# Ch 12: Graphing Derivatives & L'Hopital's Rule - AP Calculus: Homeschool Curriculum

### About This Chapter

## Who's it for?

This unit of our AP Calculus Homeschool course will benefit any student who is trying to learn about graphing derivatives and L'Hopital's Rule. There is no faster or easier way to learn about calculus. Among those who would benefit are:

- Students who require an efficient, self-paced course of study to learn to explain the properties of a function from its derivatives.
- Homeschool parents looking to spend less time preparing lessons and more time teaching.
- Homeschool parents who need a calculus curriculum that appeals to multiple learning types (visual or auditory).
- Gifted students and students with learning differences.

## How it works:

- Students watch a short, fun video lesson that covers a specific unit topic.
- Students and parents can refer to the video transcripts to reinforce learning.
- Short quizzes and a Graphing Derivatives and L'Hopital's Rule unit exam confirm understanding or identify any topics that require review.

## Graphing Derivatives and L'Hopital's Rule Unit Objectives:

- Learn to apply L'Hopital's Rule in simple and complex cases.
- Explain concavity and inflection points graphically.
- Graph the derivative from any function.
- Calculate the minimum and maximum values of a graph.
- Learn about data mining.

### 1. Graphing the Derivative from Any Function

When you know the rules, calculating the derivates of equations is relatively straightforward, although it can be tedious! What happens, though, when you don't know the function? In this lesson, learn how to graph the derivative of a function based solely on a graph of the function!

### 2. Non Differentiable Graphs of Derivatives

When I walk along a curve, I stand normal to it. That is, I stand perpendicular to the tangent. Learn how to calculate where I'm standing in this lesson.

### 3. How to Determine Maximum and Minimum Values of a Graph

What is the highest point on a roller coaster? Most roller coasters have a lot of peaks, but only one is really the highest. In this lesson, learn the difference between the little bumps and the mother of all peaks on your favorite ride.

### 4. Using Differentiation to Find Maximum and Minimum Values

If you are shot out of a cannon, how do you know when you've reached your maximum height? When walking through a valley, how do you know when you are at the bottom? In this lesson, use the properties of the derivative to find the maxima and minima of a function.

### 5. Concavity and Inflection Points on Graphs

You might not think of a cup when you think of an awesome skateboard ramp. But I'm sure a really bad ramp would give you a frown, right? Learn about cups and frowns in this lesson on concavity and inflection points.

### 6. Understanding Concavity and Inflection Points with Differentiation

Put a little more meaning behind those cups and frowns. In this lesson, use the second derivative of a function to determine if it is concave up or concave down.

### 7. Data Mining: Function Properties from Derivatives

Some shoes come with accelerometers that give a person's acceleration as a function of time. From this information, the shoe can determine roughly how fast you're going. In this lesson, learn how this works as we take the derivative of a function and glean information from it about the original function.

### 8. Data Mining: Identifying Functions From Derivative Graphs

If you saw the graph of speed as a function of time for a bicycle, a jet, and a VW bug, could you pick which vehicle produced which graph? In this lesson, try it as we match functions with their derivatives.

### 9. What is L'Hopital's Rule?

A Swiss mathematician and a French mathematician walk into a bar ... and they walk out with the famous L'Hopital's rule for finding limits. In this lesson, learn what these two mathematicians came up with and how to use it to avoid the limit of zero divided by zero!

### 10. Applying L'Hopital's Rule in Simple Cases

L'Hôpital's rule may have disputed origins, but in this lesson you will use it for finding the limits of a range of functions, from trigonometric to polynomials and for limits of infinity/infinity and 0/0.

### 11. Applying L'Hopital's Rule in Complex Cases

L'Hôpital's rule is great for finding limits, but what happens when you end up with exactly what you started with? Find out how to use L'Hôpital's rule in this and other advanced situations in this lesson.

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### Other Chapters

Other chapters within the AP Calculus AB & BC: Homeschool Curriculum course

- Functions - AP Calculus: Homeschool Curriculum
- Graphing & Functions - AP Calculus: Homeschool Curriculum
- Sequences & Series - AP Calculus: Homeschool Curriculum
- Limits - AP Calculus: Homeschool Curriculum
- Continuity - AP Calculus: Homeschool Curriculum
- Exponentials & Logarithms: Homeschool Curriculum
- Exponents & Polynomials - AP Calculus: Homeschool Curriculum
- Applications of Derivatives - AP Calculus: Homeschool Curriculum
- Calculating Derivatives & Derivative Rules - AP Calculus: Homeschool Curriculum
- Differential Equations - AP Calculus: Homeschool Curriculum
- Area Under the Curve & Integrals - AP Calculus: Homeschool Curriculum
- Integration & Integration Techniques - AP Calculus: Homeschool Curriculum
- Integration Applications - AP Calculus: Homeschool Curriculum
- Rate of Change - AP Calculus: Homeschool Curriculum
- Geometry and Trigonometry - AP Calculus: Homeschool Curriculum
- Using Scientific Calculators - AP Calculus: Homeschool Curriculum