Lecture 8.
Functions of Several Variables: Partial
Derivatives 2 (§11.3)
February 21, 2012
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Partial Derivatives: Limit Definitions
I If f is a function of two variables, its partial derivatives are the
functions fx , and fy , defined by
f (x + h, y ) − f (x, y )
fx (x, y ) = lim
h→0 h
f (x, y + h) − f (x, y )
fy (x, y ) = lim
h→0 h
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Partial Derivatives: Limit Definitions
I If f is a function of two variables, its partial derivatives are the
functions fx , and fy , defined by
f (x + h, y ) − f (x, y )
fx (x, y ) = lim
h→0 h
f (x, y + h) − f (x, y )
fy (x, y ) = lim
h→0 h
I Example: Estimate fx (3, 2) and fy (3, 2).
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Partial Derivatives: Limit Definitions
I If f is a function of two variables, its partial derivatives are the
functions fx , and fy , defined by
f (x + h, y ) − f (x, y )
fx (x, y ) = lim
h→0 h
f (x, y + h) − f (x, y )
fy (x, y ) = lim
h→0 h
I Example: Estimate fx (3, 2) and fy (3, 2).
I Practice
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives
I Slope of a tangent line
I Rate of change
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Slope of a tangent
line
I Example 2 on page 611, §11.3
I Practice: The plane x = 1 intersects the paraboloid
z = x 2 + y 2 in a parabola. Find the slope of the tangent to
the parabola at (1, 2, 5).
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
The wave heights h in the open sea depend on the speed v of the
wind and the length of time t that the wind has been blowing at
that speed. Values of the function h = f (v , t) are recorded in feet:
v (↓);t(→) 5 10 15 20 30 40 50
10 2 2 2 2 2 2 2
15 4 4 5 5 5 5 5
20 5 7 8 8 9 9 9
30 9 13 16 17 18 19 19
40 14 21 25 28 31 33 33
50 19 29 36 40 45 48 50
60 24 37 47 54 62 67 69
(a) What are the meanings of the partial derivatives ∂h/∂v and
∂h/∂t?
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
I ∂h/∂v represents the rate of change of h
when we fix t and consider h as a function of v , which describes
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
I ∂h/∂v represents the rate of change of h
when we fix t and consider h as a function of v , which describes
How quickly the wave heights change when the wind
speed changes for a fixed time duration.
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
I ∂h/∂v represents the rate of change of h
when we fix t and consider h as a function of v , which describes
How quickly the wave heights change when the wind
speed changes for a fixed time duration.
I ∂h/∂t represents the rate of change of h
when we fix v and consider h as a function of t, which describes
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
I ∂h/∂v represents the rate of change of h
when we fix t and consider h as a function of v , which describes
How quickly the wave heights change when the wind
speed changes for a fixed time duration.
I ∂h/∂t represents the rate of change of h
when we fix v and consider h as a function of t, which describes
How quickly the wave heights change when the
duration of time changes, but the wind speed is
constant.
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
The wave heights h in the open sea depend on the speed v of the
wind and the length of time t that the wind has been blowing at
that speed. Values of the function h = f (v , t) are recorded in feet:
v (↓);t(→) 5 10 15 20 30 40 50
10 2 2 2 2 2 2 2
15 4 4 5 5 5 5 5
20 5 7 8 8 9 9 9
30 9 13 16 17 18 19 19
40 14 21 25 28 31 33 33
50 19 29 36 40 45 48 50
60 24 37 47 54 62 67 69
(b) Estimate the values of fv (40, 15) and ft (40, 15). What are the
practical interpretation of these values?
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives
I
1 f (40 + 10, 15) − f (40, 15)
fv (40, 15) ≈
2 10
f (40 − 10, 15) − f (40, 15)
+
−10
1 36 − 25 16 − 25 1
= + = [1.1 + 0.9] = 1.0
2 10 −10 2
Thus, fv (40, 15) ≈ 1.0. This says that, when a 40-knot wind
has been blowing for 15 hours, the wave heights should
increase by about 1 foot for every knot that the wind speed
increases. (With the same time duration.)
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives
I Similarly,
1 f (40, 15 + 5) − f (40, 15)
ft (40, 15) ≈
2 5
f (40, 15 − 5) − f (40, 15)
+
−5
1 28 − 25 21 − 25 1
= + = [0.6 + 0.8] = 0.7
2 5 −5 2
Thus, ft (40, 15) ≈ 0.7. This says that, when a 40-knot wind
has been blowing for 15 hours, the wave heights should
increase by about 0.7 feet for every additional hour that the
wind blows.
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: Rate of change
The wave heights h in the open sea depend on the speed v of the
wind and the length of time t that the wind has been blowing at
that speed. Values of the function h = f (v , t) are recorded in feet
in the following table.
v (↓);t(→) 5 10 15 20 30 40 50
10 2 2 2 2 2 2 2
15 4 4 5 5 5 5 5
20 5 7 8 8 9 9 9
30 9 13 16 17 18 19 19
40 14 21 25 28 31 33 33
50 19 29 36 40 45 48 50
60 24 37 47 54 62 67 69
∂h
(c) What appears to be the value of the limit limt→∞ ∂t ?
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives
(c) For a fixed value of v , the function values f (v , t) appear to
incease in increasingly smaller increments, becoming nearly
constant as t increases. Thus, the correspoding rate of
change is nearly 0 as t increases, suggesting that
∂h
lim = 0.
t→∞ ∂t
Lecture 8. Functions of Several Variables: Partial Derivatives 2
Interpretations of Partial Derivatives: More Example
I Level curves are shown for a function f . Determine whether
the following partial derivatives are positive or negative at the
point P.
(a) fx
(b) fy
(c) fxx
(d) fxy
(e) fyy
Lecture 8. Functions of Several Variables: Partial Derivatives 2
