AP Calculus BC Lim 8 A Represent a Function At a Point As a Taylor Polynomial Questions

Practice AP Calculus BC questions on building Taylor polynomials from derivative values, known series, products, and compositions.

[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.

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AP Calculus BC Lim 8 A Represent a Function At a Point As a Taylor Polynomial Questions question 1