Ximera tutorial

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1A review of integration

1.1A review of differentiation

We review differentiation and integration.

1.2A review of integration

We review differentiation and integration.

1.3A review of integration techniques

We review common techniques to compute indefinite and definite integrals.

2Areas between curves

2.1Area between curves

We introduce the procedure of “Slice, Approximate, Integrate” and use it study the area of a region between two curves using the definite integral.

3Accumulated cross-sections

3.1Accumulated cross-sections

We can also use the procedure of “Slice, Approximate, Integrate” to set up integrals to compute volumes.

4Solids of revolution

4.1What is a solid of revolution?

We define a solid of revolution and discuss how to find the volume of one in two different ways.

4.2The washer method

We use the procedure of “Slice, Approximate, Integrate” to develop the washer method to compute volumes of solids of revolution.

4.3The shell method

We use the procedure of “Slice, Approximate, Integrate” to develop the shell method to compute volumes of solids of revolution.

4.4Comparing washer and shell method

We compare and contrast the washer and shell method.

5Length of curves

5.1Length of curves

We can use the procedure of “Slice, Approximate, Integrate” to find the length of curves.

6Surface area

6.1Surface areas of revolution

We compute surface area of a frustrum then use the method of “Slice, Approximate, Integrate” to find areas of surface areas of revolution.

7Applications of integration

7.1Physical applications

We apply the procedure of “Slice, Approximate, Integrate” to model physical situations.

8Integration by parts

8.1Integration by parts

We learn a new technique, called integration by parts, to help find antiderivatives of certain types of products by reexamining the product rule for differentiation.

9Trigonometric integrals

9.1Trigonometric integrals

We can use substitution and trigonometric identities to find antiderivatives of certain types of trigonometric functions.

10Trigonometric substitution

10.1Trigonometric substitution

We integrate by substitution with the appropriate trigonometric function.

11Partial fractions

11.1Rational functions

We discuss an approach that allows us to integrate rational functions.

12Improper integrals

12.1Improper Integrals

We can use limits to integrate functions on unbounded domains or functions with unbounded range.

13Sequences

13.1Sequences

We investigate sequences.

13.2Representing sequences visually

We can graph the terms of a sequence and find functions of a real variable that coincide with sequences on their common domains.

14Sequences as functions

14.1Limits of sequences

There are two ways to establish whether a sequence has a limit.

15Sums of sequences

15.1What is a series

A series is an infinite sum of the terms of sequence.

15.2Special Series

We discuss convergence results for geometric series and telescoping series.

16The divergence test

16.1The divergence test

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

17The Integral test

17.1The integral test

Certain infinite series can be studied using improper integrals.

18Alternating series

18.1The alternating series test

Alternating series are series whose terms alternate in sign between positive and negative. There is a powerful convergence test for alternating series.

19Remainders

19.1Dig-In: Estimating Series

We learn how to estimate the value of a series.

19.2Remainders for Geometric and Telescoping Series

For a convergent geometric series or telescoping series, we can find the exact error made when approximating the infinite series using the sequence of partial sums.

20Remainders

20.1Remainders for alternating series

There is a nice result for approximating the remainder of convergent alternating series.

20.2Remainders and the Integral Test

There is a nice result for approximating the remainder for series that converge by the integral test.

21Ratio and root tests

21.1The ratio test

Some infinite series can be compared to geometric series.

21.2The root test

Some infinite series can be compared to geometric series.

22Comparison tests

22.1The comparison test

We compare infinite series to each other using inequalities.

22.2The limit comparison test

We compare infinite series to each other using limits.

23Absolute and Conditional Convergence

23.1Absolute and Conditional Convergence

The basic question we wish to answer about a series is whether or not the series converges. If a series has both positive and negative terms, we can refine this question and ask whether or not the series converges when all terms are replaced by their absolute values. This is the distinction between absolute and conditional convergence, which we explore in this section.

24Approximating functions with polynomials

24.1Higher Order Polynomial Approximations

We can approximate sufficiently differentiable functions by polynomials.

25Power series

25.1Power series

Infinite series can represent functions.

26Introduction to Taylor series

26.1Introduction to Taylor series

We study Taylor and Maclaurin series.

27Numbers and Taylor series

27.1Numbers and Taylor series

Taylor series are a computational tool.

28Calculus and Taylor series

28.1Calculus and Taylor series

Power series interact nicely with other calculus concepts.

29Differential equations

29.1Differential equations

Differential equations show you relationships between rates of functions.

30Numerical methods

30.1Slope fields and Euler’s method

We describe numerical and graphical methods for understanding differential equations.

31Separable differential equations

31.1Separable differential equations

Separable differential equations are those in which the dependent and independent variables can be separated on opposite sides of the equation.

32Parametric equations

32.1Parametric equations

We discuss the basics of parametric curves.

32.2Calculus and parametric curves

We discuss derivatives of parametrically defined curves.

33Introduction to polar coordinates

33.1Introduction to polar coordinates

Polar coordinates are coordinates based on an angle and a radius.

33.2Gallery of polar curves

We see a collection of polar curves.

34Derivatives of polar functions

34.1Derivatives of polar functions

We differentiate polar functions.

35Integrals of polar functions

35.1Integrals of polar functions

We integrate polar functions.

36Working in two and three dimensions

36.1Working in two and three dimensions

We talk about basic geometry in higher dimensions.

37Vectors

37.1Vectors

Vectors are lists of numbers that denote direction and magnitude.

38Dot products

38.1The Dot Product

The dot product is an important operation between vectors that captures geometric information.

38.2Projections and orthogonal decomposition

Projections tell us how much of one vector lies in the direction of another and are important in physical applications.

39Cross products

39.1The cross product

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

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