# Taylor Approximation of e^x

Author:
Topic:
Calculus
Use to explore and generalize the Taylor polynomial for e^x
Set the function to a degree 5 polynomial centered at 0 and take the derivative of the polynomial. What do you notice? Is this inconsistent or consistent with what we know about f(x)=e^x? Set the Degree equal to 1 and center the Taylor Polynomial at 1. Compare with your finding from problem 2. Change to a Degree 3 polynomial centered at 3 different values Record the 3 different functions. Generalization the Taylor polynomial equation centered at x=a. Select “Show Error”. Explain this graph in terms of the Taylor Polynomial and (x)=e^x . How does changing the Center of the Taylor Polynomial affect the width of the graph where there is "close to zero" error? How does changing the Degree affect the width of where there is "close to zero" error? Find a single term function that approximates the error function. What do you think would happen the error graph if you let the degree go to infinity (currently this graph has an upper limit of degree 20)?