1. Linearization and Newton’s Method
4.5
Linearization and Newton’s Method

6. What you’ll learn about
Linear Approximation
Newton’s Method
Differentials
Estimating Change with Differentials
Absolute, Relative, and Percent Change
Sensitivity to Change
Essential Questions
How does approximation techniques work in
engineering and science?

7. Quick Review

8. Quick Review Solutions

9. What you’ll learn about
Linear Approximation
Newton’s Method
Differentials
Estimating Change with Differentials
Absolute, Relative, and Percent Change
Sensitivity to Change
Essential Questions
How does approximation techniques work in
engineering and science?

10. Linearization
Example Finding a Linearization 1.

11. Procedure for Newton’s Method
Guess a first approximation to a solution of the equation f(x) = 0. A graph of y = f (x) may help.
Use the first approximation to get a second, the second to get a third, and so on, using the formula

12. Using Newton’s Method Use Newton’s method to solve
Graph it to get a good approximation.
Therefore the solution appears to be –

13. Differentials Example Finding the Differential dy
Find the differential dy and evaluate dy for the given value of x and dx.

14. Differential Estimate of Change
Three types of changes that can be found Absolute, Relative and Percent Change
Absolute
True
Estimated
Relative and Percent Change
True Estimate
Relative
Percent

15. Differential Estimate of Change

16. Example Estimating Change with Differentials
The radius of a circle increases from a = 5 m to 5.1 m Use dA to estimate the increase in the circle’s area A.

17. Pg. 242, 4.5 #1-43 odd