AP Calculus AB Unit 4.6: Linearization

B There is a point P on the curve G near (2,-1) with x-coordinate 1.6. Use the line tangent to the curve at

(2,-1) to approximate the y-coordinate of point P.

Scoring Notes for Part B

- A response can earn $\mathbf{P 3}$ with $\left.\frac{d y}{d x}\right|_{(x, y)=(2,-1)}=-\frac{1}{4}, \frac{d y}{d x}=-\frac{1}{4}$, "slope is $-\frac{1}{4}$," or equivalent.

- A response that presents a linear approximation with a slope of $-\frac{1}{4}$ also earns $\mathbf{P 3}$.

- A response that declares $\left.\frac{d y}{d x}\right|_{(x, y)=(2,-1)}$ (or the slope) equal to any nonzero value $k \neq-\frac{1}{4}$ does not

earn P3. Such a response earns P4 for a presented approximation mathematically equivalent to

-1+k(-0.4).

- P4 cannot be earned with a linear approximation using a slope other than $-\frac{1}{4}$ if that slope has not

been declared to be the value of $\left.\frac{d y}{d x}\right|_{(x, y)=(2,-1)}$.

- A response does not have to present the tangent line equation but must clearly demonstrate its use at

x=1.6 in finding the requested approximation to be eligible for P4.

- A response of $-1-\frac{1}{4}(-0.4)$ earns both P3 and P4.

- A response of $-1-\frac{1}{4}(1.6-2)$ or equivalent banks P4 (i.e., subsequent errors in simplification will

not be considered in scoring for P4).

Note: An ambiguous response, such as $-1-\frac{1}{4}(1.6-2$, does not bank P4 and therefore must go on

to resolve the ambiguity with a correct final answer (e.g., -0.9 ) to earn P4.