About This Chapter
Differential Calculus & Functions - Chapter Summary
Our instructors present the operations associated with real-valued algebraic functions and discuss composite functions in these easy-to-follow lessons. Additional topics include L'Hopital's Rule, examples of even and odd functions and what inverse functions entail.
You'll also learn how to use the chain rule with complex functions and how to find the derivatives of implicit functions. A lesson on the mean value theorem is included. Once you review this chapter, you should be ready to do the following:
- Differentiate between the behavior of logarithmic and exponential functions
- Explain the limits of functions
- Define function continuity
- Use a calculation to find the derivative of trigonometric functions
- Identify the traits of functions
- Find maximum and minimum values using differentiation
- Detail Rolle's Theorem
Created by a professional instructor, our lessons are short and easy to understand. Each text or video lesson is accompanied by a multiple-choice quiz so you can test your comprehension and identify areas that need more study. If you have questions, contact one of our instructors for help. These tools are accessible on any computer, mobile phone or tablet at any time.
1. Algebra of Real-Valued Functions: Operations & Examples
This lesson will define real-valued functions using a real-world example. We will then discuss the operations involved in algebra of real-valued functions and look at examples of using each operation on functions.
2. Composite Function: Definition & Examples
One way in which we can combine functions is by forming their composition. A composite function is a larger function that is created by putting one function inside another function.
3. Behavior of Exponential and Logarithmic Functions
After watching this video lesson, you will be able to recognize exponential and logarithmic functions by looking at the end behavior of the graphs. You will also learn how the graphs change.
4. How to Determine the Limits of Functions
You know the definition of a limit. You know the properties of limits. You can connect limits and continuity. Now use this knowledge to calculate the limits of complex functions in this lesson.
5. Continuity in a Function
Travel to space and explore the difference between continuous and discontinuous functions in this lesson. Learn how determining continuity is as easy as tracing a line.
6. 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!
7. Even & Odd Functions: Definition & Examples
Even and odd functions are an aspect of line symmetry. You can use knowledge of even and odd functions to quickly identify equation symmetry. After this lesson, test your knowledge with a short quiz!
8. Inverse Functions
Inverse functions are two functions that do exactly opposite things. Check out this lesson to learn about how to write inverse functions, find inverse functions, and predict whether or not they exist.
9. Calculating Derivatives of Trigonometric Functions
The trigonometric functions show up almost everywhere that you have a repeating pattern. In this lesson, learn how to find the derivatives of the trigonometric functions.
10. Using the Chain Rule to Differentiate Complex Functions
If you've ever seen a complicated function, this lesson is for you. Most functions that we want to differentiate are complicated functions, for which no single derivative rule will work. In this lesson, learn how to use the chain rule to simplify nesting equations.
11. How to Find Derivatives of Implicit Functions
How do you define the rate of change when your function has variables that cannot be separated? Learn how implicit differentiation can be used to find dy/dx even when you don't have y=f(x)!
12. Identify Where a Function is Linear, Increasing or Decreasing, Positive or Negative
Functions do all kinds of fun things. In this lesson, learn how to identify traits of functions such as linear or nonlinear, increasing or decreasing and positive or negative.
13. 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.
14. What is the Mean Value Theorem?
Three people set off on a car trip. They all start at the same time and end at the same time. Learn what calculus says about how fast they traveled along the way as you study the Mean Value Theorem in this lesson.
15. Rolle's Theorem: A Special Case of the Mean Value Theorem
Super C, the human cannonball, is shot into the air at 35 mph, but his average vertical velocity is zero. In this lesson, you will use Rolle's theorem to explain what this means about Super C's flight.
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