1. Linear Approximation and Differentials
Lesson 4.8

2. 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)
y=f(x)
The equation of the tangent line: y = f(a) + f ‘(a)(x – a)
f(a) • a

3. 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

4. New Look at dy = rise of tangent relative to x = dx• • • x
x + x
x = dx
dy = rise of tangent relative to x = dx
y = change in y that occurs relative to x = dx

5. New Look at We know that Recall that dy/dx is NOT a quotient
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

6. 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

7. Animated Graphical View
Note how the "del y" and the dy in the figure get closer and closer

8. Try It Out Note the rules for differentials Page 274
Find the differential of 3 – 5x2 x e-2x

9. Differentials for Approximations
Consider
Use
Then with x = 25, dx = .3 obtain approximation

10. 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? x 2x x

11. Propagated Error We know that
Then dy = 6x2 dx = 6 * 3.52 * = This is the approximate propagated error for the volume

12. Propagated Error The propagated error is the dy The relative error is
sometimes called the df
The relative error is
The percentage of error
relative error * 100%

13. Assignment
Lesson 4.8
Page 276
Exercises 1 – 45 odd