Transcript Linear Approximation and Differentials Lesson 4.8 Tangent Line Approximation • Consider a tangent to a function at a point x = a y=f(x) f(a) • •The equation of.

Linear Approximation
and Differentials
Lesson 4.8
Tangent Line Approximation
• Consider a tangent to a function at a
point x = a
y=f(x)
f(a)
•
•The equation of the tangent line:
y = f(a) + f ‘(a)(x – a)
a
• Close to the point, the tangent line is an
approximation for f(x)
Tangent Line Approximation
• We claim that
f ( x1 )  f (a)  f '(a)( x1  a)
• 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
New Look at
dy
•
x
•
dy
dx
y
• x + x
x = dx
• dy = rise of tangent relative to x = dx
• y = change in y that occurs relative to
x = dx
New Look at
• We know that
 then
y
 f '( x)
x
dy
dx
y  f '( x)  x
• 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
The Differential of y
• Consider
y
dy
 f '( x) 
x
dx
• Then we can say
dy  f '( x)  dx  y
 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
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 differentials
Page 274
• Find the differential of
3 – 5x2
x e-2x
Differentials for Approximations
• Consider 25.3
• Use f ( x)  x
f  x  x   f ( x)  f '( x)  dx  x 
• Then with x = 25, dx = .3 obtain
approximation
1
2 x
 dx
Propagated Error
• Consider a rectangular box
with a square base
 Height is 2 times length
2x
of sides of base
x
x
 Given that x = 3.5
 You are able to measure with 3% accuracy
• What is the error propagated for the volume?
Propagated Error
• We know that
V  x  x  2x  2x
3
dx  3%  3.5  0.105
• Then dy = 6x2 dx
= 6 * 3.52 * 0.105 = 7.7175
This is the approximate propagated error for
the volume
Propagated Error
• The propagated error is the dy
 sometimes called the df
• The relative error is
dy
7.7175

 0.09
f ( x) 85.75
• The percentage of error
 relative error * 100%
Assignment
• Lesson 4.8
• Page 276
• Exercises 1 – 45 odd