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The domain of a function is the set of real numbers for which the function is defined. The function's range is the set of all the values that are returned when the domain is plugged into the function. (8:39)
If you need to use power rule to take the derivative of a function, but the "inside" function is not just "x", you'll need to apply chain rule, multiplying the derivative of the power function by the derivative of the "inside" function. (6:26)
Functions are discontinuous where they are undefined, or where they have holes, breaks, gaps, corners, or jumps. When you define the domain of a function, you want to make sure to indicate that the function's points of discontinuity are not included the domain. (3:31)
Sometimes you'll be asked to find various limits of a function defined by a "crazy graph". The trick is to understand that the limit is just the value the function approaches as you trace your finger along the graph toward the limit value. You may find that the left- and right-hand limits of a function are different at some points, and that the value of the function at a point is not always equal to the limit of the function there. (7:47)
The general limit of a function only exists when the left-hand limit exists, the right-hand limit exists, and the left- and right-hand limits are equal to one another. Even when the general limit doesn't exist because one of these conditions isn't met, the one-sided limits (the left-hand limit and/or the right-hand limit), can still exist independently. (9:25)
You can factor out the coefficient of a power function before using power rule to take the derivative. If you have the sum or difference of many power functions in the form of a polynomial function, you can use power rule on each term individually to find the derivative of the entire linear combination. (5:21)
The precise definition of the limit (also called the epsilon-delta definition of the limit), is just a way for us to prove that a limit exists. This is one of the hardest topics in calculus, so don't get discouraged if you need to go over it multiple times! (11:27)
TEDTalk: The Infinite Hotel, a thought experiment created by German mathematician David Hilbert, is a hotel with an infinite number of rooms. Easy to comprehend, right? Wrong. What if it's completely booked but one person wants to check in? What about 40? Or an infinitely full bus of people? Jeff Dekofsky solves these heady lodging issues using Hilbert's paradox.