Linear Approximation and Differentials
Lesson 4.8
Tangent Line Approximation
• Consider a tangent to a function at a point x = a
• Close to the point, the tangent line is an approximation for f(x)
•
a
f(a)
y=f(x)
•The equation of the tangent line:y = f(a) + f ‘(a)(x – a)
Tangent Line Approximation
• We claim that
• This is called linearization of the function at the point a.
• Recall that when we zoom in on an interval of a function far enough, it looks like a line
1 1( ) ( ) '( )( )f x f a f a x a
New Look at
• dy = rise of tangent relative to x = dx y = change in y that occurs relative to
x = dx
dydx
x
x = dx
• dy y
•• x + x
New Look at
• We know that
then
• Recall that dy/dx is NOT a quotient it is the notation for the derivative
• However … sometimes it is useful to use dy and dx as actual quantities
dydx
'( )y f xx
'( )y f x x
The Differential of y
• Consider
• Then we can say
this is called the differential of y the notation is d(f(x)) = f ’(x) * dx it is an approximation of the actual change of y
for a small change of x
'( )y dyf xx dx
'( )dy f x dx y
Animated Graphical View
• Note how the "del y" and the dy in the figure get closer and closer
Try It Out
• Note the rules for differentialsPage 274
• Find the differential of
3 – 5x2
x e-2x
Differentials for Approximations
• Consider • Use
• Then with x = 25, dx = .3 obtain approximation
25.3
( )1( ) '( )
2
f x x
f x x f x f x dx x dxx
Propagated Error
• Consider a rectangular box with a square base Height is 2 times length
of sides of base Given that x = 3.5 You are able to measure with 3% accuracy
• What is the error propagated for the volume?
xx
2x
Propagated Error
• We know that
• Then dy = 6x2 dx = 6 * 3.52 * 0.105 = 7.7175This is the approximate propagated error for the volume
32 23% 3.5 0.105
V x x x xdx
Propagated Error
• The propagated error is the dy sometimes called the df
• The relative error is
• The percentage of error relative error * 100%
7.7175 0.09( ) 85.75dyf x
Assignment
• Lesson 4.8• Page 276• Exercises 1 – 45 odd
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