Continuous functions are, in essence, functions whose graphs can be drawn without lifting up your pen. This may sound simple, but this is in fact a very rich subject. Learn how continuity is defined using limits, and about a main property of all continuous functions -- the Intermediate value theorem.
Many of you are planning on taking the Calculus AB advanced placement exam. These are example problems taken directly from previous years' exams. Even if you aren't taking the exam, these are very useful problem for making sure you understand your calculus (as always, best to pause the videos and try them yourself before Sal does).
Make your first steps in evaluating definite integrals, armed with the Fundamental theorem of calculus.
Integrals can be used to find 2D measures (area) and 1D measures (lengths). But it can also be used to find 3D measures (volume)! Learn all about it here.
Differentiating functions is not an easy task! Make your first steps in this vast and rich world with some of the most basic differentiation rules, including the Power rule. It will surely make you feel more powerful.
The chain rule sets the stage for implicit differentiation, which in turn allows us to differentiate inverse functions (and specifically the inverse trigonometric functions). This is really the top of the line when it comes to differentiation.
Differential equations are equations that include both a function and its derivative (or higher-order derivatives). For example, y=y' is a differential equation. Learn how to find and represent solutions of basic differential equations.
Riemann sums is the name of a family of methods we can use to approximate the area under a curve. Through Riemann sums we come up with a formal definition for the definite integral.
As with derivatives, solve some real world problems and mathematical problems using the power of integral calculus.
Limits are intuitive, yet elusive. Learn what they are all about and how to find limits of functions from graphs or tables of values. Learn about the difference between one-sided and two-sided limits and how they relate to each other.
Analyzing_functions_with_calculus
Let's put all of our differentiation abilities to use, by analyzing the graphs of various functions. As you will see, the derivative and the second derivative of a function can tell us a lot about the function's graph.
Some functions don't make it easy to find their integrals, but we are not ones to give up so fast! Learn some advanced tools for integrating the more troublesome functions.
Now that we have all the conceptual stuff laid down, we can start have some fun with finding limits of various functions. Some of these limits don't want you to find them so fast, but we're sure you'll get them in the end!
If f is the derivative of F, then F is an antiderivative of f. We also call F the "indefinite integral" of f. In other words, indefinite integrals and antiderivatives are, essentially, reverse derivatives. Why differentiate in reverse? Good question! Keep going and you'll find out!
Basically, a limit must be at a specific point and have a specific value in order to be defined. Nevertheless, there are two kinds of limits that break these rules. One kind is unbounded limits -- limits that approach ± infinity (you may know them as "vertical asymptotes"). The other kind is limits at infinity -- these limits describe the value a function is approaching as x goes to ± infinity (you may know them as "horizontal asymptotes").
Definite_integrals_introduction
Definite integrals are a way to describe the area under a curve. Make introduction with this intriguing concept, along with its elaborate notation and various properties.
Area_arc_length_using_calculus
Become a professional area-under-curve finder! You will also learn here how integrals can be used to find lengths of curves. The tools of calculus are so versatile!
Differentiating_common_functions
Now that you know all the important differentiation rules, let's solve some problems that involve the differentiation of various common functions.
Fundamental_theorem_of_calculus
So you've learned about indefinite integrals and you've learned about definite integrals. Have you wondered what's the connection between these two concepts? You will get all the answers right here. Beware, this is pretty mind-blowing.
Solve real world problems (and some pretty elaborate mathematical problems) using the power of differential calculus.
Covered basic differentiation? Great! Now let's take things to the next level. In this topic, you will learn general rules that tell us how to differentiate products of functions, quotients of functions, and composite functions. Anxious to find the derivative of eˣ⋅sin(x²)? You've come to the right place.
Get comfortable with the big idea of differential calculus, the derivative. The derivative of a function has many different interpretations and they are all very useful when dealing with differential calculus problems. This topic covers all of those interpretations, including the formal definition of the derivative and the notion of differentiable functions.
All video content by Khan Academy is under their license: CC by NC SA
Website created using Khan Academy Static Downloader