Applications of Differentiation

Unit 4 begins by developing understanding of average and instantaneous rates of change in problems involving motion. The unit then identifies differentiation as a common underlying structure on which to build understanding of change in a variety of contexts. Students’ understanding of units of measure often reinforces their understanding of contextual applications of differentiation. In problems involving related rates, identifying the independent variable common to related functions may help students to correctly apply the chain rule. When applying differentiation to determine limits of certain indeterminate forms using L’Hospital’s rule, students must show that the rule applies.

Students will begin applying concepts from Units 2 and 3 to scenarios encountered in the world. Students must also be able to explain how an approximated value relates to the value it’s intended to approximate.

The Big Idea Questions that drive this unit are:

1. How are problems about position, velocity, and acceleration of a particle in motion over time structurally similar to problems about the volume of a rising balloon over an interval of heights, the population of London over the 14th century, or the metabolism of a dose of medicine over time?

2. Since certain indeterminate forms seem to actually approach a limit, how can we determine that limit, provided it exists?

• Interpret a derivative as
• The slope of a tangent line to the graph of a function
• Instantaneous rate of change
• Velocity or acceleration
• Solve related rate problems
• Find the linear approximation to a function
• Use differentials in applications
• Identify indeterminate forms
• Use L'Hosptals Rule to find a limit
• Find the limit of an indeterminate form

When we look at a derivative we will look at it in two different ways. The first one will be over an interval. Anytime we look at a derivative in this format, this will be called an average rate of change. You will use the slope formula to find this.

If we want to look at the derivative at a specific point, we will need to look at what is called the instantaneous rate of change. This will involve take the derivative of f(x) and then evaluate that derivative at the specific point that is being asked about.

Use these steps to help you solve a related rate problem:

1. Read the problem carefully, perhaps several times. Pay particular attention to the rate of change you are asked to find. If possible, draw a picture to illustrate the problem.

2. Identify the variables, assign symbols to them, and label the picture.

3. Identify the rates of changes as derivatives. Be sure to list the units for each variable and each rate of change and indicate what is given and what is asked for.

4. Write an equation that relates the variables. Sometimes it is necessary to use more than one equation.

5. Differentiate both sides of the equation.

6. Substitute all known numerical values in for the variables and its derivatives and solve for the unknown rate of change.

Pythagorean Theorem

Distance Formula

Area of

• Triangle
• Rectangle
• Circle
• Trapezoid
• Square

Volume of

• Prism
• Cube
• Cylinder
• Sphere
• Cone

When finding a linear approximation, you must do the following.

1. Find f'(x)

2. Evaluate f'(x) at the point that is closest to the point they are asking you to approximate. For example, if they are asking you to approximate f(2.1), then you will evaluate f'(x) at x = 2 to find the slope at 2.

3. Use the x value above and find the y value to go with it on the original function.

4. Put the point and the slope into point-slope form.

5. Solve for y.

6. Evaluate the estimation using the tangent line that you found in step 5. This will be the linear approximation at that point.

When dealing with limits, you will sometimes get an answer that will be 0/0 or infinity over infinity. Whenever this happens, you will need to do the following.

1. See if you can use algebraic techniques to simplify the limit.

2. If step one does not work, then you will use L'Hospital's Rule. Take the derivative of the top and then take the derivative of the bottom.

3. Evaluate the new top and the new bottom at the c value. If you get 0/0 or infinity over infinity, then you will need to do step 2 again. If you get an actual number or if it does not exist, then you have finished the problem.