Mathematics 1

We explain what a continuous function is and how we precisely define continuity using limits of sequences.

Taylor approximations are used to approximate complicate functions by simple functions, that is, polynomials, locally near a given point a.

We show a little trick of how to differentiate inverse functions without knowing their formulas.

We discuss the chain rule for differentiating composite functions and verify it graphically using GeoGebra animations.

We explain and graphically illustrate the sum, product, and the quotient rule.

We present three simple rules for differentiation, motivate them graphically and prove them using differential quotients.

We explain the differentiation rule for power functions using limit calculations for difference quotients.

We define the derivative as the slope of the tangent and show how to calculate it by the limit of secant slopes.

We walk through step-by-step solutions for finding the limits of 11 example sequences.

We introduce geometric series and calculate their limits, if they exist.