Unit 4: Contextual Applications of Differentiation
Calculus BC · Unit 4 · Paper 2

Contextual Applications of Differentiation 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 1Paper 2Paper 3
Each paper is built from this unit’s 24 terms and is the same for everyone, so a teacher can assign “Unit 4, 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
1

Find the missing length at the instant

2

Speeding up test

3

Linearization

4

Rates carry signs

5

Changing direction requires a sign change

6

Decreasing at a decreasing rate

7

Units of an integral of a rate

8

Use a constraint to remove a variable

9

Justification with the Mean Value Theorem

10

Interpreting a negative rate

11

Units of a derivative

12

Optimization with a constraint

Short answer 1. Define or explain: Three-part interpretation sentence

3 pts

Short answer 2. Define or explain: Marginal cost

3 pts

Short answer 3. Define or explain: Increasing at a decreasing rate

3 pts

Short answer 4. Define or explain: Speed is not a derivative

3 pts

Free response

9 pts

CALCULATOR PERMITTED. An invasive species of plant appears in a fruit grove at time t = 0 and begins to spread. The function C defined by C(t) = 7.6·arctan(0.2t) models the number of acres in the fruit grove affected by the species t weeks after the species appears. It can be shown that C′(t) = 38/(25 + t²). (Your calculator should be in radian mode.)

A. Find the average number of acres affected by the invasive species from time t = 0 to time t = 4 weeks. Show the setup for your calculations.

B. Find the time t when the instantaneous rate of change of C equals the average rate of change of C over the time interval 0 ≤ t ≤ 4. Show the setup for your calculations.

C. Assume that the invasive species continues to spread according to the given model for all times t > 0. Write a limit expression that describes the end behavior of the rate of change in the number of acres affected by the species. Evaluate this limit expression.

D. At time t = 4 weeks, measures are taken to counter the spread. The function A, defined by A(t) = C(t) − ∫₄ᵗ 0.1·ln(x) dx, models the number of acres affected over 4 ≤ t ≤ 36. At what time t, for 4 ≤ t ≤ 36, does A attain its maximum value? Justify your answer.