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For any function f (x), the tangent is a close approximation of the function for some small distance from the tangent point.
y
x0 x a
f x f aWe call the equation of the tangent the linearization of the function.
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The linearization is the equation of the tangent line, and you can use the old formulas if you like.
Start with the point/slope equation:
1 1y y m x x 1x a 1y f a m f a
y f a f a x a
y f a f a x a
L x f a f a x a linearization of f at a
f x L x is the standard linear approximation of f at a.
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Linearization
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Example Finding a Linearization
Find the linearization of ( ) cos at / 2 and use it to approximate
cos 1.75 without a calculator.
f x x x
Since ( / 2) cos( / 2) 0, the point of tangency is ( / 2,0). The slope of the
tangent line is '( / 2) sin( / 2) 1. Thus ( ) 0 ( 1) .2 2
To approximate cos 1.75 (1.75) (1.75) 1.75 .2
f
f L x x x
f L
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Important linearizations for x near zero:
1k
x 1 kx
sin x
cos x
tan x
x
1
x
1
21
1 1 12
x x x
13 4 4 3
4 4
1 5 1 5
1 51 5 1
3 3
x x
x x
f x L x
This formula also leads to non-linear approximations:
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Differentials:
When we first started to talk about derivatives, we said that
becomes when the change in x and change in
y become very small.
y
x
dy
dx
dy can be considered a very small change in y.
dx can be considered a very small change in x.
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Estimating Change with Differentials
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Let be a differentiable function.
The differential is an independent variable.
The differential is:
y f xdxdy dy f x dx
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Example Finding the Differential dy
5
Find the differential and evaluate for the given value of and .
2 , 1, 0.01
dy dy x dx
y x x x dx
45 2
5 2 0.01
0.07
dy x dx
dy
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Examples
Find dy if
a. y = x5 + 37x
Ans: dy = (5x4 + 37) dx
b. y = sin 3x
Ans: dy = (3 cos 3x) dx
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Differential Estimate of Change
Let f(x) be differentiable at x = a. The approximate change in the value of f when x changes from a to a + dx is
df = f ‘(a) dx.
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Example
The radius r of a circle increases from a = 10 m to 10.1 m. Use dA to estimate the increase in circle’s area A.
Compare this to the true change ΔA.
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Example: Consider a circle of radius 10. If the radius increases by 0.1, approximately how much will the area change?
2A r
2 dA r dr
2 dA dr
rdx dx
very small change in A
very small change in r
2 10 0.1dA
2dA (approximate change in area)
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2dA (approximate change in area)
Compare to actual change:
New area:
Old area:
210.1 102.01
210 100.00
2.01
.01
2.01
Error
Actual Answer.0049751 0.5%