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.
The rate is decreasing
Find the missing length at the instant
Speed is not a derivative
A rate need not be constant
Related rates with implicit relations
Units of a derivative
Over- or under-estimate of a tangent line
Interpreting a negative rate
Linearization and error
How fast versus how much
Changing direction requires a sign change
Decreasing at a decreasing rate
Short answer 1. Define or explain: Units of an integral of a rate
3 ptsShort answer 2. Define or explain: Marginal cost
3 ptsShort answer 3. Define or explain: Increasing at a decreasing rate
3 ptsShort answer 4. Define or explain: Differentiate before substituting
3 ptsFree response
9 ptsCALCULATOR 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.