Calculus: Key Concepts (9/4/13) What are the 3 or 4 key concepts of calculus? What do they mean?

Limits What does it mean to say that limx  a f (x) = L ? Some limits are obvious. Example: limx  4 1/x = ? But some are not! Example: limx  0 1/x = ? Example: limh  0 ((x + h)2 – x2) / h = ? And other types of limits...?

Derivatives What do we mean by the derivative of a function f (x) at a point (a, f (a))? What’s the (analytic) definition? Note: It’s a limit!! The derivative of a function at a given point is a number. But if we free up the point, we then get the derivative function f (x).

Recall Some Derivative Facts What is the derivative function of each of these functions? xr (r any fixed real number) a x (a any fixed positive number) loga(x) sin(t) cos(t) tan(t) And what about all the derivative rules? Example: What’s the derivative function of f (x) = x e(x^2)?

Antiderivatives Given a function f (x), what is an antiderivative of f ? Why do we say “an”, not “the”? Is computing antiderivatives mechanical process like computing derivatives? Do all elementary functions have formulas for their antiderivatives?

Recall (?) Some Antiderivative Facts x r dx = (provided r  ?) 1/x dx = a x dx = ln(x) dx = sin(t) dt = cos(t) dt = tan(t) dt = e(x^2) dx =

Definite Integrals What does it mean to “integrate a function” over some part of its domain? That is, given a function f (x) defined on an interval [a, b], what does mean? How can we compute this number?

Assignment for Friday Obtain the book. Carefully review all of the ideas we discussed today. Remember to always bring your text and your clicker (oh, and your brain!) to class.