1. Taylor and MacLaurin Series
Lesson 9.7

2. Taylor & Maclaurin Polynomials
Consider a function f(x) that can be differentiated n times on some interval I
Our goal: find a polynomial function M(x)
which approximates f
at a number c in its domain
Initial requirements
M(c) = f(c)
M '(c) = f '(c)
Centered at c or expanded about c

3. Linear Approximations
The tangent line is a good approximation of f(x) for x near a
True value f(x)
Approx. value of f(x)
f'(a) (x – a)
(x – a)
f(a) a x

4. Linear Approximations
Taylor polynomial degree 1
Approximating f(x) for x near 0
Consider
How close are these?
f(.05)
f(0.4)
View Geogebra demo

5. Quadratic Approximations
For a more accurate approximation to f(x) = cos x for x near 0
Use a quadratic function
We determine
At x = 0 we must have
The functions to agree
The first and second derivatives to agree

6. Quadratic Approximations
Since
We have

7. Quadratic ApproximationsSo
Now how close are these?
View Geogebra demo

8. Taylor Polynomial Degree 2
In general we find the approximation of f(x) for x near 0
Try for a different function
f(x) = sin(x)
Let x = 0.3

9. Higher Degree Taylor Polynomial
For approximating f(x) for x near 0
Note for f(x) = sin x, Taylor Polynomial of degree 7
View Geogebra demo

10. Improved Approximating
We can choose some other value for x, say x = c
Then for f(x) = sin(x – c) the nth degree Taylor polynomial at x = c

11. Assignment
Lesson 9.7
Page 656
Exercises 1 – 5 all , 7, 9, – 29 odd