AP Calculus BC 4.6: Linearization

Practice AP Calculus BC questions on using a tangent line to approximate nearby values and determine approximation error direction.

[Maximum number: 2]

A bottle of milk is taken out of a refrigerator and placed in a pan of hot water to be warmed. The increasing function M models the temperature of the milk at time t, where M(t) is measured in degrees Celsius ( $\left.{ }^{\circ} \mathrm{C}\right)$ and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation $\frac{d M}{d t}=\frac{1}{4}(40-M)$ . At time t=0, the temperature of the milk is $5^{\circ} \mathrm{C}$ . It can be shown that M(t)<40 for all values of t.

Use the line tangent to the graph of M at t=0 to approximate M(2), the temperature of the milk at time t=2 minutes.

$\left.\frac{d M}{d t}\right|_{t=0}=\frac{1}{4}(40-5)=\frac{35}{4}$ $\left.\frac{d M}{d t}\right|_{t=0}$

1 point

The tangent line equation is $y=5+\frac{35}{4}(t-0)$ .
$M(2) \approx 5+\frac{35}{4} \cdot 2=22.5$

The temperature of the milk at time t=2 minutes is approximately 22.5° Celsius.

Approximation

1 point

Scoring notes:

- The value of the slope may appear in a tangent line equation or approximation.

- A response of $5+\frac{35}{4} \cdot 2$ is the minimal response to earn both points.

- A response of $\frac{1}{4}(40-5)$ earns the first point. If there are any subsequent errors in simplification,

the response does not earn the second point.

- In order to earn the second point the response must present an approximation found by using a

tangent line that:

○ passes through the point (0,5) and

○ has slope $\frac{35}{4}$ or a nonzero slope that is declared to be the value of $\frac{d M}{d t}$ .

- An unsupported approximation does not earn the second point.

- The approximation need not be simplified, but the response does not earn the second point if the

approximation is simplified incorrectly.

Total for part (b) 2 points

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CHA-3.F—Approximate a value on a curve using the equation of a tangent line question 1