Differential Calculus

1 - Limits

1. Limits of Sequences: We introduce sequences and carefully motivate the definition of their convergence and limits, and then use the definition to establish the limits of some basic sequences and prove some basic facts in order to get comfortable with it.
2. Properties of Limits: We prove the squeeze theorem, the fact that limits preserve order, and finally the fact that limits of sequences behave nicely with the operations of real numbers.

2 - Continuity

1. Continuity: We introduce and define the concept of continuity, establish its algebraic properties, and show that some basic functions are continuous.
2. Major Theorems: We prove the intermediate and extreme value theorems, and use them to prove the existence of certain real numbers, after which we can establish the continuity of some other functions.

3 - Differentiation

1. Differentiability: We use the problem of tangent lines to motivate and define differentiability, check some examples, and analyze some examples of functions that are not differentiable.
2. Rules of Differentiation: We prove the typical formulas for differentiation, including the chain rule and product rule, and take them for a spin.
3. Leibniz Notation: We introduce the concept of negligibility and negligible closeness, and use it to motivative and explain Leibniz notation.
4. Major Theorems: We prove Rolle's theorem, the mean value theorem, and the inverse function theorem.
5. Parametric Curves: We introduce vector arithmetic to set up parametric curves, and generalize continuity and differentiability to them.