1Ximera tutorial

1.1How to use Ximera

This course is built in Ximera.

1.2How is my work scored?

We explain how your work is scored.

Content for the First Exam

2Equations and Inequalities

2.1Needed Score?

Two young mathematicians examine an equation.

2.2Equations

We discuss solving equations.

2.3Inequalities

We discuss inequalities.

3Understanding functions

3.1Same or different?

Two young mathematicians examine one (or two!) functions.

3.2For each input, exactly one output

We define the concept of a function.

3.3Compositions of functions

We discuss compositions of functions.

3.4Inverses of functions

Here we “undo” functions.

4What is a limit?

4.1Stars and functions

Two young mathematicians discuss stars and functions.

4.2What is a limit?

We introduce limits.

5Polynomial functions

5.1How crazy could it be?

Two young mathematicians think about the plots of functions.

5.2Working with polynomials

Polynomials are some of our favorite functions.

5.3End behavior

Polynomials are some of our favorite functions.

5.4Graphs of polynomial functions

Polynomials are some of our favorite functions.

6Rational functions

6.1Will it divide?

6.2Working with rational functions

Rational functions are functions defined by fractions of polynomials.

6.3Rational equations and inequalities

Equations and inequalities with Rational Functions

6.4Graphs of rational functions

Rough graphs of rational functions

7Limit laws

7.1Equal or not?

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

7.2Continuity

Continuity is defined by limits.

7.3The limit laws

We give basic laws for working with limits.

7.4The Squeeze Theorem

The Squeeze theorem allows us to exchange difficult functions for easy functions.

8(In)determinate forms

8.1Could it be anything?

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

8.2Limits of the form zero over zero

We want to evaluate limits where the Limit Laws do not directly apply.

8.3Limits of the form nonzero over zero

We want to solve limits that have the form nonzero over zero.

8.4Practice

Try these problems.

9Using limits to detect asymptotes

9.1Zoom out

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

9.2Vertical asymptotes

We explore functions that “shoot to infinity” at certain points in their domain.

9.3Horizontal asymptotes

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

9.4Slant asymptotes

We explore functions that “shoot to infinity” at certain points in their domain.

10Continuity and the Intermediate Value Theorem

10.1Roxy and Yuri like food

Two young mathematicians discuss the eating habits of their cats.

10.2Continuity of piecewise functions

Here we use limits to ensure piecewise functions are continuous.

10.3The Intermediate Value Theorem

Here we see a consequence of a function being continuous.

10.4Practice

Try these problems.

Content for the Second Exam

11An application of limits

11.1Limits and velocity

Two young mathematicians discuss limits and instantaneous velocity.

11.2Instantaneous velocity

We use limits to compute instantaneous velocity.

12Definition of the derivative

12.1Slope of a curve

Two young mathematicians discuss the novel idea of the “slope of a curve.”

12.2The definition of the derivative

We compute the instantaneous growth rate by computing the limit of average growth rates.

13Derivatives as functions

13.1Wait for the right moment

Two young mathematicians discuss derivatives as functions.

13.2The derivative as a function

Here we study the derivative of a function, as a function, in its own right.

13.3Differentiability implies continuity

We see that if a function is differentiable at a point, then it must be continuous at that point.

14Rules of differentiation

14.1Patterns in derivatives

Two young mathematicians think about “short cuts” for differentiation.

14.2Basic rules of differentiation

We derive the constant rule, power rule, and sum rule.

15Product rule and quotient rule

15.1Derivatives of products are tricky

Two young mathematicians discuss derivatives of products and products of derivatives.

15.2The Product rule and quotient rule

Here we compute derivatives of products and quotients of functions

16Chain rule

16.1An unnoticed composition

Two young mathematicians discuss the chain rule.

16.2The chain rule

Here we compute derivatives of compositions of functions

Content for the Third Exam

17Exponential and Logarithmic Functions

17.1An interesting situation

17.2Exponential and logarithmetic functions

Exponential and logarithmic functions illuminated.

18Derivatives of exponential functions

18.1Interesting changes

Two young mathematicians think about the plots of functions.

18.2The derivative of the natural exponential function

We derive the derivative of the natural exponential function.

18.3Derivatives of exponential and logarithmetic functions

Derivatives of exponential and logarithmic functions calculated.

19Higher order derivatives and graphs

19.1Rates of rates

Two young mathematicians look at graph of a function, its first derivative, and its second derivative.

19.2Higher order derivatives and graphs

Here we make a connection between a graph of a function and its derivative and higher order derivatives.

19.3Concavity

Here we examine what the second derivative tells us about the geometry of functions.

19.4Position, velocity, and acceleration

Here we discuss how position, velocity, and acceleration relate to higher derivatives.

20Trigonometric Functions

20.1Follow the bouncing pen.

Two young mathematicians think about periodic motion.

20.2Trigonometric functions

We review trigonometric functions.

21Derivatives of trigonometric functions

21.1How fast was the pen going?

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

21.2The derivative of sine

We derive the derivative of sine.

21.3Derivatives of trigonometric functions

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

22Maximums and minimums

22.1More coffee

Two young mathematicians witness the perils of drinking too much coffee.

22.2Maximums and minimums

We use derivatives to help locate extrema.

Additional content for Autumn Final Exam

23Mean Value Theorem

23.1Let’s run to class

Two young mathematicians race to math class.

23.2The Extreme Value Theorem

We examine a fact about continuous functions.

23.3The Mean Value Theorem

Here we see a key theorem of calculus.

24Optimization

24.1A mysterious formula

Two young mathematicians discuss optimization from an abstract point of view.

24.2Basic optimization

Now we put our optimization skills to work.

25Applied optimization

25.1Volumes of aluminum cans

Two young mathematicians discuss optimizing aluminum cans.

25.2Applied optimization

Now we put our optimization skills to work.

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