index.html

NOTE: I've put here the plots from class today for e^x, sqrt(1+x) and ln(x) against several of their Taylor polynomial approximations.

Plots of e^x and it's taylor polynomial approximations at x=0:

Linear approximation:

[Graphics:HTMLFiles/index_2.gif]

Out[1]=

In[2]:=

Degree 2 approximation:

[Graphics:HTMLFiles/index_5.gif]

Out[2]=

In[3]:=

Degree 3 approximation:

[Graphics:HTMLFiles/index_8.gif]

Out[3]=

In[4]:=

Degree 4 approximation:

[Graphics:HTMLFiles/index_11.gif]

Degree 7 approximation:

[Graphics:HTMLFiles/index_14.gif]

Taylor polynomial approximations for Sqrt(1 + x) at x=0:

(looking at the following Taylor polynomials, what would you guess is the radius of convergence for the full Taylor series for Sqrt(1+x) at x=0?)

Degree 1 approximation:

[Graphics:HTMLFiles/index_2.gif]

Degree 2 approximation:

[Graphics:HTMLFiles/index_5.gif]

Degree 3 approximation:

[Graphics:HTMLFiles/index_8.gif]

Degree 4 approximation:

[Graphics:HTMLFiles/index_11.gif]

Degree 5 approximation:

[Graphics:HTMLFiles/index_14.gif]

Taylor Polynomial Approximations for Ln(x) at x=1:

(Looking at the following Taylor polynomials for Ln(x) at x=1, what would you guess is the radius of convergence for the full Taylor series for Ln(x) at x=1??)

Degree 1 approximation:

[Graphics:HTMLFiles/index_19.gif]

Degree 2 approximation:

[Graphics:HTMLFiles/index_25.gif]

Degree 3 approximation:

[Graphics:HTMLFiles/index_28.gif]

Degree 4 approximation:

[Graphics:HTMLFiles/index_31.gif]

Degree 5 approximation:

[Graphics:HTMLFiles/index_34.gif]

Degree 8 approximation:

[Graphics:HTMLFiles/index_37.gif]

Created by Mathematica (April 10, 2006)