Taylor Polynomial for Approximating Functions

1. Section 12.10 The Taylor Approximation to a Function

2. TAYLOR POLYNOMIAL OF ORDER n The Taylor polynomial of order n based at a, Pn(x), for the function f is the nth partial sum of the Taylor series at a for f. Thus,

3. MACLAURIN POLYNOMIALS When a = 0 in the Taylor Polynomial of order n, we call it the Maclaurin polynomial of order n. That is, the Maclaurin polynomial of order n is the nth partial sum of the Maclaurin series for a function f.

4. TAYLOR’S THEOREM WITH REMAINDER Let f be a function whose (n + 1)st derivative f(n+1)(x) exists for each x in an open interval I containing a. Then, for each x in I, whose remainder term (or error) Rn(x) is given by the formula and c is some point between x and a.

5. USEFUL TOOLS FORBOUNDING |Rn(x)| • The triangle inequality. |a±b| ≤ |a| + |b| • The fact that a fraction gets larger as its denominator gets smaller. • The fact that a fraction gets larger as its numerator gets larger. • |sin x| ≤ 1; |cos x| ≤ 1 It is usually impossible to get an exact value for Rn(x). So, we usually bound |Rn(x)|. Our primary tools are: