Two young mathematicians discuss stars and functions.

Visual introduction to Limits

Visual introduction to One-sided Limits

We discuss how to use tables to determine limits and the inherent danger to using this method.

We introduce the notation and formal algebraic definitions for limits.

Here we see a dialogue where students discuss combining limits with arithmetic.

We give basic laws for working with limits.

Two young mathematicians discuss the eating habits of their cats.

The limit of a continuous function at a point is equal to the value of the function at that point.

The limit of a continuous function at an endpoint to determine continuity at endpoints.

Here we see a consequence of a function being continuous.

Here we use limits to check whether piecewise functions are continuous.

Here we see a consequence of a function being continuous.

Two young mathematicians investigate the arithmetic of large and small numbers.

What does it mean to take a limit “At Infinity”?

We introduce indeterminate forms and discuss what makes one form indeterminate compared to another.

We want to evaluate limits for which the Limit Laws do not apply.

What can be said about limits that have the form nonzero over zero?

Two young mathematicians discuss what curves look like when one “zooms out.”

We explore functions that “shoot to infinity” near certain points.

We explore functions that behave like horizontal lines as the input grows without bound.

Two young mathematicians discuss limits and instantaneous velocity.

We use limits to compute instantaneous velocity.

We discuss how we will tackle the next phase of calculus; the derivative!

We consider the derivative from an analytic point of view; digging into the algebraic notation and manipulation for instantaneous rates of change.

We consider the derivative from an analytic point of view; digging into the algebraic notation and manipulation for instantaneous rates of change.

We discuss and justify computing derivatives with variable input and introduce the idea of derivatives as functions.

We discuss how continuity and differentiability are interrelated.

We develop the rules needed to split functions across addition and subtraction signs, which allows us to take derivatives term by term.

We develop the power rule, otherwise known as the polynomial rule, to take derivatives of terms with the form $x^n$ for any real value $n$.

We develop the technique for decomposing the product of functions for derivatives, introducing the “Product Rule”.

We introduce the quotient rule as a way to decompose a quotient of functions when taking a derivative.

We discuss and ultimately develop a rule that allows us to take derivatives of compositions of functions. The so-called Chain Rule.

We discuss how to tackle the problem of applying chain rules when you have functions with several layers of composition.

We discuss how to take a derivative of an implicitly defined function.

We develop the rules to differentiate exponential functions.

We develop the rules needed to differentiate logarithmic functions.

We discuss a new differentiation technique, useful for functions with large number of products - Logarithmic Differentiation.

We discuss how to find and evaluate local extrema of a function using derivatives.

We develop the Extreme Value Theorem, a way to know when an absolute extrema must exist without doing calculus.

We discuss how to find and evaluate absolute extrema of a function using derivatives.

We present the visual interpretation of concavity and how it is determined by the second derivative.

We discuss how to determine where a function is concave up or concave down.

We discuss the real world and visual interpretation of a point of inflection.

We discuss how to find points of inflection of a function using the second derivative.

We discuss how to use the second derivative to classify extrema of a function.

We discuss one of the original uses for calculus, how position, velocity, and acceleration are related.

We discuss how to use our tools to create detailed sketches of functions beyond the ability of precalc tools.

We discuss how to visualize linear approximation as a tangent line application.

We discuss to actually apply linear approximation to approximate a value.

We discuss how to solve real world style problems that involve interrelated rates of change.

We discuss how to model and then optimize that model, for real world style problems.

We introduce a new notation for arbitrarily many terms being added together in preparation for Riemann Sums later.

We discuss the antiderivative at a conceptual level before we dive into the mechanics.

We develop rules to quickly determine antiderivatives of most of our core functions.

We discuss how to approximate the area under a curve.

We discuss the various types of Riemann Approximation Endpoint Methods.

We discuss how to approximate the area under a curve.

We develop the algebraic techniques to actually compute the perfect approximation of area under the curve.

We introduce the idea of the indefinite integral - all the antiderivatives!

“Indefinite Integrals are a class of functions” - What?

We present the first fundamental theorem of calculus

We present the second fundamental theorem of calculus

We introduce the method of U-Sub as a way to unravel a chain rule.

We give some hints and tips on how to see and predict where a U-Sub might help.