1. Linearization, Newton’s Method
Section 4.5a

2. Starting with a Calculator Exploration…
Let
1. Find the line tangent to the graph of at
2. Set and Zoom in on the
two graphs at
The line tangent to a curve
lies close to the curve near the
point of tangency…

3. y x Slope = Point-slope equation for the tangent line:x
Point-slope equation for the tangent line:
Thus, the tangent is the graph of the linear function
 As long as the line remains close to the graph of f, L(x) gives
a good approximation of f (x)…

4. Definition: Linearization
If is differentiable at x = a, then the approximating
function
is the linearization of f at a.
The approximation is the standard
linear approximation of at a. The point x = a is the
center of the approximation.

5. Practice Problems Let’s take a look at the graph!!!
For each of the following, find the linearization of the function at
the given point.
Let’s take a look
at the graph!!!

6. Practice Problems Again, consider the graph…
For each of the following, find the linearization of the function at
the given point.
Again, consider
the graph…

7. Newton’s Method Newton’s method is a numerical technique for
approximating a zero of a function with zeros of
its linearizations… (note: this is a technique
utilized by many calculators…)

8. y x The process of Newton’s method:
To find a solution of an equation f(x) = 0, we begin with an initial
estimate x , found either by looking at the graph or by guessing. 1 y
Root
sought x
1st
Approximations

9. y x The process of Newton’s method:
Then we use the tangent curve at that point to approximate the
curve  the point where the tangent hits the x-axis gives the
next approximation, x . y 2
Root
sought x
2nd
1st
Approximations

10. y x The process of Newton’s method:
We continue this process, using each approximation to generate
the next  usually, we will eventually get close enough to the
actual zero to stop. y
Root
sought x
4th
3rd
2nd
1st
Approximations

11. y
Root
sought x
Point-slope equation for the tangent to the curve at

12. y
Root
sought x
We can find where it crosses the x-axis by setting y = 0:

13. y x This value of x is the next approximation!!! Root sought
Solve for x:

14. Procedure for newton’s method
1. Guess a first approximation to a solution of the
equation A graph may help.
2. Use the first approximation to get a second, the
second to get a third, and so on, using the formula:

15. Practice Problems
Use Newton’s method to estimate the solution. Make your
answer accurate to 6 decimal places.
The graph of the function suggests that x = – 0.3 is a good first
approximation: 1

16. Practice Problems Final Answer:
Use Newton’s method to estimate the solution. Make your
answer accurate to 6 decimal places.
Let and
Store into
Then store into and press the “Enter” key
over and over. Watch as the numbers converge to the zero of f.
Use different values for your initial approximation, and repeat
steps 2 and 3…
Final Answer: