MAT 1332: CALCULUS FOR LIFE SCIENCES
JING LI
Contents
1.
Review: Functions of several variables I: Introduction
1
1.1.
Functions of two or more independent variables
1
1.2.
The level set
1
1.3.
Limits and continuity
1
2.
Function of several variables II: Partial derivatives
1
2.1.
Definition
1
2.2.
Geometric interpretation of partial derivatives
3
2.3.
Linear approximation
7
1.
Review: Functions of several variables I: Introduction
1.1.
Functions of two or more independent variables.
•
Domain
•
Range
1.2.
The level set.
1.3.
Limits and continuity.
2.
Function of several variables II: Partial derivatives
2.1.
Definition.
Suppose that the response of an organism depends on a number of independent
variables. To investigate this dependency, a common experimental design is to measure the response
when changing one variable while keeping all other variable fixed.
This experimental design illustrates the idea behind partial derivatives.
Suppose we want to
know how the function
f
(
x, y
) changes when
x
and
y
change. Instead of changing both variables
simultaneously, we might get an idea of how
f
(
x, y
) depends on
x
and
y
when we change one variable
while keeping the other variable fixed.
To illustrate this we look at
Example 1.
f
(
x, y
) =
x
2
y
We want to know how
f
(
x, y
) changes if we change one variable, say
x
, and keep the other variable,
in this case,
y
, fixed. We fixed
y
=
y
0
, then the change of
f
with respect to
x
is simply the derivative
of
f
with respect to
x
when
y
=
y
0
. That is,
d
dx
f
(
x, y
0
) =
d
dx
x
2
y
0
= 2
xy
0
.
Such a derivative is called a partial derivative.
Date
: 20100322.
1
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JING LI
(1)
Functions of two variables
Definition.
Partial derivative
Suppose that
f
is a function of two independent variables
x
and
y
.
•
The
partial derivative of
f
with respect to
x
is defined by
∂f
(
x, y
)
∂x
= lim
h
→
0
f
(
x
+
h, y
)

f
(
x, y
)
h
•
The
partial derivative of
f
with respect to
y
is defined by
∂f
(
x, y
)
∂y
= lim
h
→
0
f
(
x, y
+
h
)

f
(
x, y
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 Spring '07
 MUNTEANU
 Continuity, Derivative, Limits, Calculus for Life Sciences, Jing Li

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