Math Notes Part 2

Linearization
o
Review
The tangent to
y
=
f
(
x
) at
x
=
a
is
y
=
f
(a) +
f
1
(
a
)(
x
–
a
)
o
For values of
x
real close to a, the tangent line is a good approximation of
y
=
f
(
x
)
That is,
f
(
x
)
≈
f
(
a
) +
f
1
(
a
)(
x
–
a
) if
x
is close to
a
The
linearization
of f(
x
) at
x
= a is: L(
x
) =
f
(
a
) +
f
1
(
a
)(
x
–
a
)
o
Definitions
The differential of
x
is any real number of
dx
The differential of
y
is
dy
=
f
1
(
x
)
dx
o
=
→
f1x
lim∆x
0∆y∆x
So, for a very small
∆
x
,
≈
f1x
∆y∆x
•
So
f
1
(
x
)
∆
x
=
∆
y
o
Let
dx
=
∆
x
Then we have
f
1
(
x
)
dx
≈
∆
y
•
So
if ∆
x
,
is small
dy
≈
∆
y
We can approximate that the actual change is:
•
=
+
 ( )
y∆y
fx
∆x
f x
by
=
dy
f1xdx

Implicit Differentiation (2.3)
o
If
y
is not defined explicitly in terms of
x
, you’ll need to use implicit
differentiation

Related Rates (3.6)
o
To solve related rates of change
Draw a picture
Figure out which rate(s) of change you know and which you are trying to
find
Write an equation that relates the variables involved
Differentiate both sides with respect to time
Plug in specific values
Attach the correct units

Graphing (3.1 and 3.2)
o
First Derivative Test for Rise and Fall
If
f
1
(
x
) changes from
positive to negative at a point, then the
y
value of
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 Winter '08
 Poe,Laurie
 Math, Calculus, Approximation, Derivative, critical numbers, Local max/min values

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