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

2Review of Limits

2.1Guess the Value

Two young mathematicians think about limits.

2.2Review Limits.

Review methods of evaluating limits.

3Review of differentiation

3.1Review Derivatives BreakGround

Two young mathematicians think about derivatives.

3.2Review Derivatives

Review differentiation.

4Linear approximation

4.1Replacing curves with lines

Two young mathematicians discuss linear approximation.

4.2Linear approximation

We use a method called “linear approximation” to estimate the value of a (complicated) function at a given point.

4.3Explanation of the product and chain rules

We give explanation for the product rule and chain rule.

5Concepts of graphing functions

5.1What’s the graph look like?

Two young mathematicians discuss how to sketch the graphs of functions.

5.2Concepts of graphing functions

We use the language of calculus to describe graphs of functions.

6Computations for graphing functions

6.1Wanted: graphing procedure

Two young mathematicians discuss how to sketch the graphs of functions.

6.2Computations for graphing functions

We will give some general guidelines for sketching the plot of a function.

7Implicit differentiation

7.1Standard form

Two young mathematicians discuss the standard form of a line.

7.2Implicit differentiation

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

7.3Finding dx dy

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

8Logarithmic differentiation

8.1Multiplication to addition

Two young mathematicians think about derivatives and logarithms.

8.2Logarithmic differentiation

We use logarithms to help us differentiate.

Content for the Second Exam

9Inverse Trigonometric Functions

9.1Inv Trig Function BreakGround

Two young mathematicians think about trigonometric functions.

9.2Inverse trigonometric functions

We review trigonometric functions.

10Derivatives of inverse trigonometric functions

10.1Derivatives of inverse trigonometric functions BreakGround

Two young mathematicians think about the plots of functions.

10.2Derivatives of inverse trigonometric functions

We derive the derivatives of inverse trigonometric functions using implicit differentiation.

10.3The Inverse Function Theorem

We see the theoretical underpinning of finding the derivative of an inverse function at a point.

11More than one rate

11.1A changing circle

Two young mathematicians discuss a circle that is changing.

11.2More than one rate

Here we work abstract related rates problems.

12Applied related rates

12.1Pizza and calculus, so cheesy

Two young mathematicians discuss tossing pizza dough.

12.2Applied related rates

We solve related rates problems in context.

13L’Hopital’s rule

13.1A limitless dialogue

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

13.2L’Hopital’s rule

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

14L’Hopital’s rule for other forms

14.1Indeterminate mutterings

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

14.2L’Hopital’s rule for other forms

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

15Antiderivatives

15.1Jeopardy! Of calculus

Two young mathematicians discuss a ‘Jeopardy!’ version of calculus.

15.2Basic antiderivatives

We introduce antiderivatives.

15.3Falling objects

We study a special type of differential equation.

Content for the Third Exam

16Approximating the area under a curve

16.1What is area?

Two young mathematicians discuss the idea of area.

16.2Approximating area with rectangles

We introduce the basic idea of using rectangles to approximate the area under a curve.

17Area approximations in sigma notation

17.1So many rectangles.

A dialogue where students discuss area approximations.

17.2Sigma Notation

17.3Area approximations with sigma notation

18Definite integrals

18.1Computing areas

Two young mathematicians discuss cutting up areas.

18.2The definite integral

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

19Properties of the definite integral

19.1Computing areas

Two young mathematicians discuss cutting up areas.

19.2Properties of the definite integral

Properties of the definite integral

20First Fundamental Theorem of Calculus

20.1What’s in a calculus problem?

Two young mathematicians discuss what calculus is all about.

20.2The First Fundamental Theorem of Calculus

The rate that accumulated area under a curve grows is described identically by that curve.

21Second Fundamental Theorem of Calculus

21.1A secret of the definite integral

Two young mathematicians discuss what calculus is all about.

21.2The Second Fundamental Theorem of Calculus

The accumulation of a rate is given by the change in the amount.

21.3A tale of three integrals

At this point we have three “different” integrals.

22Applications of integrals

22.1What could it represent?

Two young mathematicians discuss whether integrals are defined properly.

22.2Applications of integrals

We give more contexts to understand integrals.

Additional content for the Final Exam

23The idea of substitution

23.1Geometry and substitution

Two students consider substitution geometrically.

23.2The idea of substitution

We learn a new technique, called substitution, to help us solve problems involving integration.

24Working with substitution

24.1Integrals are puzzles!

Two young mathematicians discuss how tricky integrals are puzzles.

24.2Working with substitution

We explore more difficult problems involving substitution.

24.3The Work-Energy Theorem

Substitution is given a physical meaning.

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