Ximera tutorial

How to use Ximera

This course is built in Ximera.

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We explain how your work is scored.

Content for the First Exam

Equations and Inequalities

Needed Score?

Two young mathematicians examine an equation.

Equations

We discuss solving equations.

Inequalities

We discuss inequalities.

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.

What is a limit?

Stars and functions

Two young mathematicians discuss stars and functions.

What is a limit?

We introduce limits.

Polynomial functions

How crazy could it be?

Two young mathematicians think about the plots of functions.

Working with polynomials

Polynomials are some of our favorite functions.

End behavior

Polynomials are some of our favorite functions.

Graphs of polynomial functions

Polynomials are some of our favorite functions.

Rational functions

Will it divide?

Working with rational functions

Rational functions are functions defined by fractions of polynomials.

Rational equations and inequalities

Equations and inequalities with Rational Functions

Graphs of rational functions

Rough graphs of rational functions

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” near certain points.

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.

Content for the Second Exam

An application of limits

Limits and velocity

Two young mathematicians discuss limits and instantaneous velocity.

Instantaneous velocity

We use limits to compute instantaneous velocity.

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.

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

Content for the Third Exam

Exponential and Logarithmic Functions

An interesting situation

Exponential and logarithmetic functions

Exponential and logarithmic functions illuminated.

Derivatives of exponential functions

Interesting changes

Two young mathematicians think about the plots of functions.

The derivative of the natural exponential function

We derive the derivative of the natural exponential function.

Derivatives of exponential and logarithmetic functions

Derivatives of exponential and logarithmic functions calculated.

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 make a connection between a graph of a function and its derivative and 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.

Trigonometric Functions

Follow the bouncing pen.

Two young mathematicians think about periodic motion.

Trigonometric functions

We review trigonometric functions.

Derivatives of trigonometric functions

How fast was the pen going?

Two young mathematicians think about the rate of change of periodic motion.

The derivative of sine

We derive the derivative of sine.

Derivatives of trigonometric functions

We use the chain rule to unleash the derivatives of the trigonometric functions.

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.

Additional content for Autumn Final 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.

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.

Content for the First Exam

Review of Limits

Guess the Value

Two young mathematicians think about limits.

Review Limits.

Review methods of evaluating limits.

Review of differentiation

Review Derivatives BreakGround

Two young mathematicians think about derivatives.

Review Derivatives

Review differentiation.

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.

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.

Implicit differentiation

Standard form

Two young mathematicians discuss the standard form of a line.

Implicit differentiation

In this section we differentiate equations that contain more than one variable on one side.

Finding dx dy

In this section we differentiate equations without expressing them in terms of a single variable.

Logarithmic differentiation

Multiplication to addition

Two young mathematicians think about derivatives and logarithms.

Logarithmic differentiation

We use logarithms to help us differentiate.

Content for the Second Exam

Inverse Trigonometric Functions

Inv Trig Function BreakGround

Two young mathematicians think about trigonometric functions.

Inverse trigonometric functions

We review trigonometric functions.

Derivatives of inverse trigonometric functions

Derivatives of inverse trigonometric functions BreakGround

Two young mathematicians think about the plots of 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.

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 solve related rates problems in context.

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.

L’Hopital’s rule for other forms

Indeterminate mutterings

Two young mathematicians consider a way to compute limits using derivatives.

L’Hopital’s rule for other forms

We use derivatives to give us a “short-cut” for computing limits.

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.

Content for the Third Exam

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.

Area approximations in sigma notation

So many rectangles.

A dialogue where students discuss area approximations.

Sigma Notation

Area approximations with sigma notation

Definite integrals

Computing areas

Two young mathematicians discuss cutting up areas.

The definite integral

Definite integrals arise as the limits of Riemann sums, and compute net areas.

Properties of the definite integral

Computing areas

Two young mathematicians discuss cutting up areas.

Properties of the definite integral

Properties of the definite integral

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.

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.

Additional content for the Final Exam

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.

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