We will see several cases where this is needed in this section.
Integration Strategy — In this section we give a general set of guidelines for determining how to evaluate an integral. The guidelines give here involve a mix of both Calculus I and Calculus II techniques to be as general as possible. Improper Integrals — In this section we will look at integrals with infinite intervals of integration and integrals with discontinuous integrands in this section. Collectively, they are called improper integrals and as we will see they may or may not have a finite i.
Determining if they have finite values will, in fact, be one of the major topics of this section. Comparison Test for Improper Integrals — It will not always be possible to evaluate improper integrals and yet we still need to determine if they converge or diverge i. So, in this section we will use the Comparison Test to determine if improper integrals converge or diverge. Approximating Definite Integrals — In this section we will look at several fairly simple methods of approximating the value of a definite integral. It is not possible to evaluate every definite integral i.
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These methods allow us to at least get an approximate value which may be enough in a lot of cases. Probability — Many quantities can be described with probability density functions. For example, the length of time a person waits in line at a checkout counter or the life span of a light bulb. None of these quantities are fixed values and will depend on a variety of factors. In this section we will look at probability density functions and computing the mean think average wait in line or average life span of a light blub of a probability density function.
Parametric Equations and Curves — In this section we will introduce parametric equations and parametric curves i. We will graph several sets of parametric equations and discuss how to eliminate the parameter to get an algebraic equation which will often help with the graphing process.
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Arc Length with Parametric Equations — In this section we will discuss how to find the arc length of a parametric curve using only the parametric equations rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation. We will derive formulas to convert between polar and Cartesian coordinate systems.
We will also look at many of the standard polar graphs as well as circles and some equations of lines in terms of polar coordinates. We will also discuss using this derivative formula to find the tangent line for polar curves using only polar coordinates rather than converting to Cartesian coordinates and using standard Calculus techniques. Area with Polar Coordinates — In this section we will discuss how to the area enclosed by a polar curve.
We will also discuss finding the area between two polar curves. Arc Length with Polar Coordinates — In this section we will discuss how to find the arc length of a polar curve using only polar coordinates rather than converting to Cartesian coordinates and using standard Calculus techniques.
Arc Length and Surface Area Revisited — In this section we will summarize all the arc length and surface area formulas we developed over the course of the last two chapters. Sequences — In this section we define just what we mean by sequence in a math class and give the basic notation we will use with them. We will focus on the basic terminology, limits of sequences and convergence of sequences in this section.
More on Sequences — In this section we will continue examining sequences. We will determine if a sequence in an increasing sequence or a decreasing sequence and hence if it is a monotonic sequence.
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Series — The Basics — In this section we will formally define an infinite series. We will also give many of the basic facts, properties and ways we can use to manipulate a series. We will also briefly discuss how to determine if an infinite series will converge or diverge a more in depth discussion of this topic will occur in the next section.
We will illustrate how partial sums are used to determine if an infinite series converges or diverges. We will also give the Divergence Test for series in this section. Special Series — In this section we will look at three series that either show up regularly or have some nice properties that we wish to discuss. Integral Test — In this section we will discuss using the Integral Test to determine if an infinite series converges or diverges. The Integral Test can be used on a infinite series provided the terms of the series are positive and decreasing.
A proof of the Integral Test is also given. In order to use either test the terms of the infinite series must be positive. Proofs for both tests are also given. Alternating Series Test — In this section we will discuss using the Alternating Series Test to determine if an infinite series converges or diverges.
The Alternating Series Test can be used only if the terms of the series alternate in sign. A proof of the Alternating Series Test is also given. Absolute Convergence — In this section we will have a brief discussion on absolute convergence and conditionally convergent and how they relate to convergence of infinite series.
Ratio Test — In this section we will discuss using the Ratio Test to determine if an infinite series converges absolutely or diverges. The Ratio Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Ratio Test is also given. Root Test — In this section we will discuss using the Root Test to determine if an infinite series converges absolutely or diverges.
The Root Test can be used on any series, but unfortunately will not always yield a conclusive answer as to whether a series will converge absolutely or diverge. A proof of the Root Test is also given. Strategy for Series — In this section we give a general set of guidelines for determining which test to use in determining if an infinite series will converge or diverge.
A summary of all the various tests, as well as conditions that must be met to use them, we discussed in this chapter are also given in this section. Estimating the Value of a Series — In this section we will discuss how the Integral Test, Comparison Test, Alternating Series Test and the Ratio Test can, on occasion, be used to estimating the value of an infinite series. Power Series — In this section we will give the definition of the power series as well as the definition of the radius of convergence and interval of convergence for a power series.
We will also illustrate how the Ratio Test and Root Test can be used to determine the radius and interval of convergence for a power series. Power Series and Functions — In this section we discuss how the formula for a convergent Geometric Series can be used to represent some functions as power series. To use the Geometric Series formula, the function must be able to be put into a specific form, which is often impossible. However, use of this formula does quickly illustrate how functions can be represented as a power series.
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We also discuss differentiation and integration of power series. This will work for a much wider variety of function than the method discussed in the previous section at the expense of some often unpleasant work. It will save and restore your Xmax Xmin Ymax and Ymin and turn the axes back on as long as the program is ran to completion. Right Triangle Solver This program calculates answers for right triangles, given two pieces of information. This program displays a menu at the bottom of the screen, with a prompt directly above the menu options.
There are as many as five menu options to choose from at any one time. These keys are used to select the menu option desired by the user. Above the prompt, a right triangle is drawn, with angle labels, and then side labels. When selected, the angle or length can be entered adjacent to where the angle or side is located. This allows the user to see that he is indeed putting the known information in the correct location.
The left arrow key can also be used to backspace. This version times out after 1 minute to save your batteries. Right Triangle Solver This program shows you a picture of a right triangle with sides and angles labled. Then it asks you which variables you were given, and then solvers for the unkown.
Program AND picture are required in order for this program to work correctly. When you run the program, all you have to do is select whether you need the length of the hypotenuse, or the length of the missing leg. If you choose the leg, you will have to give the length of the other leg and the hypotenuse.
If you choose the hypotenuse, you will have to give the lengths of the two legs.
Related Quick Review of Topics in Trigonometry: Sum, Differences & Product Formulas (Quick Review Notes)
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