AP Calculus BC 10.11.2: Taylor Approximations

Practice AP Calculus BC questions on evaluating a Taylor polynomial near its center to approximate a function value accurately.

[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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LIM-8.B—Approximate function values using a Taylor polynomial question 1