MATH 1B Lecture Notes - Lecture 20: Taylor Series
Document Summary
Math 1b: calculus - lecture 20: using taylor series. Q: use the second order taylor polynomial of x=4 for f(x) = sqrt(x) to estimate sqrt(4. 1). Place a bound on the error for your estimate. Remember: a taylor series is the sum from n=0 to of [(f(n)(a)(x-a)n) / n! A: we have f(x) = x1/2. f"(x) = x. 3/2. f""(x) = - x f"""(x) = (3x. So the 2nd order taylor polynomial for f(x)=sqrt(x) is: So we obtain an estimate for sqrt(4. 1), namely: By taylor"s formula, there exists c between 4 and 4. 1 such that the sqrt(4. 1) - p2(4. 1) So, as c is between 4 and 4. 1 and f"""(x) is decreasing on [4,4. 1], we have: Q: use the 3rd order taylor polynomial at x=0 for f(x) = sin(x) to estimate sin(0. 1).
