1. Warm up
Construct the Taylor polynomial of degree 5 about x = 0 for the function f(x)=ex.
Graph f and your approximation function for a graphical comparison.
To check for accuracy, find f(1) and P5(1).

2. Taylor Polynomials
The polynomial Pn(x) which agrees at x = 0 with f and its n derivatives is called a Taylor Polynomial at x = 0.
Taylor polynomials at x = 0 are called Maclaurin polynomials.
Go over “Before the Lesson” problems

3. Polynomials not centered at x = 0
Suppose we want to approximate f(x) = ln x by a Taylor polynomial. The function is not defined for x < 0.
How can we write a polynomial to approximate a function about a point other than x = 0?

4. Polynomials not centered at x = 0
We modify the definition of a Taylor approximation of f in two ways.
The graph of P must be shifted horizontally. This is accomplished by replacing x with x – a.
The function value and the derivative values must be evaluated at x = a rather than at x = 0.

5. Taylor Polynomial of degree n approximating f(x) near x = a
Construct the Taylor polynomial of degree 4 approximating the function f(x) = ln x for x near 1.

6. Replace ?? with last term in
How does the graph look?
Graph y1 = ln x
Graph Taylor polynomial of degree 4 approximating ln x for x near 1:
Graph each of the following one at a time to see what is happening around x = 1.
y5 = y4(x) + ??
y6 = y5(x) + ??
Y7 = y6(x) + ??
Replace ?? with last term in
the Taylor polynomial
of next degree

7. Conclusions
Taylor polynomials centered at x = a give good approximations to f(x) for x near a. Farther away, they may or may not be good.
The higher the degree of the Taylor polynomial, the larger the interval over which it fits the function closely.

8. Taylor Polynomials to Taylor Series
Recall the Taylor polynomials centered at x = 0 for cos x:
The more terms we added the better the approximation.

9. Taylor Series or Taylor expansion
For an infinite number of terms we can represent the whole sequence by writing a Taylor series for cos x:
How would represent the series for ex?

10. Taylor Series for sin x
To get the Taylor series for sin x take the derivative of both sides.

11. Taylor expansions
About x = 0
About x = 1 ■