Lessons
In this lesson we are going to go through the logical framework that defines a definite integral. The main goal of this lesson is to show that when \[\lim_{||P\,||\to 0}\,L(P)\] and \[\lim_{||P\,||\to 0}\,U(P)\] are equal, then they are equal to \[\int_a^bf(x)\,dx.\] We also provide three equivalent definitions of a definite integral.
We review the concepts of Trigonometry and the Pythagorean Theorem. This lesson focuses on being able to determine the length of all three sides and both angles of a right triangle.
We review the Quadratic Equation and find the two values of $x$ for which $ax^2+bx+c=0$ is true.
The slope of a straight line is its "rise" over its "run". The value of the slope does not depend on where you calculate it.
In this lesson we define the union and intersection of two sets. A union of two sets is another set whose elements are in either of the original two sets. An intersection of two sets is another set whose elements have to be in both of the original two sets.
We describe natural numbers, integers, rational numbers and real numbers.
In this lesson we expand the idea of continuity to be more than at a single point. A function that is uniformly continuous over some interval is continuous at every point in that interval. We also show that continuous functions are uniformly continuous.
We define what it means for a set of numbers to be bounded.
In this lesson we discuss a subtle point not needed by non-mathematics majors. Because we need to know the value of $f(x)$ in the domain $(a-\delta,a)\cup (a,a+\delta)$ to prove
\[
\lim_{x\to a}\,f(x)=L,
\]
the number $a$ must be inside the domain and not just in it. We define the difference between in and inside.
