Calculus AB · Unit 7 · Paper 2

Differential Equations unit test

A test on this unit alone, marked as a percentage and a letter grade — for the test your class is actually sitting, rather than for May. Answer everything, then submit once: seeing the answer to question 3 before attempting question 4 makes the final percentage meaningless.

Paper 1 Paper 2 Paper 3

Each paper is built from this unit’s 28 terms and is the same for everyone, so a teacher can assign “Unit 7, Paper 2” and every student sits the identical test. Multiple choice is marked objectively; the written sections you mark yourself against the model answer and rubric.

Suggested time 36 min 33 points0/17 attempted

Exponentiating a sum gives a product

Why the constant matters

Interpreting k

Separation requires a product form

Domain of a particular solution

Solutions cannot cross an equilibrium

Slope field

Newton’s law of cooling

Reading equilibrium solutions

Half-life and doubling time

dy/dt = ky solution

Half-life from k

Short answer 1. Define or explain: Newton's law of cooling

3 pts

Short answer 2. Define or explain: Verifying a proposed solution

3 pts

Short answer 3. Define or explain: Sign of the constant in exponential models

3 pts

Short answer 4. Define or explain: k as a continuous rate

3 pts

Free response

9 pts

NO CALCULATOR. A pie is taken from a hot oven and put on a table. The internal temperature of the pie at time t minutes can be modeled by the function H that satisfies the differential equation dH/dt = −(1/15)(H − 20), where H(t) is measured in degrees Celsius and H(0) = 75. For t > 0, it is known that 20 < H(t) < 75. A proposed slope-field figure shows short segments at grid points for 0 < t < 30 and 25 < H < 75 in which EVERY segment has positive slope, with the segments steeper at larger values of H and the slopes constant along each horizontal row.

A. Explain why the figure described could not be a slope field for the differential equation dH/dt = −(1/15)(H − 20).

B. Find the slope of the line tangent to the graph of H at time t = 0. Show the work that leads to your answer.

C. It can be shown that d²H/dt² = (1/225)(H − 20). The line tangent to the graph of H at time t = 0 is used to approximate H(5). Is this approximation an overestimate or an underestimate for the actual value of H(5)? Give a reason for your answer.

D. Use separation of variables to find an expression for H(t), the particular solution to the given differential equation with initial condition H(0) = 75.