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Wize University Calculus 1 Textbook > Applications of Differentiation

Taylor Polynomial Remainder

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Taylor Polynomial Remainder

Since Taylor Polynomials are only evaluated to a finite nn value, they are approximations. Thus, there is some error when using a Taylor Polynomial. The error is the difference in the exact answer and the Taylor polynomial:
Rn(x)=f(x)−Tn(x)\boxed{R_n(x)=f(x)-T_n(x)}
where nn is "evaluated at infinity”. Clearly, we cannot evaluate n at ∞\infty. However, we have the following result:

Taylor’s Remainder Theorem

The error of the nth degree Taylor Polynomial approximation, Rn(x)R_n(x) ,is given by
Rn(x)=f(n+1)(c)(n+1)!(x−a)n+1\boxed{R_n(x)=\frac{f^{(n+1)}(c)}{(n+1)!}(x-a)^{n+1}}

where cc is an unknown number between xx and aa.

Bounding Error

To find a bound on the error RnR_n, we can maximize f(n+1)(c)f^{(n+1)}(c) If we let
M=∣max{f(n+1)(c)}∣ \boxed{M=|\text{max}\{f^{(n+1)}(c)\}|}
then
∣Rn(x)∣≤M(n+1)!∣x−a∣n+1\boxed{\lvert R_n(x)\rvert\leq \frac{M}{(n+1)!}\lvert x-a\rvert^{n+1}}


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Procedure for Taylor Remainder

  1. Determine the function f(x)f(x), the degree of the approximation nn, and the point at which it is centered about aa.
  2. Find a point cc, between aa and the xx in question, that maximizes f(n+1)(c)f^{(n+1)}(c) .
  3. Use the equation above to compute a bound on the error.
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Example: Taylor Polynomial Remainder

A function f(x)f(x) has a 3rd derivative equal to 9(1−x) \dfrac{9}{(1-x)}. The second degree Taylor Polynomial T2(x)  centered at  a=0T_2(x)\, \, \text{centered at }\,a=0 is used to approximate f(0.1)f(0.1). Find the upper bound for the error given by the Taylor Polynomial.


∣Rn(x)∣≤M(n+1)!∣x−a∣n+1  ,M=|max{f(n+1)(c)}∣\displaystyle \lvert R_n(x)\rvert\leq \frac{M}{(n+1)!}\lvert x-a\rvert^{n+1} \ \ , M=\text{|max}\{f^{(n+1)}(c)\}|

R2(x)≤∣M3!∣∣x∣3 R_2(x) \leq \left|\dfrac{M}{3!}\right||x|^3


f′′′(x)=91−xf'''(x) = \dfrac{9}{1-x}

a=0 ,x=0.1  ⟹  0≤c≤0.1a=0 \ , x=0.1 \implies 0\le c \le 0.1

f′′′(c)=91−cas c  increases ,1−c  decreases   ⟹  f′′′(c) increases  ⟹  choose c=0.1 to maximize f′′′(x) on [0,0.1]M=f′′′(0.1)=91−0.1=10R2(x)≤∣103!∣∣x∣3=∣106∣∣x∣3=53∣x∣3R2(x)≤53∣x∣3R2(0.1)≤53(0.1)3R2(0.1)≤1600\begin{array}{l} f'''(c) = \dfrac{9}{1-c} \\ \\ \text{as } c \ \text{ increases }, 1-c \ \text{ decreases }\implies f'''(c) \text{ increases} \\ \\ \implies \text{choose } c=0.1 \ \text{to maximize } f'''(x) \text{ on } [0,0.1] \\ \\ M=f'''(0.1) = \dfrac{9}{1-0.1}=10 \\ \\ \\ R_2(x) \leq \left|\dfrac{10}{3!}\right||x|^3 =\left|\dfrac{10}{6}\right||x|^3=\dfrac{5}{3}|x|^3 \\ \\ R_2(x) \leq \dfrac{5}{3}|x|^3 \\ \\ R_2(0.1) \leq \dfrac{5}{3}(0.1)^3 \\ \\ \boxed{R_2(0.1) \leq \dfrac{1}{600}} \end{array}

Thus the upper bound for the error is 1600\dfrac{1}{600}.

Practice: Taylor Polynomial Remainder

Use the remainder theorem check if the following inequality is true or false:

ln⁡(89)≤−19\displaystyle\ln\left(\frac{8}{9}\right)\le-\frac{1}{9}

Let f(x)=ln⁡(1−x2)f(x)=\ln(1-x^2).
Extra Practice
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Use the second order Taylor polynomial for f(x)=xf(x)=\sqrt{x} at x=16x=16 to approximate 17\sqrt{17}. What is a good bound on the error in this approximation?
Estimate the size of the error made in the linear approximation to estimate 99.9\sqrt{99.9}.
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value?
Find the second order Taylor polynomial T2(x)T_2(x) for f(x)=xf(x)=\sqrt{x} at x=16x=16. Use this to approximate 17\sqrt{17}. What is a good bound on the error in this approximation?
Consider f(x)=1+xf(x) = \sqrt{1+x}.
Taylor polynomials
Find the degree 4 Taylor polynomial for f(x) = sin(3x) centered at x=π/6.x=\pi/6. What is the lowest error bound within [0,π/6][0,\pi/6]?
Concept Clarifier
Find the fifth order Taylor polynomial for y=sin⁡ xy=\sin\ x centred about x0=πx_0=\pi. What is the minimum error bound for such Taylor polynomial in the interval x∈[0,3π]x \in [0,3\pi]?
If f(x)=e−2x+3x2f(x)=e^{-2x}+3x^2 approximate f(0.1)f(0.1) ? Find the maximum error for approximating positive numbers?
Find the second order Taylor polynomial T2(x)T_2(x) for f(x)=xf(x)=\sqrt{x} at x=16x=16. Use this to approximate 17\sqrt{17}. What is a good bound on the error in this approximation?
Let f(x)=cos⁡(3x)+x−1f(x)=\cos{(3x)}+x-1. Approximate f(π3+15)f(\frac{\pi}{3}+\frac{1}{5}) using linear approximation. Is this overestimation or underestimation of the actual value? Estimate the max error bound for this approximation.
Consider f(x)=1+xf(x) = \sqrt{1+x}.