# 4.3: Applications of Derivatives

**At Grade**Created by: CK-12

## Related Rates

CONTENT

This lesson is a kind of tour through implicit differentiation. It may be best to understand and teach these ideas based upon the idea of differentiation as a kind of operator. That is, if we are given some equality among any number of different variables like:

then we can operate on both sides of this equation with the derivative operator

This equality should be interpreted as explaining how the rates of change for the side lengths

Alternatively, we could have operated on both sides with the derivative operator

PROCESS

Related rates are readily applicable to real-world problems. For example, how does the radius of a balloon or a car tire change as air is pumped into it? How fast does the amount of carbon-dioxide in the atmosphere increase for a given rate human population increase? How much does a star become brighter as we increase its magnification? All of these questions are questions of related rates, and can be used to make this material more interesting.

A nice way to teach this is to have students actually experiment with the formulas. For example, the students could be split into groups of three to study the formula for a right triangle. One member would be the referee and would need some kind of a stopwatch. The other two students would stand in one corner and walk away from each other along the wall measuring the length to the next corner in steps. Ideally they should coordinate so that their step size is about the same.

The distances along the wall and the diagonal distance between the students at opposite corners should be recorded. Then the referee should count or clap off seconds and the other two students should walk along the wall back towards the corner. One should walk a rate of one step per second and the other at a rate of one step every two seconds. After

where

PRODUCTS

Word problems are the classic approach to testing this material, but it should be recognized that some students are much better at these than others. For the students who struggle putting words into equations, it may be best to test this material purely using equations. Questions like the following are good for these students:

1. Suppose the variables

a. What is

b. When

2. The volume of a sphere

a. Find a formula for

b. If the radius of a unit sphere

## Extrema and the Mean Value Theorem

CONTENT

In order to teach effectively about extrema, Rolle’s Theorem, and the mean value theorem it is necessary to use geometric intuition and good pictures throughout. The language in the description of a **maximum** and a **minimum** is quite complicated but the concept could not be clearer. Forgetting endpoints for a moment, a function has a maximum at*interior* maxima and minima.

The key to understanding everything in this lesson is that the function goes from increasing to decreasing at an interior maximum, and from decreasing to increasing at an interior minimum. This means that the derivative goes from positive to negative in the former, and from negative to positive in the latter. However, for a continuous derivative this is only possible if it is zero at the point in question. That is, a differentiable maximum/minimum must have horizontal tangent line with

Rolle’s Theorem is not proven in the book but the idea is extremely geometric and the opportunity to show a simple picture in class should not be missed. The point is that if a nice (differentiable) function over

The mean value theorem is proven by creating a function from *change* between

PROCESS

Teaching this material can be a little tricky since it is largely conceptual. The best technique may be to have students pair off so that one person will first explain how to prove that if a point is an interior extrema then its derivative is zero. The other person will then explain how to prove the mean value theorem. The partner who is explaining should have only paper and pencil whereas the other person can help him/her along with the help of the book. This will help students become more comfortable with the logical progression of these proofs.

PRODUCTS

Students can be tested on this material by finding maxima and minima and by sketching a great many plots. This will be particularly useful with the next few chapters in mind. By having the goal in mind that they are to use derivatives in order to plot functions, students will be far more likely to understand what follows.

## The First Derivative Test

CONTENT

The content of this chapter really is purely geometric and has the best chance at getting through to students in this way. It is quite simple, when the derivative is positive the function is increasing and when it is negative, it is decreasing. This is actually precisely what the derivative measures, whether a function is increasing or decreasing. Furthermore, it should be quite intuitive that a function increases to a maximum then decreases away from it, and that the opposite is true of a minimum.

PROCESS

This material can be conveyed to students by having them think before studying the formal ideas. They should consider some complicated looking function like

We find that its derivative:

is zero whenever

Students should then be asked how they can tell if

This plot makes it clear that

PRODUCTS

A very nice little riddle is solved by the function above. Consider asking students to solve the problem:

*Using the following steps instead of your calculator, determine which is bigger:**or**?*

a. Start with the equation

i. Answer:

b. Now manipulate this inequality so that one side involves only

i. Answer:

c. Now notice that by replacing

i. Answer: We solved this above, the function

d. Now for the punch line, what is the value of this function at its critical point/s?

i. Answer: At

e. How do these value/s compare with the function’s value at

i. Answer:

f. What are the numerical values of

i. Answer:

## The Second Derivative Test

CONTENT

Notice that in order to determine whether critical values were maxima, minima, or saddle points we appealed to how the derivative was changing. If the derivative was positive beforehand and then negative after, i.e. decreasing, then the point was a maximum. If on the other hand the derivative was negative beforehand and then positive after, i.e. increasing, then the point was a minimum. If on the other hand the sign did not change, i.e. the derivative was constant; the point was a saddle point. The quick way to check how something is changing is to take its derivative. So if we want to see how the derivative is changing, we should look at its derivative: The second derivative!

When the second derivative is positive the first derivative is increasing and this gives the function a *concave up shape*. If this is true at a critical point then the critical point must be a minimum. When the second derivative is negative the first derivative is decreasing and this gives the function a *concave down* shape. If this is true at a critical point then the critical point must be a maximum. If the second derivative is zero at a critical point then the function may have a maximum, minimum, or saddle point there.

PROCESS

Recall the complicated function we examined earlier:

we used its first derivative:

to see that it had a critical point at

From this we see easily that:

So the second derivative is positive at the critical point

PRODUCTS

Students can be tested on this material with a large variety of word problems asking them to formulate a function and then classify its critical points. For students who do not work well with word problems it may be better to simply give functions directly and ask students to classify their critical points using the first and second derivatives. As a look ahead, again, it is useful to have students draw plots of the functions as well, using the information that they have determined.

## Limits at Infinity

CONTENT

In order to understand the behavior of functions, it is often important to look more closely near points they are not defined, and examine how they behave after they run off your graph paper. A function like

tell us that to the left of the origin the function tends down towards negative infinity and to the right coming in towards zero it tends up to positive infinity.

We can also look at how functions behave for very very large values of

Although we see clearly that

That is to say that in this limit, for large

PROCESS

With an eye towards what follows, students may find it useful to practice this material by dividing into small groups and analyzing a function. Each group could be given a function like

PRODUCTS

This material is readily tested in a way that leads nicely towards using derivatives and limits to graph functions. Students should be given some function with a variety of critical points and discontinuities and asked to consider how the function behaves near these points. This should be done by taking limits if the derivative is not defined or one and two derivatives where it is defined and checking for their signs. Then the student should be asked to analyze how the function behaves for values of

## Analyzing the Graph of a Function

CONTENT

This section is essentially a culmination of the ideas that students have been considering thus far. By putting together all of the tools they have been given, it should be much easier to get a feeling for how a function’s plot will look ahead of time.

PROCESS

This section should be taught by having students repeat the kinds of exercises they have done earlier. The class could be split into pairs and each team could thoroughly analyze some complicated function by finding all the critical points, classifying each, and examining the limits at discontinuities and with each tail. It would be nice if each pair could present their function to the class on the board so that everyone could see the variety of functions.

PRODUCTS

This material is best tested by having students thoroughly analyze at least one function with some interesting behavior. The best functions are usually rational functions.

## Optimization

CONTENT

Optimization problems are probably the best way to make calculus seem important. This content should be described as one of the most important things that you hope to convey to students in the course. The tools they have learned thus far will be consolidated in this section to provide an enormous application.

PROCESS

The best way to teach this material is to provide students with a simple question. Suppose you are CEO of Starbucks and are trying to decide how expensive a small coffee should cost to obtain the most profit. Clearly if the coffee is free the profit will be zero. On the other hand, if the price is too high nobody will buy it and the profit will also be zero. So there must be some *intermediate* price that is not too high and not too low so that the profit is the greatest. This is a problem of optimization, and it can be solved using calculus.

If students are interested, tell them that by analyzing years and years of data you have modeled the profit

It is clear that this function satisfies our expectations that profit

In order to determine the optimal price, we must use calculus. We take a derivative of the function to see that:

In order to find the maximum, we would like to look for critical points. That is, we want to look for prices

Now, this equation cannot be solved by algebra or by any other exact means. So we must resort to something like Newton’s Method. So we let:

and we calculate that:

If we guess that a good price for coffee might be close to

So the derivative

at

PRODUCTS

There are some great problems in optimization for students who are good at word problems and for students who are not. Word problems can be avoided by asking for specific properties of an explicit function or by simply presenting a word problem with the equations explicitly written out.

## Approximation Errors

CONTENT

This lesson is best explained by first saying that essentially any function that we’ll use in calculus is exactly equal to its Taylor Polynomial as

where the dots are meant to imply that this summation goes on and on without ever ending. The fact is that the terms are guaranteed to eventually get smaller and smaller because of the factorial in the denominator, no matter how big

An approximation just means that instead of taking *all* the infinite number of terms, we cut off the series somewhere. The remaining infinite terms added up are called the “tail” of the series and the approximation is good when this tail is small. That is because the tail represents the difference between the actual function and the approximate function. Thus, we make the approximation better and better when we take more terms or when

PROCESS

Students could be split into groups of three that are each given a function and a value for **artist**, one will be the **derivator**, and one will be the **calculator**.

The calculator will begin by making a table with a series of points near

Then the artist will work with this table to draw a careful graph of the function on graph paper near the point

The calculator will calculate and fill in the column of the table for the same

PRODUCTS

Students can be tested on this material by having them create approximations for functions with various degrees of accuracy.

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## Date Created:

Feb 23, 2012## Last Modified:

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