chemical reaction, the temperature T, entropy S, Gibbs free energy G and enthalpy H are related by G = H-TS. Show that .
60. For the chemical reaction of exercise 59, show that . Chemists measure the enthalpy of a reaction by measuring this rate of change.
61. Suppose that three resistors are in parallel in an electrical circuit. If the resistances are R1, R2 and R3 ohms, respectively, then the net resistance in the circuit equals R = Compute and interpret the partial derivative . Given this partial derivative, explain how to quickly write down the partial derivatives and .
62. The ideal gas law relating pressure, temperature and volume is for some constant c. Show that .
63. A process called tag-and-recapture is used to estimate populations of animals in the wild. First, some number T of the animals are captured, tagged and released into the wild. Later, a number S of the animals are captured, of which t are tagged. The estimate of the total population is then . Compute P(100, 60, 15); the proportion of tagged animals in the recapture is = . Based on your estimate of the total population, what proportion of the total population has been tagged? Now compute and use it to estimate
how much your population estimate would change if one more recaptured animal was tagged.
64. Suppose that L hours of labor and K dollars of investment by a company results in a productivity of P = L0.75K0.25 . Compute the marginal productivity of labor, defined by and the marginal productivity of capital, defined by
65. For the function

use the limit definitions of partial derivatives to show that f xy(0, 0) = -1 but f yx(0, 0) = 1. Determine which assumption in Theorem 3.1 is not true.
66. For , show that [Note that we have previously shown that this function is not continuous at (0, 0).]
67. Suppose that f (x, y) is a function with continuous second-order partial derivatives. Consider the curve obtained by intersecting the surface z = f (x, y) with the plane y = y0. Explain
how the slope of this curve at the point x = x0 relates to . Relate the concavity of this curve at the point x = x0 to .
68. As in exercise 67, develop a graphical interpretation of .
69. In exercises 67 and 68, you interpreted the second-order partial derivatives f xx and f yy as concavity. In this exercise, you will develop a geometric interpretation of the mixed partial derivative f xy. (More information can be found in the article “What is f xy?'' by Brian McCartin in the March 1998 issue of the journal PRIMUS.) Start by using Taylor's Theorem (see section 8.7) to show that
[Hint: Treating y as a constant, you have f (x+h, y) = f (x, y)+hfx(x, y)+h2g(x, y) for some function g(x, y). Similarly expand the other terms in the numerator.] Therefore, for small h and k, , where f 0 = f (x, y), f 1 = f (x+h, y), f 2 = f (x, y+k) and f 3 = f (x+h, y+k). The four points P0 = (x, y, f 0), P1 = (x+h, y, f 1), P2 = (x, y+k, f 2) and P3 = (x+h, y+k, f 3) determine a parallelepiped as shown in the figure.


Recalling that the volume of a parallelepiped formed by vectors a, b and c is given by a( b c), show that the volume of this box equals |(f 0-f 1-f 2+f 3)hk|. That is, the volume is approximately equal to | f xy(x, y)|(hk)2. Conclude that the larger f xy(x, y) is, the greater the volume of the box and hence, the further the point P3 is from the plane determined by the points P0, P1 and P2. To see what this means graphically, start with the function f (x, y) = x2+y2 at the point (1, 1, 2). With h = k = 0.1, show that the points (1, 1, 2), (1.1, 1, 2.21), (1, 1.1, 2.21) and (1.1, 1.1, 2.42) all lie in the same plane. The derivative f xy(1, 1) = 0 indicates that at the point (1.1, 1.1, 2.42), the graph does not curve away from the plane of the points (1, 1, 2), (1.1, 1, 2.21) and (1, 1.1, 2.21). Contrast this to the behavior of the function f (x, y) = x2+xy at the point (1, 1, 2). This says that f xy measures the amount of curving of the surface as you sequentially change x and y by small amounts.
70. A ball, such as a baseball, flying through the air encounters air resistance in the form of air drag. The magnitude of the drag force is typically the product of a number (called the drag coefficient) and the square of the velocity. The drag coefficient is not actually a constant. The figure (reprinted from Keep Your Eye on the Ball by Watts and Bahill) shows experimental data for the drag coefficient as a function of the roughness of the ball (measured by , where is the size of the bumps on the ball and D is the diameter of the ball) and the Reynolds number (Re, which is proportional to velocity). We'll call the drag coefficient f , rename and v = Re and consider f (u, v). Use the graph to estimate and and interpret each partial derivative. All golf balls have “dimples'' that make the surface of the golf ball rougher. Explain why a golf ball with dimples, traveling at a velocity corresponding to a Reynolds number of about 0.9 105, will fly much farther than a ball with no dimples.


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