In this lecture, we will discuss:
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Biopesticide (2).pptx .This slides helps to know the different types of biop...
Function of several variables
1. Faculty of Sciences (Section V)
Lebanese University
Function of Several Variables
Dr. Kamel ATTAR
attar.kamel@gmail.com
F 2020 - 2021 F
2. 2Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
1 Functions of one variable
Definition-Domain, Range and Graph
Examples
2 Functions of SeveraL VariabLes
Definition
Domains and Ranges
Examples
Functions of two variables
Functions of three variables
Graphs, Level Curves and Level Surfaces
Level Curves and Level Surfaces
Examples
Graph
Graphing with Traces (z-Axis Traces)
Exercises
3 Limits for Functions of Two Variables
Definition
Exercises
Dr. Kamel ATTAR | Function of Several Variables |
3. 3Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Example
Exercises
Two-Path Test for Nonexistence of a Limit
Test for Nonexistence of a Limit
Exercises
Continuity
Definition
Exercises
4 Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Geometrical interpretation
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives with respect to x and y
Exercises
The chain rule
Chain Rule for Functions of Two Independent variables
Examples
Chain Rule for Functions of Three Independent variables
Dr. Kamel ATTAR | Function of Several Variables |
4. 4Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Examples
Exercises
Dr. Kamel ATTAR | Function of Several Variables |
5. 5Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Function of one variable
Definition
A function f of one variable is a rule that assigns to each variable x a unique
number denoted by y = f(x). We write
f : D ⊂ R 7−→ R
x 7−→ y = f(x)
I The domain is the set of all possible values of x. (set of inputs).
I The range is the set of all possible value of y from the domain. (set of
outputs)
I The graph is the set of all possible points (x, y) produced by the function.
Geometrically, the graph is a curve in plan.
Dr. Kamel ATTAR | Function of Several Variables |
6. 6Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Example
Function Domain Range
y = x + 2 R =] − ∞, ∞[ R =] − ∞, ∞[
y = 3x2
− 7 R [−7, +∞[
y = sin x R [−1, 1]
y =
1
x
] − ∞, 0[∪]0, +∞[ ] − ∞, 0[∪]0, +∞[
y =
√
x [0, +∞[ [0, +∞[
y = ln x ]0, +∞[ ] − ∞, +∞[
Dr. Kamel ATTAR | Function of Several Variables |
7. 7Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Example
Find the domain of the following functions:
f(x) =
p
1 − 2x , g(x) = −
7
x
, h(x) = 4 − x2
and u(x) =
1
x − 2
Dr. Kamel ATTAR | Function of Several Variables |
8. 8Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
9. 9Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
10. 10Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
11. 11Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
12. 12Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
13. 13Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
14. 14Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition-Domain, Range and Graph
Examples
Dr. Kamel ATTAR | Function of Several Variables |
15. 15Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Definition
• A function of two variables is a rule that assigns a real number f(x, y) to
each pair of real numbers (x, y) in the domain D of the function f. we write
f : D ⊂ R2
7−→ R
(x, y) 7−→ z = f(x, y)
• A function of three variables is a rule that assigns a real number f(x, y, z)
to each triple of real numbers (x, y, z) in the domain D. we write
f : D ⊂ R3
7−→ R
(x, y, z) 7−→ w = f(x, y, z)
• A function of n-variables is a rule that assigns a real number
f(x1, x2, · · · , xn) to each n-tuples of real numbers (x1, x2, · · · , xn)
f : D ⊂ Rn
7−→ R
(x1, x2, · · · , xn) 7−→ w = f(x1, x2, · · · , xn)
Dr. Kamel ATTAR | Function of Several Variables |
16. 16Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Definition (Domain)
The domain of a function is assumed to be the largest set for which the
defining rule generates real numbers.
Definition (Range)
The range consists of the set of output values for the dependent variable.
f : D ⊂ Rn
7−→ R
(x1, x2, · · · , xn) 7−→ w = f(x1, x2, · · · , xn)
Domain of f : The set D is the function’s domain.
Range of f : The set of w−values taken on by f is the function’s range.
Input variables : We call the xj ’s the function’s input variables.
Output variable : We call w the function’s output variable.
Dr. Kamel ATTAR | Function of Several Variables |
17. 17Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) =
p
4 − x2 − y2
Dr. Kamel ATTAR | Function of Several Variables |
18. 18Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) =
p
4 − x2 − y2
Figure:
g(x) =
√
x , Dg = [0, +∞[
Dr. Kamel ATTAR | Function of Several Variables |
19. 19Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) =
p
4 − x2 − y2
Figure:
g(x) =
√
x , Dg = [0, +∞[
Domain:
4 − x2
− y2
≥ 0
4 ≥ x2
+ y2
Df = {(x, y) ∈ R2
: x2
+ y2
≤ 4}
Dr. Kamel ATTAR | Function of Several Variables |
20. 20Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) =
p
4 − x2 − y2
Figure:
g(x) =
√
x , Dg = [0, +∞[
Domain:
4 − x2
− y2
≥ 0
4 ≥ x2
+ y2
Df = {(x, y) ∈ R2
: x2
+ y2
≤ 4}
Dr. Kamel ATTAR | Function of Several Variables |
21. 21Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) = ln(1 − x − y)
Dr. Kamel ATTAR | Function of Several Variables |
22. 22Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) = ln(1 − x − y)
Figure:
g(x) = ln(x) , Dg =]0, +∞[
Dr. Kamel ATTAR | Function of Several Variables |
23. 23Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) = ln(1 − x − y)
Figure:
g(x) = ln(x) , Dg =]0, +∞[
Domain:
1 − x − y > 0
1 > x + y
Df = {(x, y) ∈ R2
: x+y < 1}
Dr. Kamel ATTAR | Function of Several Variables |
24. 24Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: f(x, y) = ln(1 − x − y)
Figure:
g(x) = ln(x) , Dg =]0, +∞[
Domain:
1 − x − y > 0
1 > x + y
Df = {(x, y) ∈ R2
: x+y < 1}
Dr. Kamel ATTAR | Function of Several Variables |
25. 25Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Example (Functions of two variables)
Function Domain Range
z = ex
cos y R2
Entire plane ] − ∞, ∞[
z =
p
y − x2 y ≥ x2
[0, ∞[
z =
1
xy
xy 6= 0 ] − ∞, 0[∪]0, ∞[
z = sin xy R2
Entire plane [−1, 1]
Example
Find the domain of the function f(x, y) =
ln(x − 3)
p
y + 2
x2 − 4
.
Dr. Kamel ATTAR | Function of Several Variables |
26. 26Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Example (Functions of three variables)
Function Domain Range
w = x2
cos y sin z R3
Entire space ] − ∞, ∞[
w =
p
x2 + y2 + z2 R3
Entire space [0, ∞[
w =
1
x2 + y2 + z2
(x, y, z) 6= (0, 0, 0) ]0, ∞[
w = xy ln z R2
∪ R∗
+ Half-space z > 0 ] − ∞, ∞[
Dr. Kamel ATTAR | Function of Several Variables |
27. 27Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Definition
Level curve (2 varaibles): The set of points in the plane where a function
f(x, y) has a constant value f(x, y) = c is called a level curve
of f.
Level surface (3 variables): The set of points (x, y, z) in space where a
function of three independent variables has a constant value
f(x, y, z) = c is called a level surface of f.
Dr. Kamel ATTAR | Function of Several Variables |
28. 28Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Dr. Kamel ATTAR | Function of Several Variables |
29. 29Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Example
The level surfaces of f(x, y, z) =
p
x2 + y2 + z2 is described as below
Dr. Kamel ATTAR | Function of Several Variables |
30. 30Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Definition
Graph in R2
: The set of all points (x, y, f(x, y) in space, for (x, y) in the
domain of f, is called the graph of f. The graph of f is also called
the surface z = f(x, y).
Graph in R3
: The set of all points (x, y, z, f(x, y, z) in space, for (x, y, z) in
the domain of f, is called the graph of f.
Dr. Kamel ATTAR | Function of Several Variables |
31. 31Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
32. 32Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
33. 33Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
34. 34Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
35. 35Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
36. 36Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z = f(x, y) =
p
x2 + y2 .
z ∈ {0, 1, 2, 3, 4}
Dr. Kamel ATTAR | Function of Several Variables |
37. 37Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
38. 38Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
39. 39Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
40. 40Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
41. 41Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
42. 42Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Sketch the graph of
z =
p
9 − x2 − y2 .
z ∈ {0, 1, 2, 3}
Dr. Kamel ATTAR | Function of Several Variables |
43. 43Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Domains and Ranges
Level Curves and Level Surfaces
Graph
Graphing with Traces (z-Axis Traces)
Figure: z =
p
x2 + y2 − 1 Figure: z = e−x2
−y2
+1
Dr. Kamel ATTAR | Function of Several Variables |
44. 44Ú74
Exercises
¬ Let f(x, y) = ln(x + y − 1)
a) Evaluate f(1, 1) and f(e, 1).
b) Find and sketch the domain of f
Find and sketch the domain of the following functions
a) h(x, y) = x−3y
x+3y
b) g(x, y) =
p
x2 + y2 − 1 + ln(4 − x2
− y2
)
® Draw several level curves of the following functions
a) u(x, y) =
x2
+ y2
y
b) v(x, y) =
p
x2 + y2
Dr. Kamel ATTAR | Function of Several Variables |
45. 45Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Limits for Functions of Two Variables
Definition
We say that a function f(x, y) approaches the limit L as (x, y) approaches
(x0, y0), if and only if
lim
(x,y)→(x0,y0)
f(x, y) = L .
We simply substitute the x and y values of the point being approached into the
functional expression to find the limiting value.
Example
(a) lim
(x,y)→(0,1)
x − xy + 3
x2y + 5xy − y3
=
0 − (0)(1) + 3
(0)2(1) + 5(0)(1) − (1)3
= −3
(b) lim
(x,y)→(3,−4)
p
x2 + y2 =
q
(3)2 + (−4)2 =
√
25 = 5
Dr. Kamel ATTAR | Function of Several Variables |
46. 46Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Exercises A:
Find the limits
1. lim
(x,y)→(0,0)
3x2
− y2
+ 5
x2 + y2 + 2
2. lim
(x,y)→(0,4)
x
√
y
3. lim
(x,y)→(0,ln 2)
ex−y
4. lim
(x,y)→(3,4)
p
x2 + y2 − 1
5. lim
(x,y)→(2,−3)
1
x
+
1
y
2
6. lim
(x,y)→(0,0)
cos
x2
+ y3
x + y + 1
!
7. lim
(x,y)→(1,1)
ln |1 + x2
y2
| 8. lim
(x,y)→(0,0)
ey
sin x
x
Dr. Kamel ATTAR | Function of Several Variables |
47. 47Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Example
Find lim
(x,y)→(0,0)
x2
− xy
√
x −
√
y
Solution
Dr. Kamel ATTAR | Function of Several Variables |
48. 48Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Example
Find lim
(x,y)→(0,0)
x2
− xy
√
x −
√
y
Solution
Since the denominator
√
x −
√
y approaches 0 as (x, y) → (0, 0), we cannot
calculate the limit. If we multiply numerator and denominator by
√
x +
√
y,
however, we produce an equivalent fraction whose limit we can find:
lim
(x,y)→(0,0)
x2
− xy
√
x −
√
y
= lim
(x,y)→(0,0)
(x2
− xy)(
√
x +
√
y)
(
√
x −
√
y)(
√
x +
√
y)
= lim
(x,y)→(0,0)
x(x − y)(
√
x +
√
y)
x − y
= lim
(x,y)→(0,0)
x(
√
x +
√
y) = 0 .
Dr. Kamel ATTAR | Function of Several Variables |
49. 49Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Exercises B:
Find the limits by rewriting the fractions first
1. lim
(x,y)→(1,1)
x2
− 2xy + y2
x − y
2. lim
(x,y)→(1,1)
x2
− y2
x − y
3. lim
(x,y)→(1,1)
xy − y − 2x + 2
x − 1
4. lim
(x,y)→(2,−4)
y + 4
x2y − xy + 4x2 − 4x
5. lim
(x,y)→(2,2)
x + y − 4
√
x + y − 2
6. lim
(x,y)→(2,0)
p
2x − y − 2
2x − y − 4
7. lim
(x,y)→(4,3)
√
x −
p
y + 1
x − y − 1
Dr. Kamel ATTAR | Function of Several Variables |
50. 50Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Theorem (Two-Path Test for Nonexistence of a Limit)
If a function f(x, y) has different limits along two different paths in the domain
of f as (x, y) approaches (x0, y0), then lim
(x,y)→(x0,y0)
f(x, y) does not exist.
Example
Show that the function
f(x, y) =
2x2
y
x4 + y2
has no limit as (x, y) approaches (0, 0).
Dr. Kamel ATTAR | Function of Several Variables |
51. 51Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Solution
Let’s take the curve y = x2
, the function has the constant value
lim
(x,y=x2)→(0,0)
f(x, y) = lim
(x,y=x2)→(0,0)
2x2
x2
x4 + x2
= lim
(x,y=x2)→(0,0)
2x4
2x4
= 1 .
Now we take y = 2x2
, we obtain
lim
(x,y)→(0,0)
f(x, y) = lim
(x,y=2x2)→(0,0)
2x2
(2x2
)
x4 + 4x4
= lim
(x,y=2x2)→(0,0)
4x4
5x4
= 4/5
Then,
lim
(x,y=2x2)→(0,0)
f(x, y) 6= lim
(x,y=x2)→(0,0)
f(x, y)
By the two-path test, f has no limit as (x, y) approaches (0, 0).
Dr. Kamel ATTAR | Function of Several Variables |
52. 52Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Exercises C:
Show that the limits do not exist.
1. lim
(x,y)→(1,1)
xy2
− 1
y − 1
2. lim
(x,y)→(1,−1)
xy + 1
x2 − y2
.
Answer:
1. y = x and y =
1
x
2. y = −
1
x
and y = −
1
x2
Dr. Kamel ATTAR | Function of Several Variables |
53. 53Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Definition
A function f(x, y) is Continuous at the point (x0, y0) if
1. f is defined at (x0, y0),
2. lim
(x,y)→(x0,y0)
f(x, y) exists
3. lim
(x,y)→(x0,y0)
f(x, y) = f(x0, y0)
A function is continuous if it is continuous at every point of its domain.
Dr. Kamel ATTAR | Function of Several Variables |
54. 54Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Definition
Test for Nonexistence of a Limit
Continuity
Exercises D:
1. Show that
f(x, y) =
x2
− xy
√
x −
√
y
, (x, y) 6= (0, 0)
0 , (x, y) = (0, 0)
Is continuous at every point.
2. Show that
f(x, y) =
2xy
x2 + y2
, (x, y) 6= (0, 0)
0 , (x, y) = (0, 0)
Is continuous at every point except the origin.
Dr. Kamel ATTAR | Function of Several Variables |
55. 55Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Partial Derivatives with respect to x
Definition
The partial derivative of f(x, y) with respect to x at the point (x0, y0) is
d
dx
f(x, y)
56.
57.
58. (x0,y0)
= lim
h→0
f(x0 + h, y0) − f(x0, y0)
h
,
Dr. Kamel ATTAR | Function of Several Variables |
59. 56Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Geometrical interpretation
The slope of the curve z = f(x, y0) at
the point P = (x0, y0, f(x0, y0)) in the
plane y = y0 is the value of the par-
tial derivative of f with respect to x at
(x0, y0). The tangent line to the curve
at P is the line in the plane y = y0 that
passes through P with this slope. The
partial derivative
df
dx
at (x0, y0) gives
the rate of change of f with respect to
x when y is held fixed at the value y0.
Dr. Kamel ATTAR | Function of Several Variables |
60. 57Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Partial Derivatives with respect to y
Definition
The partial derivative of f(x, y) with respect to y at the point (x0, y0) is
d
dy
f(x, y)
64. 58Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Geometrical interpretation
The slope of the curve z = f(x0, y) at the
point P = (x0, y0, f(x0, y0)) in the plane
x = x0 is the value of the partial deriva-
tive of f with respect to y at (x0, y0). The
tangent line to the curve at P is the line in
the plane x = x0 that passes through P
with this slope. The partial derivative df
dy
at
(x0, y0) gives the rate of change of f with
respect to y when x is held fixed at the
value x0.
Dr. Kamel ATTAR | Function of Several Variables |
65. 59Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Dr. Kamel ATTAR | Function of Several Variables |
66. 60Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
Find the values of ∂
∂x
f and ∂
∂y
f at the point (4, −5) if
f(x, y) = x2
+ 3xy + y − 1 .
Solution
To find ∂
∂x
f, we treat y as a constant and differentiate with respect to x:
∂
∂x
f =
∂
∂x
(x2
+ 3xy + y − 1) = 2x + 3y .
The value of ∂
∂x
f at (4, −5) is 2(4) + 3(−5) = −7.
Dr. Kamel ATTAR | Function of Several Variables |
67. 61Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Solution
To find ∂
∂y
f, we treat x as a constant and differentiate with respect to y:
∂
∂y
f =
∂
∂y
(x2
+ 3xy + y − 1) = 3x + 1 .
The value of ∂
∂y
f at (4, −5) is 3(4) + 1 = 13.
Dr. Kamel ATTAR | Function of Several Variables |
68. 62Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
Find ∂f
∂y
as a function if f(x, y) = y sin xy .
Solution
We treat x as a constant and differentiate with respect to y:
∂f
∂y
=
∂
∂y
(y sin xy)
= y
∂
∂y
sin xy + sin xy
∂
∂y
(y)
= (y cos xy)
∂
∂y
(xy) + sin xy
= xy cos xy + sin xy .
Dr. Kamel ATTAR | Function of Several Variables |
69. 63Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
Find ∂f
∂x
and ∂f
∂y
as a functions if f(x, y) =
2y
y + cos x
.
Solution
We treat x as a constant and differentiate with respect to y:
∂f
∂y
=
(y + cos x) ∂
∂y
(2y) − 2y ∂
∂y
(y + cos x)
(y + cos x)2
=
(y + cos x)(2) − 2y
(y + cos x)2
=
2 cos x
(y + cos x)2
Dr. Kamel ATTAR | Function of Several Variables |
70. 64Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
The plane x = 1 intersects the paraboloid z = x2
+ y2
in a parabola. Find the
slope of a tangent to the parabola at (1, 2, 5).
Solution
The slope is the value of the partial derivative ∂z
∂y
at (1, 2):
∂z
∂y
79. (1,2)
= 2(2) = 4 .
Dr. Kamel ATTAR | Function of Several Variables |
80. 65Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Solution
As a check we can treat the parabola as the graph of the single-variable
function
z = (1)2
+ y2
= 1 + y2
in the plane x = 1 and ask for the slope at y = 2. The slope calculated now as
an ordinary derivative, is
∂z
∂y
89. y=2
= 2(2) = 4 .
Dr. Kamel ATTAR | Function of Several Variables |
90. 66Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Exercises A:
Find
∂f
∂x
and
∂f
∂y
.
1. f(x, y) = 2x2
− 3y − 4 2. f(x, y) = x2
− xy + y2
3. f(x, y) = (x2
− 1)(y + 2) 4. f(x, y) = (xy − 1)2
5. f(x, y) = (2x − 3y)3
6. f(x, y) =
p
x2 + y2
7. f(x, y) = 1
x+y
8. f(x, y) = x
x2+y2
9. f(x, y) = x+y
xy−1
10. f(x, y) = ex+y+1
11. f(x, y) = ln(x + y) 12. f(x, y) = exy
ln y
Dr. Kamel ATTAR | Function of Several Variables |
91. 67Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Chain Rule for Functions of Two Independent variables
Theorem
If f(x, y) is differentiable and if x = x(t), y = y(t) are differentiable functions
of t, then the composite w = f x(t), y(t)
is a differentiable function of t and
df
dt
= fx (x(t), y(t)) · x0
(t) + fy (x(t), y(t)) · y0
(t) ,
or
df
dt
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
Dr. Kamel ATTAR | Function of Several Variables |
92. 68Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Dr. Kamel ATTAR | Function of Several Variables |
93. 69Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
Let f be a function defined as
f(x, y) = xy
1. Use the Cahin rule to find the derivative of f with respect to t along the
path x = cos t and y = sin t.
2. What is the derivative’s value at t = π
2
.
Dr. Kamel ATTAR | Function of Several Variables |
94. 70Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Solution
1. We apply the Chain Rule to find ∂f
∂t
as follows:
∂f
∂t
=
∂f
∂x
∂x
∂t
+
∂f
∂y
∂y
∂t
=
∂(xy)
∂x
∂
∂t
(cos t) +
∂(xy)
∂y
∂
∂t
(sin t)
= (y)(− sin t) + (x)(cos t)
= (sin t)(− sin t) + (cos t)(cos t)
= − sin2
t + cos2
t
= cos 2t .
2.
∂f
∂t
101. 71Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Chain Rule for Functions of Three Independent
variables
Theorem
If f(x, y, z) is differentiable and x, y, and z are differentiable functions of t,
then f is a differentiable function of t and
∂f
∂t
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
Dr. Kamel ATTAR | Function of Several Variables |
102. 72Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Dr. Kamel ATTAR | Function of Several Variables |
103. 73Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
Example
Find
∂f
∂t
if
f(x, y, z) = xy + z , x = cos t , y = sin t , z = t .
Solution
∂f
∂t
=
∂f
∂x
dx
dt
+
∂f
∂y
dy
dt
+
∂f
∂z
dz
dt
= (y)(− sin t) + (x)(cos t) + (1)(1)
= (sin t)(− sin t) + (cos t)(cos t) + 1
= − sin2
t + cos2
t + 1 = 1 + cos 2t .
Dr. Kamel ATTAR | Function of Several Variables |
104. 74Ú74
Functions of one variable
Functions of SeveraL VariabLes
Limits for Functions of Two Variables
Partial Derivatives of a Function of Two Variables
Partial Derivatives with respect to x
Partial Derivatives with respect to y
Geometrical interpratation of partial derivatives wit
Chain Rule for Functions of Two Independent varia
Chain Rule for Functions of Three Independent va
In Exercise 1 − 6
(a) Express
∂f
∂t
as a function of t, both by using the Chain Rule and by
expressing f in terms of t and differentiating directly with respect to t.
(b) Evaluate
∂f
∂t
at the given value t .
1. f(x, y) = x2
+ y2
, x = cos t , y = sin t ; t = π
2. f(x, y) = x2
+ y2
, x = cos t + sin t , y = cos t − sin t ; t = 0
3. f(x, y) =
x
z
+
y
z
, x = cos2
t , y = sin2
t z = 1/t ; t = 3
4. f(x, y) = ln x2
+y2
+z2
, x = cos t , y = sin t z = 4
√
t ; t = 3
5. f(x, y) = 2yex
−ln z , x = ln(t2
+1) , y = tan−1
t z = et
; t = 1
6. f(x, y) = z − sin xy , x = t , y = ln t z = et−1
; t = 1.
Dr. Kamel ATTAR | Function of Several Variables |