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Differentiation
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2.8
Related Rates
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3. Objectives
Examine related variables.
Solve related-rate problems.

4. Related Variables

5. Related Variables
In this section, you will study problems involving variables that are changing with respect to time. If two or more such variables are related to each other, then their rates of change with respect to time are also related.
For instance, suppose that x and y are related by the equation
y = 2x.

6. Related Variables
If both variables are changing with respect to time, then their rates of change will also be related.
In this simple example, you can see that because y always has twice the value of x, it follows that the rate of change of y with respect to time is always twice the rate of change of x with respect to time.

7. Example 1 – Examining Two Rates That Are Related
The variables x and y are differentiable functions of t and are related by the equation
y = x2 + 3.
When x = 1, dx/dt = 2. Find dy/dt when x = 1.
Solution:
Use the Chain Rule to differentiate both sides of the equation with respect to t.

8. Example 1 – Solution
cont’d
When x = 1 and dx/dt = 2, you have

9. Solving Related-Rate Problems

10. Example 2 – Changing Area
A pebble is dropped into a calm pool of water, causing ripples in the form of concentric circles, as shown in the photo.
The radius r of the outer ripple is increasing at a constant rate of 1 foot per second.
When the radius is 4 feet, at what rate is the total area A of the disturbed water changing?
Total area increases as the outer radius increases

11. Example 2 – Solution
The variables r and A are related by the equation for the area of a circle, A =  r 2.
To solve this problem, use the fact that the rate of change of the radius is given by dr/dt.
Equation:
Given rate:
Find:

12. Example 2 – Solution
cont’d
Using this model, you can proceed as in Example 1.

13. Example 2 – Solution When r = 4 and dr/dt = 1, you have
cont’d
When r = 4 and dr/dt = 1, you have
When the radius is 4 feet, the area is changing at a rate of 8 square feet per second.

14. Solving Related-Rate Problems
In Example 2, note that the radius changes at a constant rate (dr/dt = 1 for all t), but the area changes at a nonconstant rate.

15. Solving Related-Rate Problems
The solution shown in Example 2 illustrates the steps for solving a related-rate problem.

16. Solving Related-Rate Problems
In Step 2 of the guidelines, note that you must write an equation that relates the given variables.
To help you with this step, reference tables that summarize many common formulas are included in the appendices.
For instance, the volume of a sphere of radius r is given by the formula

17. Solving Related-Rate Problems
The table below lists examples of the mathematical models for some common rates of change that can be used in the first step of the solution of a related-rate problem.

18. Example 4 – Increasing Production
A company is increasing the production of a product at the rate of 200 units per week. The weekly demand function is modeled by
p = 100 – 0.001x
where p is the price per unit and x is the number of units produced in a week. Find the rate of change of the revenue with respect to time when the weekly production is 2000 units.
Will the rate of change of the revenue be greater than $20,000 per week?

19. Example 4 – Solution
Because production is increasing at a rate of 200 units per week, you know that at time t the rate of change is dx/dt = 200.
So, the problem can be stated as shown.
Given rate:
Find:

20. Example 4 – Solution
cont’d
To find the rate of change of the revenue, you must find an equation that relates the revenue R and the number of units produced x.
Equation:
By differentiating both sides of the equation with respect to t, you obtain

21. Example 4 – Solution Using x = 2000 and dx/dt = 200, you have
cont’d
Using x = 2000 and dx/dt = 200, you have
No, the rate of change of the revenue will not be greater than $20,000 per week.