# Ch 12: AP Calculus - Graphing Derivatives & L'Hopital's Rule: Tutoring Solution

### About This Chapter

## How it works:

- Begin your assignment or other AP calculus work.
- Identify the graphing derivatives and L'Hopital's Rule concepts that you're stuck on.
- Find fun videos on the topics you need to understand.
- Press play, watch and learn!
- Complete the quizzes to test your understanding.
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## Who's it for?

This chapter of our AP calculus tutoring solution will benefit any student who is trying to learn about graphing derivatives and L'Hopital's Rule for AP calculus and earn better grades. This resource can help students including those who:

- Struggle with understanding how to determine maximum value on a graph, explain concavity, identify functions from derivative graphs, apply L'Hopital's rule in simple cases or any other graphing derivatives and L'Hopital's Rule in AP calculus topic
- Have limited time for studying
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## Why it works:

**Engaging Tutors:**We make learning about graphing derivatives and L'Hopital's Rule for AP calculus simple and fun.**Cost Efficient:**For less than 20% of the cost of a private tutor, you'll have unlimited access 24/7.**Consistent High Quality:**Unlike a live calculus tutor, these video lessons are thoroughly reviewed.**Convenient:**Imagine a tutor as portable as your laptop, tablet or smartphone. Learn about graphing derivatives and L'Hopital's Rule on the go!**Learn at Your Pace:**You can pause and rewatch lessons as often as you'd like, until you master the material.

## Learning Objectives

- Graph the derivative of any function.
- Describe what happens at the kinks of non-differentiable graphs of derivatives.
- Determine the minimum and maximum values of a graph.
- Find minimum and maximum values using differentiation.
- Identify concavity and inflection points of graphs.
- Determine concavity and inflection points using differentiation.
- Describe function properties from derivatives.
- Use derivative graphs to identify functions.
- Become familiar with L'Hopital's rule.
- Apply L'Hopital's rule in simple and complex cases.

### 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: Tutoring Solution course

- Functions in AP Calculus: Tutoring Solution
- Graphing and Functions in AP Calculus: Tutoring Solution
- Sequences and Series in AP Calculus: Tutoring Solution
- Limits in AP Calculus: Tutoring Solution
- Continuity in AP Calculus: Tutoring Solution
- Exponentials and Logarithms in AP Calculus: Tutoring Solution
- Exponents and Polynomials in AP Calculus: Tutoring Solution
- Applications of Derivatives in AP Calculus: Tutoring Solution
- AP Calculus - Derivatives Calculations & Rules: Tutoring Solution
- Differential Equations in AP Calculus: Tutoring Solution
- Area Under the Curve and Integrals in AP Calculus: Tutoring Solution
- AP Calculus - Integration: Tutoring Solution
- Integration Applications in AP Calculus: Tutoring Solution
- Rate of Change in AP Calculus: Tutoring Solution
- Geometry and Trigonometry in AP Calculus: Tutoring Solution
- How to Use Scientific Calculators for AP Calculus: Tutoring Solution