MAT 21C Lecture 11: MAT 21C – Lecture 11 – Taylor Series and Remainder Theorem
MAT 21C – Lecture 11 – Taylor Series and Remainder Theorem
• A Taylor series can be described with a function, f(x) with derivatives of all
orders. The nth Taylor coefficient of f(x) at x = a is
• The Taylor series of f(x) at x = a is
. The Taylor polynomial of
degree n of f(x) at x = a is
• The question is how well does approximate f(x) when x is close to a?
• Example: Consider the Taylor Series at x = 0 for f(x) = sin(x). Approximate sin(0.1)
and sin(1) using Taylor polynomials.
n
0
sin(x)
f(0) = 0
1
cos(x)
f’ =
2
-sin(x)
f’’ =
3
-cos(x)
f’’’ = -1
4
sin(x)
5
cos(x)
6
-sin(x)
7
-cos(x)
The Taylor series of the 7th order is
The Taylor polynomial of degree 2n + 1 is
. Taylor coefficients are
;
If x = 0.1, then
and
The actual value of
sin. = .…
If x = 1, then
and
The actual value of sin = .…
• Taylor’s Theore with Reaider:
for some
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MAT 21C Full Course Notes
Verified Note
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Document Summary
Mat 21c lecture 11 taylor series and remainder theorem n sin(x) cos(x) sin(x) cos(x) The nth taylor coefficient of f(x) at x = a is (cid:1855)=(cid:4666)(cid:4667): the taylor series of f(x) at x = a is (cid:4666) (cid:4667). =(cid:2868) (cid:4666) (cid:4667)=(cid:1858)(cid:4666)(cid:4667)+ (cid:1858) (cid:4666)(cid:4667)(cid:4666) degree n of f(x) at x = a is (cid:4666)(cid:4667)= (cid:4666)(cid:4667)! =(cid:2868) (cid:4667)+ (cid:2869)(cid:2870)!(cid:1858) (cid:4666)(cid:4667)(cid:4666) (cid:4667)(cid:2870)+ +(cid:2869)! (cid:1858)(cid:4666)(cid:4667)(cid:4666) (cid:4667: the question is how well does (cid:4666)(cid:4667) approximate f(x) when x is close to a? (cid:1858)(cid:4666)(cid:882)(cid:4667) (cid:1858)(cid:4666)(cid:4667) (cid:1858)(cid:2872)(cid:4666)(cid:882)(cid:4667)=(cid:882) (cid:1858)(cid:2873)(cid:4666)(cid:882)(cid:4667)=(cid:883) (cid:1858)(cid:2874)(cid:4666)(cid:882)(cid:4667)=(cid:882) (cid:1858)(cid:2875)(cid:4666)(cid:882)(cid:4667)= (cid:883) The taylor series of the 7th order is (cid:2869)(cid:2871)!(cid:2871)+ (cid:2869)(cid:2873)!(cid:2873)+(cid:2869)(cid:2875)! (cid:2875)+ . The taylor polynomial of degree 2n + 1 is (cid:4666)(cid:4667)= (cid:2869)(cid:2871)!(cid:2871)+(cid:2869)(cid:2873)!(cid:2873) (cid:2869)(cid:2875)! (cid:2875)+ If x = 0. 1, then (cid:2869)(cid:4666)(cid:882). (cid:883)(cid:4667)=(cid:882). (cid:883); (cid:2871)(cid:4666)(cid:882). (cid:883)(cid:4667)=(cid:882). (cid:883) (cid:2869)(cid:2871)!(cid:4666)(cid:882). (cid:883)(cid:4667)(cid:2871)=(cid:882). (cid:882)(cid:891)(cid:891)(cid:890)(cid:885)(cid:885)(cid:885) and (cid:2873)(cid:4666)(cid:882). (cid:883)(cid:4667)=(cid:882). (cid:883) (cid:2869)(cid:2871)!(cid:4666)(cid:882). (cid:883)(cid:4667)(cid:2871)+(cid:2869)(cid:2873)!(cid:4666)(cid:882). (cid:883)(cid:4667)(cid:2873)=(cid:882). (cid:882)(cid:891)(cid:891)(cid:890)(cid:885)(cid:885)(cid:886)(cid:883)(cid:888)(cid:888)(cid:888) the actual value of. If x = 1, then (cid:2869)(cid:4666)(cid:883)(cid:4667)=(cid:883); (cid:2871)(cid:4666)(cid:883)(cid:4667)=(cid:883) (cid:2869)(cid:2871)!(cid:4666)(cid:883)(cid:4667)(cid:2871)=(cid:882). (cid:890)(cid:885)(cid:885)(cid:885)(cid:885)(cid:885)(cid:885) and (cid:2873)(cid:4666)(cid:883)(cid:4667)=(cid:883) (cid:2869)(cid:2871)!(cid:4666)(cid:883)(cid:4667)(cid:2871)+(cid:2869)(cid:2873)!(cid:4666)(cid:883)(cid:4667)(cid:2873)=(cid:882). (cid:890)(cid:886)(cid:883)(cid:888)(cid:888)(cid:888)(cid:888)(cid:888)(cid:888) .
