Problems with Rates of Change

A specific type of problem, that typically appears in the free response sections of the AP calculus AB test, defines the rate of change in time of a function. This can be the rate of accumulation of water in a lake or barrel, the rate at which calories are burn, etc. The rate of change of the function f(t) is defined with the help of a graph or an analytical expression. While the problems can be designed to look quite different, the defining characteristic is that the derivative of a function of time is known.

A large number of types of questions may be asked:
1. One type of problems may ask to calculate the time when the function f is increasing or decreasing at its fastest rate in a specific time interval. This type of questions can be answered by finding the maximum or minimum of f '(t).

2. The problem may ask the value f the function f(t) at a specific time in the interval. This question can be asked by integrating f '(t) from a point in time when the value of function f is known, f(t₀). Example.

3. The problem may ask if function f(t) is increasing or decreasing during a time interval or at a specific time. Example.

4. The problem may ask to determine the points of inflection of function f(t).

5. The problem may ask to determine the minima or maxima of function f(t).

Examples of problems with rates of change of functions

A checking account has $120,000 at time t=0. During the interval 0 ≤ t ≤ 18 months, money is deposited into the checking account at a rate d(t) = 2,000(t+1)sin(t/4) dollars/month. At the same time, money is withdrawn from the checking account at the rate w(t) = 10,000/(t+1) dollars/month. Is the dollar amount increasing at t=9 months? How many dollars are in the account at time t=18 months?

Solution:
d(9) - w(9) = 2000(9 + 1)sin(9/4) - 10000/(9 + 1)
d(9) - w(9) = 14,561 $/month is a positive value so money is deposited into the account at a higher rate than it is withdrawn at t = 9 months.
At t = 18 months, the sum available in the account is:

Let f be a function derivable on the (0, .6) interval. If f '(x) = 3 + ln(x + 1) and f(.1) = 2, what is f(.6)?.

Solution:


Using the calculator , f(.6) = 3.647

A particle is moving along the x axis so that the velocity v is given by the differentiable function whose graph is shown below. The graph has horizontal tangents at t = t₁, t = t₃, t = t₅ and t = t₇.
a) Is the speed increasing or decreasing on the (t₁, t₂) interval?
b) At what times is the acceleration of the particle equal to zero?
c) At what time is the particle farthest to the right?