AP Calculus BC 10.11 Finding Taylor Polynomial Approximations of Functions Questions

Review Taylor polynomials by constructing local approximations from derivatives and evaluating them near the centre, with terms kept to the requested degree.

Covered syllabus points

LIM-8.A—Represent a function at a point as a Taylor polynomial LIM-8.B—Approximate function values using a Taylor polynomial

[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.

Write the second-degree Taylor polynomial for f about x=1.

B Write the second-degree Taylor polynomial for f about x=1.

$f^{\prime}(1)=2$ and $f^{\prime \prime}(1)=-9$	Two terms	Point 4 (P4)
$P_{2}(x)=-1+2(x-1)-\frac{9}{2}(x-1)^{2}$	Remaining term	Point 5 (P5)

Scoring Notes for Part B

- P4 and P5 can be earned with an answer consistent with incorrect values of $f^{\prime}(1)$ and $f^{\prime \prime}(1)$

imported from part A.

- Any terms of degree greater than two or " $+\ldots$ " does not earn P5.

- A response of $-1+2(x-1)-\frac{9}{2}(x-1)^{2}$ earns P4 and P5, regardless of any subsequent algebraic

simplification.

- A response that does not present the polynomial as powers of (x-1) but instead presents a correct

expanded/simplified form of the polynomial (e.g., $-\frac{9}{2} x^{2}+11 x-\frac{15}{2}$ ) earns P4 but not P5.

[Maximum number: 2]

Let y=f(x) be the particular solution to the differential equation $\frac{d y}{d x}=y \cdot(x \ln x)$ with initial condition f(1)=4. It can be shown that $f^{\prime \prime}(1)=4$ .

Write the second-degree Taylor polynomial for f about x=1. Use the Taylor polynomial to approximate f(2).

Write the second-degree Taylor polynomial for f about x=1. Use the Taylor polynomial to
approximate f(2).

$f^{\prime}(1)=\left.\frac{d y}{d x}\right|_{(x, y)=(1,4)}=4 \cdot(1 \ln 1)=0$

The second-degree Taylor polynomial for f about x=1 is
$f(1)+f^{\prime}(1)(x-1)+\frac{f^{\prime \prime}(1)}{2!}(x-1)^{2}=4+0(x-1)+\frac{4}{2}(x-1)^{2}.$

Polynomial

1 point

$f(2) \approx 4+2(2-1)^{2}=6$

Approximation

1 point

Scoring notes:

- The first point is earned for $4+\frac{4 \cdot \ln 1}{1!}(x-1)^{1}+\frac{4}{2!}(x-1)^{2}$ or any correctly simplified equivalent expression. A term involving (x-1) is not necessary. The polynomial must be written about (centered at) x=1.

- If the first point is earned, the second point is earned for just " 6 " with no additional supporting work.

- If the polynomial is never explicitly written, the first point is not earned. In this case, to earn the second point supporting work of at least " 4+2(1) " is required.

Total for part (a) 2 points

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Covered syllabus points

Question 1

Question 2