About This Chapter
Graphing Derivatives and L'Hopital's Rule - Chapter Summary and Learning Objectives
Put simply, calculus is the mathematical study of change. That can really come in handy when everything seems to be in flux all the time! When you graph derivatives, you visually represent a mathematical function as its input changes. In the real world, this can be used to calculate data like financial profit, value depreciation or energy efficiency - things you're probably going to want to know when you hit the job market. This series of lessons on graphing derivatives and L'Hopital's Rule will help you learn about the following:
- How to graph derivatives from any function
- What happens when kinks pop up in the process
- The maximum and minimum values of a graph
- Features like concavity and inflection points
- L'Hopital's rule and its applications
|Graphing the Derivative from Any Function||See how to start with any function and graph the derivative.|
|Non-Differentiable Graphs of Derivatives||Learn what happens at kinks.|
|How to Determine Maximum and Minimum Values of a Graph||Discover how to determine maximum and minimum points and values on a graph.|
|Using Differentiation to Find Maximum and Minimum Values||Learn to use differentiation to determine maximum and minimum points on a range.|
|Concavity and Inflection Points on Graphs||See concavity and inflection points on a graph.|
|Understanding Concavity and Inflection Points with Differentiation||Gain an understanding of how to use differentiation to determine concavity and inflection points.|
|Data Mining: Function Properties From Derivatives||See an explanation of the properties of a function from its derivative.|
|Data Mining: Identifying Functions From Derivative Graphs||Discover how to identify functions from derivative graphs.|
|What Is L'Hopital's Rule?||Learn what L'Hopital's rule is and how it is used.|
|Applying L'Hopital's Rule in Simple Cases||Explore the use of L'Hopital's rule for simple cases.|
|Applying L'Hopital's Rule in Complex Cases||See how to apply L'Hopital's rule to 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 within the Math 104: Calculus course
- Graphing and Functions
- Vectors in Calculus
- Geometry and Trigonometry
- How to Use a Scientific Calculator
- Rate of Change
- Calculating Derivatives and Derivative Rules
- Applications of Derivatives
- Area Under the Curve and Integrals
- Integration and Integration Techniques
- Integration Applications
- Differential Equations
- Studying for Math 104