I ran across a problem in my Calculus II homework that I could use some help on.
Consider the function f(x)=e^x
Find the equation of a line p(x)=A+Bx with the property p(0)=f(0), and p ' (0)=f ' (0)
Find p by setting up and solving these equations for A and B
Next find the approximating quadratic, p2(x) = A + Bx + Cx2 such that,
p2(0) = f(0); p2 ' (0) = f ' (0); and P2 " (0) = f " (0):
by solving the equations for A, B, and C.
Set up and find p4(x) explicitly for f(x)=e^x by extending the conditions as in the problems above
Write your polynomials involving "n!". Discuss how you could guess p3 and p5, as well as why "n!" keeps showing up.
Thank you for your help!