Tutorial
This course is built in Ximera.

Content for the First Exam

Understanding functions
Same or different?
Two young mathematicians examine one (or two!) functions.
For each input, exactly one output
We define the concept of a function.
Compositions of functions
We discuss compositions of functions.
Inverses of functions
Here we “undo” functions.
Review of famous functions
How crazy could it be?
Two young mathematicians think about the plots of functions.
Polynomial functions
Polynomials are some of our favorite functions.
Rational functions
Rational functions are functions defined by fractions of polynomials.
Trigonometric functions
We review trigonometric functions.
Exponential and logarithmetic functions
Exponential and logarithmic functions illuminated.
What is a limit?
Stars and functions
Two young mathematicians discuss stars and functions.
What is a limit?
We introduce limits.
Limit laws
Equal or not?
Here we see a dialogue where students discuss combining limits with arithmetic.
Continuity
Continuity is defined by limits.
The limit laws
We give basic laws for working with limits.
The Squeeze Theorem
The Squeeze theorem allows us to exchange difficult functions for easy functions.
(In)determinate forms
Could it be anything?
Two young mathematicians investigate the arithmetic of large and small numbers.
Limits of the form zero over zero
We want to evaluate limits where the Limit Laws do not directly apply.
Limits of the form nonzero over zero
We want to solve limits that have the form nonzero over zero.
Practice
Try these problems.
Using limits to detect asymptotes
Zoom out
Two young mathematicians discuss what curves look like when one “zooms out.”
Vertical asymptotes
We explore functions that “shoot to infinity” at certain points in their domain.
Horizontal asymptotes
We explore functions that behave like horizontal lines as the input grows without bound.
Slant asymptotes
We explore functions that ‘‘shoot to infinity’’ at certain points in their domain.
Continuity and the Intermediate Value Theorem
Roxy and Yuri like food
Two young mathematicians discuss the eating habits of their cats.
Continuity of piecewise functions
Here we use limits to ensure piecewise functions are continuous.
The Intermediate Value Theorem
Here we see a consequence of a function being continuous.
Practice
Try these problems.
An application of limits
Limits and velocity
Two young mathematicians discuss limits and instantaneous velocity.
Instantaneous velocity
We use limits to compute instantaneous velocity.

Content for the Second Exam

Definition of the derivative
Slope of a curve
Two young mathematicians discuss the novel idea of the “slope of a curve.”
The definition of the derivative
We compute the instantaneous growth rate by computing the limit of average growth rates.
Derivatives as functions
Wait for the right moment
Two young mathematicians discuss derivatives as functions.
The derivative as a function
Here we study the derivative of a function, as a function, in its own right.
Differentiability implies continuity
We see that if a function is differentiable at a point, then it must be continuous at that point.
Rules of differentiation
Patterns in derivatives
Two young mathematicians think about “short cuts” for differentiation.
Basic rules of differentiation
We derive the constant rule, power rule, and sum rule.
The derivative of the natural exponential function
We derive the derivative of the natural exponential function.
The derivative of sine
We derive the derivative of sine.
Product rule and quotient rule
Derivatives of products are tricky
Two young mathematicians discuss derivatives of products and products of derivatives.
The Product rule and quotient rule
Here we compute derivatives of products and quotients of functions
Chain rule
An unnoticed composition
Two young mathematicians discuss the chain rule.
The chain rule
Here we compute derivatives of compositions of functions
Derivatives of trigonometric functions
We use the chain rule to unleash the derivatives of the trigonometric functions.
Higher order derivatives and graphs
Rates of rates
Two young mathematicians look at graph of a function, its first derivative, and its second derivative.
Higher order derivatives and graphs
Here we look at graphs of higher order derivatives.
Concavity
Here we examine what the second derivative tells us about the geometry of functions.
Position, velocity, and acceleration
Here we discuss how position, velocity, and acceleration relate to higher derivatives.
Implicit differentiation
Standard form
Two young mathematicians discuss the standard form of a line.
Implicit differentiation
In this section we differentiate equations without expressing them in terms of a single variable.
Derivatives of inverse exponential functions
We derive the derivatives of inverse exponential functions using implicit differentiation.
Logarithmic differentiation
Multiplication to addition
Two young mathematicians think about derivatives and logarithms.
Logarithmic differentiation
We use logarithms to help us differentiate.
More than one rate
A changing circle
Two young mathematicians discuss a circle that is changing.
More than one rate
Here we work abstract related rates problems.
Applied related rates
Pizza and calculus, so cheesy
Two young mathematicians discuss tossing pizza dough.
Applied related rates
We work related rates problems in context.
Derivatives of inverse functions
We can figure it out
Two young mathematicians discuss the derivative of inverse functions.
Derivatives of inverse trigonometric functions
We derive the derivatives of inverse trigonometric functions using implicit differentiation.
The Inverse Function Theorem
We see the theoretical underpinning of finding the derivative of an inverse function at a point.
Maximums and minimums
More coffee
Two young mathematicians witness the perils of drinking too much coffee.
Maximums and minimums
We use derivatives to help locate extrema.
Concepts of graphing functions
What’s the graph look like?
Two young mathematicians discuss how to sketch the graphs of functions.
Concepts of graphing functions
We use the language of calculus to describe graphs of functions.
Computations for graphing functions
Wanted: graphing procedure
Two young mathematicians discuss how to sketch the graphs of functions.
Computations for graphing functions
We will give some general guidelines for sketching the plot of a function.

Content for the Third Exam

Mean Value Theorem
Let’s run to class
Two young mathematicians race to math class.
The Extreme Value Theorem
We examine a fact about continuous functions.
The Mean Value Theorem
Here we see a key theorem of calculus.
Linear approximation
Replacing curves with lines
Two young mathematicians discuss linear approximation.
Linear approximation
We use a method called ‘‘linear approximation’’ to estimate the value of a (complicated) function at a given point.
Explanation of the product and chain rules
We give explanation for the product rule and chain rule.
Optimization
A mysterious formula
Two young mathematicians discuss optimization from an abstract point of view.
Basic optimization
Now we put our optimization skills to work.
Applied optimization
Volumes of aluminum cans
Two young mathematicians discuss optimizing aluminum cans.
Applied optimization
Now we put our optimization skills to work.
L’Hopital’s rule
A limitless dialogue
Two young mathematicians consider a way to compute limits using derivatives.
L’Hopital’s rule
We use derivatives to give us a ‘‘short-cut’’ for computing limits.
Differential equations
Modeling the spread of infectious diseases
Two young mathematicians discuss differential equations.
Differential equations
We study equations with that relate functions with their rates.
Antiderivatives
Jeopardy! Of calculus
Two young mathematicians discuss a ‘Jeopardy!’ version of calculus.
Basic antiderivatives
We introduce antiderivatives.
Falling objects
We study a special type of differential equation.
Approximating the area under a curve
What is area?
Two young mathematicians discuss the idea of area.
Approximating area with rectangles
We introduce the basic idea of using rectangles to approximate the area under a curve.
Antiderivatives and area
Meaning of multiplication
A dialogue where students discuss multiplication.
Relating velocity and position, antiderivatives and areas
We see a connection between approximating antiderivatives and approximating areas.
Definite integrals
Computing areas
Two young mathematicians discuss cutting up areas.
The definite integral
Definite integrals compute signed area.
First Fundamental Theorem of Calculus
What’s in a calculus problem?
Two young mathematicians discuss what calculus is all about.
The First Fundamental Theorem of Calculus
The rate that accumulated area under a curve grows is described identically by that curve.
Second Fundamental Theorem of Calculus
A secret of the definite integral
Two young mathematicians discuss what calculus is all about.
The Second Fundamental Theorem of Calculus
The accumulation of a rate is given by the change in the amount.
A tale of three integrals
At this point we have three “different” integrals.

Additional content for the Final Exam

Applications of integrals
What could it represent?
Two young mathematicians discuss whether integrals are defined properly.
Applications of integrals
We give more contexts to understand integrals.
The idea of substitution
Geometry and substitution
Two students consider substitution geometrically.
The idea of substitution
We learn a new technique, called substitution, to help us solve problems involving integration.
Working with substitution
Integrals are puzzles!
Two young mathematicians discuss how tricky integrals are puzzles.
Working with substitution
We explore more difficult problems involving substitution.
The Work-Energy Theorem
Substitution is given a physical meaning.

You can download a Certificate as a record of your successes.