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Rate of Change

If we are pumping gas into a balloon, both the volume and the radius of the balloon are increasing and their rates of increase are related to each other. We may wish to study the relationship among these parameters. For example, if we know that the volume of balloon is increasing at a certain rate, then we may wish to know how that affects the rate of change of the radius. This consideration gives rise to related rates problems. Here is a general procedure for solving related rates problems.

General procedure for solving related rates problems
Step 1: Read the given problem carefully and, if appropriate, draw a diagram.
Step 2: Represent the given information and unknowns by mathematical symbols.
Step 3: Write an equation involving the rate of change to be determined. If the equation contains more than one variable, it may be necessary to reduce the equation to one variable.
Step 4: Differentiate each term of the equation which is obtained in above step with respect to time.
Step 5: Substitute all the known rates of change and known values into the equation which is obtained in the above step, and solve it for the desired rate of change.
Step 6: Write the answer and indicate the units of measure.

Ex 1: If a snowball melts so that its surface area decreases at a rate of 1 cm2/min, then find the rate at which the diameter decreases when the diameter is 12 cm.

Sol:

Step 1: Define the variables. Let 'A' be the surface area of the snowball; 'D' be the diameter of snowball; 't' be the time in minutes.
Step 2: Given: = 1 cm2/min; Find: at D = 12 cm.
Step 3: Set up an equation: A = πD2.
Step 4: Differentiate both sides with respect to time 't'.
Step 5: Substitute the known values in above equation.
Step 6: Thus, the diameter of snowball decreases at the rate of 0.013 cm/min.

Ex 2: A boat is pulled into a dock by a rope attached to the bow of the boat and passing through a pulley on the dock that is 1 m higher than the bow of the boat. If the rope is pulled in at a rate of 2 m/s, how fast is the boat approaching the dock when it is 10 m from the dock ?
Sol:

Step 1: Define the variables. Let 'L' is the length of the rope; 'x' is the distance of the boat from the dock and 't' be the time in seconds.
Step 2: Given: = 2 m/sec; Find: at x = 10 m.
Step 3: Set up an equation: .
Step 4: Differentiate both sides with respect to time 't'.
Step 5: Substitute the known values in above equation.
Step 6: Thus, the boat is approaching the dock at 2.01 m/s.
Average rate of change

If y = f(x), then the average rate of change in y
between x = x1 and x = x2 is defined as

Geometrically, the average rate of change of y w.r.t. x is the slope of secant line joining (x1, f(x1)) and (x2, f(x2)) which are the points lying on the graph of y = f(x)
The units of average rate of change of a function are the units of 'y' per unit of the variable 'x'.

Instantaneous rate of change of a function f at x = x0:
If y = f(x), then instantaneous rate of change of a function f at x = x0 is defined as

which is equal to f '(x0)
i.e., instantaneous rate of change of f at x = x0 is f '(x0).

Rectilinear motion

The motion of a particle in a line is called rectilinear motion.
It is customary to represent the line of motion by a co-ordinate axis.
We choose a reference point (origin), a positive direction (to the right of origin) and a unit of distance on the line.

The rectilinear motion is described by
s = f(t) where f(t) is the rule connecting 's' and 't'
and 's' is the coordinate of the particle for the amount of time 't' that elapsed since the motion began.



If a particle moves according to the rule s = f(t)
where s is the displacement of the particle at time 't', then
is the average rate of change in s between 't' and t + Δt.
i.e., =

Since the rate of change of displacement is the velocity,
we call as the average velocity of the function s = f(x) between the time 't' and 't + Δt'.