About This Chapter
Principles of Calculus - Chapter Summary
In this chapter, you will begin with lessons on how to determine the limits of functions, compare discontinuous and continuous functions and how to use the chain rule. The concept of tangents will be introduced, in addition to instructions on how to find the normal line to a curve and how to discover the minima and maxima. Our instructors will explain how L'Hopital's rule came about and how it can be used to find limits.
Your study will continue with a discussion on integration by parts and how to use several common integration techniques to solve intervals. The best way to utilize the partial fractions technique is also presented. You will review the definition for definite integrals and the derivation of the formula for total surface area and volume of a sphere, frustum and cylinder. The chapter concludes with instructions about how to use the separation of variables technique. Once you have completed these lessons, you should be able to:
- Calculate the limits of complex functions and differentiate between continuous and discontinuous functions
- Utilize the chain rule to simplify equations
- Define tangents
- Determine the maxima and minima of a function
- Use integration to break up a big interval
- Understand the relationship between Riemann sums and definite intervals
- Recognize the formulas for the surface area of a sphere, frustum, cylinder and cone
- Utilize the separation of variables technique to solve equations
Our mobile-friendly lessons and quizzes are available to review 24 hours a day, seven days a week. Use the assessment quizzes to test how well you've understood the lesson and material. If you want to review just one section of a video, use the video tabs feature in the Timeline section to skip directly to the part of the video you'd like to watch. Each lesson is accompanied by a printable lesson transcript so you can study the material offline at any time.
1. 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.
2. 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.
3. 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.
4. Tangents: Definition & Properties
In this lesson, we will cover the tangent as it is used in calculus. You will learn the process of finding the tangents of curves and will also find out what defines tangents in calculus.
5. Finding the Normal Line to a Curve: Definition & Equation
Finding the normal line to a curve requires several steps, but they aren't hard. In this lesson, learn what the steps are and what each step does so you can better understand the process.
6. Finding Minima & Maxima: Problems & Explanation
One of the most important practical uses of higher mathematics is finding minima and maxima. This lesson will describe different ways to determine the maxima and minima of a function and give some real world examples.
7. 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!
8. Using Integration By Parts
Your mother may have warned you not to bite off more than you can chew. The same thing is true with integration. In this lesson, learn how integration by parts can help you split a big interval into bite-sized pieces!
9. How to Use Trigonometric Substitution to Solve Integrals
In this lesson, we use each of the common integration techniques to solve different integrals. It's not always obvious which technique will be the easiest, so being familiar with an arsenal of methods might save you a lot of work!
10. How to Integrate Functions With Partial Fractions
In this lesson, learn how to integrate complicated fractions by using the partial fractions technique. That is, you will turn a complicated fraction into something a bit easier to integrate by finding partial fractions!
11. Definite Integrals: Definition
Explore how driving backwards takes you where you've already been as we define definite integrals. This lesson will also teach you the relationship between definite integrals and Riemann sums. Then, discover how an integral changes when it is above and below the x-axis.
12. Derivation of Formula for Total Surface Area of the Sphere by Integration
The total surface area of a sphere is found using an equation. In this lesson, we derive this equation using the arclength definition, radius relationships, and integral calculus.
13. Derivation of Formula for Volume of the Sphere by Integration
In this lesson, we derive the formula for finding the volume of a sphere. This formula is derived by integrating differential volume elements formed by slicing the sphere into cylinders with a differential thickness.
14. Separation of Variables to Solve System Differential Equations
In this lesson, we discuss how to solve some types of differential equations using the separation of variables technique. We'll ponder the dastardly deeds of a mad scientist, using his chemical concoction as an example for how to use separation of variables.
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