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### RELATED RATES AND OPTIMIZATION TUTORIAL

This tutorial shows how calculus can be used to solve real problems. These are just some of the applications of the concepts covered in the previous tutorials.

# Related Rates

Related rates problems require us to find the rate of change of one value, given the rate of change of a related value. We must find an equation that associates the two values and apply the chain rule to differentiate each side of the equation with respect to time.. Suppose we have a variable v. The derivative, dv/dt would be the rate of change of v.

When solving related rates problems, we should follow the steps listed below.

1) Draw a diagram. This is the most helpful step in related rates problems. It allows us to visualize the problem.

2) Assign variables to each quantitiy in the problem that is a function of time. Each of these values will have some rate of change over time.

3) List all information that is given in the problem and the rate of change that we are trying to find.

4) Write an equation that associates the variables with one another. If there are variables for which we are not given the rates of change (except for the rate of change that we are trying to determine), we must find some relation from the nature of the question that allows us to write these variables in terms of variables for which the rates of change are given. We must then substitute these relations into the main equation.

Note: For an example of this situation, see example #3 below.

5) Using the chain rule, differentiate each side of the equation with respect to time.

6) Substitute all given information into the equation and solve for the required rate of change.

Note: It is important to wait until the equation has been differentiated to substitute information into the equation. If values are substituted too early, it can lead to an incorrect answer.

# Examples

1 | Find the rate of change
2 | Find the rate of change
3 | Find the rate of change when given more variables than rates of change

# Maximum and Minimum Values

A function f has an absolute maximum at c if f(c) ³ f(x), for all x in the domain of f. At x=c, the graph reaches its highest point. The number f(c) is called the maximum value of f.

A function f has an absolute minimum at c if f(c) £ f(x), for all x in the domain of f. At x=c, the graph reaches its lowest point. The number f(c) is called the minimum value of f.

Together, the maximum and minimum values are called the extreme values of the function f.

A function f(x), defined on the open interval (a, b) has a local maximum at a point c in (a, b), if f(c) ³ f(x) for all x near c. This means that there is an interval around c (possibly very small), such that f(c) ³ f(x) for all x in the interval.

A function f(x), defined on the open interval (a, b) has a local minimum at a point c in (a, b), if f(c) £ f(x) for all x near c. This means that there is an interval around c (possibly very small), such that f(c) £ f(x) for all x in the interval.

Suppose we have a function defined on a closed interval [c, d]. A local maximum or minimum can not occur at the endpoints of this interval because the definition requires that the point is contained in some open interval (a, b). Since the function is not defined for some open interval around either c or d, a local maximum or local minimum cannot occur at this point. An absolute maximum or minimum can occur, however, because the definition requires that the point simply be in the domain of the function.

# Fermat's Theorem

If f has a local maximum or minimum at c and f'(c) exists, then f'(c) = 0.

The converse of this theorem is not true. If f'(c) = 0, then c is not necessarily a maxiumum or minimum value. There may also be a maximum or minimum value when f'(c) does not exist.

A critical number of a function f is a number c in the domain of f, such that f'(c)=0 or f'(c) does not exist.

Using the definition of a critical number, we can rephrase Fermat's Theorem as : If f has a local maximum or minimum at c, then c is a critical number of f.

# Closed Interval Method

To find the absolute maximum or minumum values of a continuous function f on a closed interval [a, b], we first find the critical numbers of the function in (a, b) and calculate the value of the function at each critical number. Next we find the values of the function at the endpoints of the interval. We then compare all of these values. The largest value is the absolute maximum of the function on the interval [a, b], while the smallest value is the absolute minimum.

# Examples

4 | Find the critical numbers of the function
5 | Find the absolute maximum and minimum of the function on [a, b]

# Optimization

The concept of maximum and minimum values allows us to solve optimization problems. These problems are one of the most practical applications of differential calculus. They allow us to find the optimal way to perform some task.

To solve optimization problems, we follow the steps listed below

1) Draw a diagram, if necessary, to help visualize the problem.

2) Assign variables to the quantity to be optimized and all other unknown quantities given in the question.

3) Write an equation that associates the optimal quantity to the other variables. If the optimal quantity is expressed in terms of more than one variable, we must eliminate the extra variables. We use the nature of the question to find some relation between the variables and substitute these relations into the equation for the optimal quantity. The optimal quantity equation should be in terms of only one variable so that it has the form f(x).

4) Find the absolute maximum or minimum of f(x), depending on the question. If the domain of f is closed, use the closed interval method.

# Examples

6 | Find two numbers whose sum is S and whose product is a maximum
7 | Find the dimensions of the box that minimize surface area

For more practice with the concepts covered in this tutorial, visit the Related Rates and Optimization Problems page at the link below. The solutions to the problems will be posted after these chapters are covered in your calculus course.

To test your knowledge of these application problems, try taking the general related rates and optimization test on the iLrn website or the advanced related rates and optimization test at the link below.

Related Rates and Optimization Problems
General Related Rates and Optimization Test on iLrn
Advanced Related Rates and Optimization Test

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MATH 1036
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Alex Karassev
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