Section 9.3
Partial Derivatives
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Recall - Contours and Level Curves
Example
g 9x y
2
2
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Example
Boxes 1 and 2 on page 361
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
Visualizing
Tangent Plane
Notation and Rules
RECALL
Example from Instructor’s manual from 9.3
𝑓′(10) is between
65−60
10−0
= 0.5 and
68−65
20−10
= 0.3 yielding a best estimate 0.4
℉
min
.
At 12:10PM, the temperature is increasing at a rate of 0.4 °F per minute.
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
EXTENSION
TO FUNCTIONS
TWOfrom
VARIABLES
Example from
Instructor’s OF
manual
9.3
The following table gives the number of calories burned per minute B = f(s,w) for someone who is
rollerblading, as a function of the person's weight, w, and speed, s.
w\s
8 mph
9 mph
10 mph
11 mph
120 lbs
4.2
5.8
7.4
8.9
140 lbs
5.1
6.7
8.3
9.9
160 lbs
6.1
7.7
9.2
10.8
180 lbs
7.0
8.6
10.2
11.7
200 lbs
7.9
9.5
11.1
12.6
Estimate fw(160,10) and fs(160,10) and interpret your answers.
𝑓𝑤 (160, 10) is between
𝑓𝑠 (160, 10) is between
9.2−8.3
160−140
9.2−7.7
10−9
= 0.045 and
= 1.5 and
10.2−9.2
180−160
10.8−9.2
11−10
= 0.050 yielding an estimate of 0.0475 cal/min per lb.
= 1.6 yielding an estimate of 1.55 cal/min per mph.
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
PROBLEMS
Use the level curves of f (x, y) in Figure 9.4 to estimate
(a) fx (2, 1)
(b) fy (1, 2)
𝑓𝑥 2, 1 ≈ −0.8
𝑓𝑦 1, 2 ≈ −2.2
PROBLEMS
Use the level curves of f (x, y) in Figure 9.4 to estimate
(a) fx (2, 1)
(b) fy (1, 2)
Approximations
Box on page 363 and Problem 17
𝑓 52, 108 ≈ 5.67 + 0.60 52 − 50 − 0.15 108 − 100 = 5.67
Applied Calculus, 3/E by Deborah Hughes-Hallet
Copyright 2006 by John Wiley & Sons. All rights reserved.
PROBLEMS
Problems 1 – 2 concern the contour diagram for a function f (x, y) in Figure 9.3. At
the point P, which of the following is true?
(a)
(b)
(c)
(d)
fx > 0, fy > 0
fx > 0, fy < 0
fx < 0, fy > 0
fx < 0, fy < 0
PROBLEMS
Figure 9.5 shows level curves of f (x, y). At which of the following points is one or
both of the partial derivatives, fx, fy, approximately zero? Which one(s)?
(a)
(b)
(c)
(d)
(1, –0.5)
(–0.4, 1.5)
(1.5, –0.4)
(–0.5, 1)
(a) At (1, –0.5), the level curve is approximately
horizontal, so fx ≈ 0, but fy ≠ 0.
(b) At (–0.4, 1.5), neither are approximately zero.
(c) At (1.5, –0.4), neither are approximately zero.
(d) At (–0.5, 1), the level curve is approximately
vertical, so fy ≈ 0, but fx ≠ 0.
PROBLEMS
Figure 9.6 is a contour diagram for f (x, y) with the x and y axes in the usual directions.
Is fx (P) positive, negative, or zero? Is fxx (P) positive, negative, or zero? Is fy (P)
positive, negative, or zero?
At P, the values of f are increasing at an
increasing rate as we move in the positive xdirection, so fx (P) > 0, and fxx (P) > 0.
At P, the values of f are not changing as we
move in the positive y-direction, so fy (P) = 0.
PROBLEMS
Figure 9.7 is a contour diagram for f (x, y) with the x
and y axes in the usual directions. At the point P, if x
increases, what is true of fx (P)? If y increases, what is
true of fy (P)?
(a)
(b)
(c)
(d)
(e)
Have the same sign and both increase.
Have the same sign and both decrease.
Have opposite signs and both increase.
Have opposite signs and both decrease.
None of the above.
We have fx (P) > 0 and decreasing, and fy (P) < 0 and decreasing, so the answer is (d).