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’(0) = 1
2
-sin(x)
f’’(0) = 0
3
-cos(x)
f’’’(0) = -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(0.1) = 0.0998334166468…
If x = 1, then
and
The actual value of sin(1) = 0.84147…
• Taylor’s Theorem with Remainder:
for some
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MAT 21C Full Course Notes
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