Ximera tutorial

How to use Ximera

This course is built in Ximera.

How is my work scored?

We explain how your work is scored.

1Sequences

1.1Sequences

A sequence is an ordered list of numbers.

2Sequences as functions

2.1Sequences as functions

A sequence can be thought of as a function from the integers to the real numbers. There are two ways to establish whether a sequence has a limit.

3Sums of sequences

3.1Series

A series is summation of a sequence.

4Integral and divergence tests

4.1The integral test

Infinite sums can be studied using improper integrals.

4.2The divergence test

If an infinite sum converges, then its terms must tend to zero.

5Ratio and root tests

5.1The ratio test

Some infinite series can be compared to geometric series.

5.2The root test

Some infinite series can be compared to geometric series.

6Comparison tests

6.1The comparison test

We compare infinite series to each other using inequalities.

6.2The limit comparison test

We compare infinite series to each other using limits.

7Alternating series

7.1The alternating series test

Alternating series have nice properties.

8Approximating functions with polynomials

8.1Approximating functions with polynomials

We can approximate smooth functions with polynomials.

9Power series

9.1Power series

Infinite series can represent functions.

10Introduction to Taylor series

10.1Introduction to Taylor series

We study Taylor and Maclaurin series.

11Numbers and Taylor series

11.1Numbers and Taylor series

Taylor series are a computational tool.

12Calculus and Taylor series

12.1Calculus and Taylor series

Power series interact nicely with other calculus concepts.

13Parametric equations

13.1Parametric equations

We discuss the basics of parametric curves.

13.2Calculus and parametric curves

We discuss derivatives and integrals of parametric curves.

14Introduction to polar coordinates

14.1Introduction to polar coordinates

Polar coordinates are a special type of parametric curves.

14.2Gallery of polar curves

We see a collection of polar curves.

15Derivatives of polar functions

15.1Derivatives of polar functions

We differentiate polar functions.

16Integrals of polar functions

16.1Integrals of polar functions

We integrate polar functions.

17Working in two and three dimensions

17.1Working in two and three dimensions

We talk about basic geometry in higher dimensions.

17.2Drawing a sphere

Learn how to draw a sphere.

Vector-valued functions

18Vectors

18.1Vectors

Vectors are lists of numbers that denote direction and magnitude.

19Dot products

19.1The dot product

The dot product measures how aligned two vectors are with each other.

20Cross products

20.1The cross product

The cross product is a special way to multiply two vectors in three-dimensional space.

21Lines and curves in space

21.1Lines and curves in space

Vector-valued functions are parameterized curves.

22Calculus and vector-valued functions

22.1Calculus and vector-valued functions

With one input, and vector outputs, we work component-wise.

23Motion and paths in space

23.1Motion and paths in space

We interpret vector-valued functions as paths of objects in space.

23.2Parameterizing by arc length

We find a new description of curves that trivializes arc length computations.

24Normal vectors

24.1Unit tangent and unit normal vectors

We introduce two important unit vectors.

24.2Planes in space

We discuss how to find implicit and explicit formulas for planes.

24.3Parametric plots

Tangent and normal vectors can help us plot make interesting parametric functions.

24.4Drawing a torus

Learn how to draw a torus.

Functions of several variables

25Functions of several variables

25.1Functions of several variables

We introduce functions that take vectors or points as inputs and output a number.

26Continuity of functions of several variables

26.1Continuity

We investigate what continuity means for functions of several variables.

27Partial derivatives

27.1Partial derivatives

We introduce partial derivatives.

28The gradient

28.1The gradient

We introduce the gradient vector.

29Linear approximation

29.1Linear approximation

We extend the ideas of linear approximation to functions of several variables.

30Chain rule for functions of several variables

30.1The chain rule

We investigate the chain rule for functions of several variables.

31Taylor polynomials

31.1Taylor polynomials

We introduce Taylor polynomials for functions of several variables.

32Quadric surfaces

32.1Quadric surfaces

We will get to know some basic quadric surfaces.

32.2Drawing paraboloids

Learn how to draw an elliptic and a hyperbolic paraboloid.

33Maximums and minimums

33.1Maxima and minima

We see how to find extrema of functions of several variables.

34Constrained optimization

34.1Constrained optimization

We learn to optimize surfaces along and within given paths.

35Lagrange multipliers

35.1Lagrange multipliers

We give a new method of finding extrema.

36Multiple integrals

36.1Integrals over trivial regions

We study integrals over basic regions.

36.2Integrals with trivial integrands

We study integrals over general regions with and integrand equaling one.

37Common coordinates

37.1Polar coordinates

We integrate over regions in polar coordinates.

37.2Cylindrical coordinates

We integrate over regions in cylindrical coordinates.

37.3Spherical coordinates

We integrate over regions in spherical coordinates.

38Computations and interpertations

38.1Surface area

We compute surface area with double integrals.

38.2Mass, moments, and center of mass

We use integrals to model mass.

38.3Computations and interpretations

We practice more computations and think about what integrals mean.

Vector-valued functions of several variables

39Vector fields

39.1Vector fields

We introduce the idea of a vector at every point in space.

40Line integrals

40.1Line integrals

We accumulate vectors along a path.

41Green’s Theorem

41.1Curl and line integrals

Green’s Theorem is a fundamental theorem of calculus.

41.2Green’s theorem as a planimeter

A planimeter computes the area of a region by tracing the boundary.

41.3Divergence and line integrals

Divergence measures the rate field vectors are expanding at a point.

42The shape of things to come

42.1Surface integrals

We generalize the idea of line integrals to higher dimensions.

42.2Divergence theorem

We introduce the divergence theorem.

42.3Stokes’ theorem

We introduce Stokes’ theorem.

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