Problem : Compute the Taylor polynomial at 0 of degree 3 for f (x) = sin(e ^{x}) .
The first few derivatives are
f'(x) | = | cos(e ^{x})e ^{x} | |
f''(x) | = | cos(e ^{x})e ^{x} - sin(e ^{x})e ^{2x} | |
f'''(x) | = | f''(x) - [sin(e ^{x})2e ^{2x} + cos(e ^{x})e ^{3x}] |
sin(1) + cos(1)x + + x ^{3} |
Problem : Approximate log(0.8) using the Taylor polynomial of degree 3 at x = 1 for log(x) .
The desired Taylor polynomial is
(x - 1) - + |
Problem : Find the Taylor series for sin(x) + cos(x) .
Adding term-by-term, we have
1 + x - - + + ^{ ... } |