Math 152 – Spring 2016
Section 10.9
1 of 4
Section 10.9 – Applications of Taylor Polynomials
Suppose f (x) equals its Taylor Series
f (x) =
∞
X
f (n) (a)
(x − a)n
n!
n=0
and recall the Taylor Polynomials Tn (x) are
Tn (x) =
n
X
f (i) (a)
i=0
i!
= f (a) +
Note. Assume f (x) =
(x − a)i
f 0 (a)
f 00 (a)
f (n) (a)
(x − a) +
(x − a)2 + · · · +
(x − a)n
1!
2!
n!
∞
P
n=0
f (n) (a)
n! (x
− a)n .
1. The Taylor polynomials Tn (x) provide an approximation to the original function
f (x).
2. T1 (x) = f (a) + f 0 (a)(x − a) is the tangent line to f (x) at a.
3. As n increases Tn (x) becomes a better approximation to f (x).
4. Tn (x) provides better approximations closer to a, where the Taylor polynomials
are centered.
Taylor Polynomials for f (x) = ex :
T1 (x) = 1 + x
1
T2 (x) = 1 + x + x2
2
1
T3 (x) = 1 + x + x2 +
2
1
T4 (x) = 1 + x + x2 +
2
Example 1. Approximate the function f (x) =
2 at a = 8.
1 3
x
6
1 3
1
x + x4
6
24
√
3
x by a Taylor polynomial of degree
Math 152 – Spring 2016
Section 10.9
2 of 4
Example 2. Approximate g(x) = sin x by a fourth degree Taylor polynomial centered
at π/4. Use T4 (x) to approximate sin 40◦ .
Example 3. Find the Taylor polynomial T3 (x) for the function f (x) = x cos 2x at 0.
Math 152 – Spring 2016
Section 10.9
3 of 4
x
Example 4. Find the Taylor polynomial T4 (x) for the function f (x) = x2 e− 2 .
Example 5. Approximate f (x) = ln(2 + 3x) by a third degree Taylor polynomial
centered at 2.
Math 152 – Spring 2016
Section 10.9
4 of 4
Example 6. Find the Taylor polynomial T2 (x) for the function f (x) = sin−1 x centered
at a = .5.
Example 7. Approximate f (x) = tan x by a Taylor polynomial with degree 2 at π/3
Question: How good is the approximation? How large does n need to be?
Answer: Recall, Rn (x) = f (x) − Tn (x). We need to estimate the value of |Rn (x)| =
|f (x) − Tn (x)|. Three ways to do this:
1. Calculator/Computer: We can graph |Rn (x)| and estimate the error.
2. If the series is Alternating, we can use the Alternating Estimation Theorem.
3. In all cases, we can use Taylor’s Inequality from 10.7, which says that if |f (n+1) (x)| ≤
M , then
M
|Rn (x)| ≤
|x − a|n+1
(n + 1)!