Contextual Applications 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.
Position from velocity
Average velocity versus instantaneous
Related rates with a cone or sphere
Position needs an initial condition
Units of a definite integral
Particle at rest vs changing direction
Speed is not a derivative
Cone volume relation
Sphere volume and surface area
Common related-rates error
Rectilinear motion
Linearization formula
Short answer 1. Define or explain: Decreasing at a decreasing rate
3 ptsShort answer 2. Define or explain: Related rates procedure
3 ptsShort answer 3. Define or explain: Rate versus change versus amount
3 ptsShort answer 4. Define or explain: The rate is decreasing
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.