Review linearization by using tangent-line models to estimate nearby function values and assess approximation error, especially when a calculator value is unavailable.
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 ( ∘C) and t is the number of minutes since the bottle was placed in the pan. M satisfies the differential equation dtdM=41(40−M). At time t=0, the temperature of the milk is 5∘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.
dtdMt=0=41(40−5)=435dtdMt=0
1 point
The tangent line equation is y=5+435(t−0).
M(2)≈5+435⋅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+435⋅2 is the minimal response to earn both points.
- A response of 41(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 435 or a nonzero slope that is declared to be the value of dtdM.
- 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