4.4 Linearization and Differentials
Linearization• Linearization uses the concept that when
you zoom in to the graph at a point the y-values of the curve are very close to the y-values of the tangent line
3
(3, f(3) 1f x x 1 5
4 4y x
Linearization• We can approximate
without a calculator by using the tangent line to f(x) at (3, f(3)) which is
3
(3, f(3) 1f x x 1 5
4 4y x
3.2 3 .2 3.2f
1 5
4 4y x
Linearization• We can approximate
without a calculator by using the tangent line
3.2 1 3.2 4.2f
1 5
4 4y x
1 53.2 3.2
4 4.25 3.2 1.25
2.05
y
3.2 1 3.2
4.2
2.049
f
Approximate value
Actual value
LinearizationNote that the closer x is to 3 the better the
approximation
x approximation, y(x) actual value, f(x)
3.2 2.05 f(3.2) = 2.0493
3.1 2.025 f(3.1) = 2.0248
3.01 2.0025 f(3.01) = 2.002498
3.001 2.00025 f(3.001) = 2.00024998
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
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.
To find the linearization of f(x) at the point (a, f(a))
Important linearizations for x near zero:
1k
x 1 kx
sin x
cos x
tan x
x
1
x
f x L x
Example
a) Find the linearization L(x) of f(x) at x = a.
b) Compare f(a + 0.1) to L(a + 0.1)
2
6@ 2f x a
x
Example
• Find the linearization of
at x = 0. Use it to approximate:
( ) 1f x x
) 1.2
) 1.05
) 1.005
a
b
c
Example
• Find the linearization of y = cosx at x = 0. Use it to approximate cos(0.5).
Example
• Choose a linearization with center not at a but nearby so that the derivative will be easy to evaluate.
3
2
1) ; 8.5
2) ( ) 1 2 ; 0.9
3) ( ) sin ; 3.2
f x x a
f x x a
f x x a
Example
• Use linearization to find the value of
at x = 0.12 to the nearest hundredth. 4 sinf x x
Hw: p. 229/1-5,11,13,14,37
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.
Let be a differentiable function.
The differential is an independent variable.
The differential is:
y f xdxdy dy f x dx
dy is dependent on x and dx
Examples: find the differential
51) 37
2) sin3
3) tan2
4)1
y x x
y x
d x
xdx
Examples: a) Find dyb) Evaluate dy for the given values of x and
dx
1. y = x2 – 4x ; x = 3, dx = 0.5
2. y = xlnx; x = 1, dx = 0.1
• Differentials can be used to estimate the change in y when x = a changes to
x = a + dx
True Change Estimated Change y f f a dx f a 'dy df f a dx
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)
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
Original Area
Error
Actual Answer.0049751 0.5%
0.01%.0001.01
100
Example
• The earth’s radius is 3959 + 0.1 miles. What effect does a tolerance of + 0.1 miles have on the estimate of the earth’s surface area?
Example
• Estimate the volume of material in a cylindrical shell with a height of 15m, radius 10m and shell thickness 0.2m.
HW: p.229/19-26,31-35,38,39,42
Top Related