AP Calculus BC 10.12 Taylor Error Bounds Overview

Review Taylor error bounds by choosing Lagrange or alternating-series estimates and checking the required conditions before stating the maximum error.

Covered syllabus points

LIM-8.C—Determine the error bound associated with a Taylor polynomial approximation

[Maximum number: 2]

Let y=f(x) be the particular solution to the differential equation $\frac{d y}{d x}=(3-x) y^{2}$ with initial

condition f(1)=-1.

The second-degree Taylor polynomial for f about x=1 is used to approximate f(1.1). Given

that $\left|f^{\prime \prime \prime}(x)\right| \leq 60$ for all x in the interval $1 \leq x \leq 1.1$ , use the Lagrange error bound to show

that this approximation differs from f(1.1) by at most 0.01.

C The second-degree Taylor polynomial for f about x=1 is used to approximate f(1.1). Given that

$\left|f^{\prime \prime \prime}(x)\right| \leq 60$ for all x in the interval $1 \leq x \leq 1.1$ , use the Lagrange error bound to show that this

approximation differs from f(1.1) by at most 0.01.

$\left\|f(1.1)-P_{2}(1.1)\right\| \leq \frac{\max _{1 \leq x \leq 1.1}\left\|f^{\prime \prime \prime}(x)\right\|}{3!}\|1.1-1\|^{3} \leq \frac{60}{6}(0.1)^{3}=0.01$	Form of error bound	Point 6 (P6)
		Analysis	Point 7 (P7)

Scoring Notes for Part C

- P6 is earned for presenting either $\frac{\max _{1 \leq x \leq 1.1}\left|f^{\prime \prime \prime}(x)\right|}{3!}|1.1-1|^{3}$ or $\frac{60}{6}(0.1)^{3}$ . Subsequent errors in

simplification will not earn P7.

- To earn P7, a response must have earned P6 and must explicitly connect the error bound with 0.01;

for example by communicating Error $\leq 0.01$ , Error Bound =0.01, or equivalent.

- A response that declares the error is equal to 0.01 (or any equivalent form of this value) does not

earn P7.

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Covered syllabus points

10.12 Lagrange Error Bound question 1