Taylor and Maclaurin Polynomials

Find the 7th Maclaurin polynomial for f(x) = sin(x).

Here is a Powerpoint Presentation showing the 1st through the 23rd Maclaurin polynomials approximating the sine function.

Find the 7th Taylor polynomial centered at pi for f(x) = sin(x).

Find the Taylor polynomial of degree 5, P₅(x), centered at a = 1 for f(x) = ln(x).

ln(x)     P₅(x)

ln(x)     P₁₀(x)

Find the 3rd Maclaurin polynomial for f(x) = tan(x) and use it to approximate tan(0.2).

Taylor and Maclaurin Series

Find the Maclaurin series and Taylor series centered at pi for f(x) = sin(x).

From the patterns apparent in examples 1 and 2 above we can deduce that the Maclaurin series is

and the Taylor series centered at pi is

Find the Maclaurin series for f(x) = e^x.

Example 7

Find the power series centered at zero (Maclaurin series) representation for

g(x) = sin(3x²) and use this series to approximate g(0.5).

Here is the graph of y = sin(3x²)

over the interval [-3.07,3.07].

EXTRA CREDIT

MORE TERMS WERE SUMMED THAN NECESSARY IN THE COMPUTATION ON THE BOTTOM LINE AT THE LEFT.  WHAT WOULD BE THE MINIMUM NUMBER OF TERMS NEEDED TO ACHIEVE THE INDICATED ACCURACY.