It might seem like an age-old question, but have you ever pondered over 'When am I ever going to use this in real life' in your math class? To stumble on the applications of basic arithmetic or finances are obvious, but when it comes to calculus, we often find ourselves scratching our heads with confusion. 6th grade english coach mcdonald's health class website list.
- What Do You Know Ap Calculus Frq
- What Do You Know Ap Calculus College Board
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There's one thing I need to tell you: Of course, you may not sit down and solve a tricky differential equation on a daily basis, but calculus is still all around you. Did you know that because of the ability to model and control systems, it gives us extraordinary power over the material world?
Calculus is the language of engineers, scientists, and economists. The work of these professionals has a huge impact on our daily life – from your microwaves, cell phones, TV, and car to medicine, economy, and national defense.
What is Calculus?
You'll learn the basics from the ground up, starting with how calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. Later lessons go in-depth to make sure you learn exactly what you need to know in order to confidently ace the AP Calculus exam. How do I prepare for the course? The AP Calculus AB exam can be intimidating if you don't know what to expect. Knowing how the exam will be formatted and what types of questions it will ask can go a long way toward helping you feel more prepared and confident on test day. You can learn calculus in a very mechanical, follow this procedure in this situation, sort of way. If this is ok with you, all you need is a firm grounding in algebra and trigonometry. Calculus for Beginners and Artists Chapter 0: Why Study Calculus? Chapter 1: Numbers Chapter 2: Using a Spreadsheet Chapter 3: Linear Functions Chapter 4: Quadratics and Derivatives of Functions Chapter 5: Rational Functions and the Calculation of Derivatives Chapter 6: Exponential Functions, Substitution and the Chain Rule. AP Calculus – Final Review Sheet When you see the words. This is what you think of doing 1. Find the zeros Find roots. Set function = 0, factor or use quadratic equation if quadratic, graph to find zeros on calculator 2. Show that f x is even Show that (−)= ( ) symmetric to y-axis 3.
British Scientist Sir Isaac Newton (1642-1727) invented this new field of mathematics. ‘Calculus' is a Latin word, which means ‘stone.' Romans used stones for counting. For the counting of infinitely smaller numbers, Mathematicians began using the same term, and the name stuck.
Though it was proved that some basic ideas of Calculus were known to our Indian Mathematicians, Newton & Leibnitz initiated a new era of mathematics. There are a large number of applications of calculus in our daily life. It can't be possibly illustrated in a single view. I'm trying to describe them shortly here.
Application in Engineering
- An Architect Engineer uses integration in determining the amount of the necessary materials to construct curved shape constructions (e.g. dome over a sports arena) and also to measure the weight of that structure. Calculus is used to improve the architecture not only of buildings but also of important infrastructures such as bridges.
- In Electrical Engineering, Calculus (Integration) is used to determine the exact length of power cable needed to connect two substations, which are miles away from each other.
- Space flight engineers frequently use calculus when planning for long missions. To launch an exploratory probe, they must consider the different orbiting velocities of the Earth and the planet the probe is targeted for, as well as other gravitational influences like the sun and the moon.
Application in Medical Science
- Biologists use differential calculus to determine the exact rate of growth in a bacterial culture when different variables such as temperature and food source are changed.
Application in Physics
- In Physics, Integration is very much needed. For example, to calculate the Centre of Mass, Centre of Gravity and Mass Moment of Inertia of a sports utility vehicle.
- To calculate the velocity and trajectory of an object, predict the position of planets, and understand electromagnetism.
Application in Statistics
- Statisticians use calculus to evaluate survey data to help develop business plans for different companies. Because a survey involves many different questions with a range of possible answers, calculus allows a more accurate prediction for the appropriate action.
Application in Research Analysis
- An operations research analyst will use calculus when observing different processes at a manufacturing corporation. By considering the value of different variables, they can help a company improve operating efficiency, increase production, and raise profits.
Application in Graphics
- A graphics artist uses calculus to determine how different three-dimensional models will behave when subjected to rapidly changing conditions. It can create a realistic environment for movies or video games.
Application in Chemistry
- It is used to determine the rate of a chemical reaction and to determine some necessary information of Radioactive decay reaction.
Let's take a look at some other examples.
- Credit card companies use calculus to set the minimum payments due on credit card statements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance.
- Doctors and lawyers use calculus to help build the discipline necessary for solving complex problems, such as diagnosing patients or planning a prosecution case.
- The field of epidemiology — the study of the spread of infectious disease — relies heavily on calculus. It can be used to determine how far and fast a disease is spreading, where it may have originated from and how to best treat it.
For most professions, this topic is very important. So, this is why you can't dismiss it as just another subject. Despite its mystique as a more complex branch of mathematics, it touches our lives each day, in ways too numerous to calculate.
You can find tips from our experts here to become super at calculus.
More gamesunblocked games. Best of luck!
Show Mobile NoticeShow All NotesHide All Notes
What Do You Know Ap Calculus Frq
You appear to be on a device with a 'narrow' screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my 'class notes', they should be accessible to anyone wanting to learn Calculus II or needing a refresher in some of the topics from the class.
These notes do assume that the reader has a good working knowledge of Calculus I topics including limits, derivatives and basic integration and integration by substitution.
Calculus II tends to be a very difficult course for many students. There are many reasons for this.
The first reason is that this course does require that you have a very good working knowledge of Calculus I. The Calculus I portion of many of the problems tends to be skipped and left to the student to verify or fill in the details. If you don't have good Calculus I skills, and you are constantly getting stuck on the Calculus I portion of the problem, you will find this course very difficult to complete.
The second, and probably larger, reason many students have difficulty with Calculus II is that you will be asked to truly think in this class. That is not meant to insult anyone; it is simply an acknowledgment that you can't just memorize a bunch of formulas and expect to pass the course as you can do in many math classes. There are formulas in this class that you will need to know, but they tend to be fairly general. You will need to understand them, how they work, and more importantly whether they can be used or not. As an example, the first topic we will look at is Integration by Parts. The integration by parts formula is very easy to remember. However, just because you've got it memorized doesn't mean that you can use it. You'll need to be able to look at an integral and realize that integration by parts can be used (which isn't always obvious) and then decide which portions of the integral correspond to the parts in the formula (again, not always obvious).
Finally, many of the problems in this course will have multiple solution techniques and so you'll need to be able to identify all the possible techniques and then decide which will be the easiest technique to use.
Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
- Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus II have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn't covered in class.
- Because I want these notes to provide some more examples for you to read through, I don't always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.
- Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible when writing these up, but the reality is that I can't anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I've not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
- This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.
Here is a listing (and brief description) of the material that is in this set of notes.
Integration Techniques - In this chapter we will look at several integration techniques including Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions and Partial Fractions. We will also look at Improper Integrals including using the Comparison Test for convergence/divergence of improper integrals.Integration by Parts – In this section we will be looking at Integration by Parts. Of all the techniques we'll be looking at in this class this is the technique that students are most likely to run into down the road in other classes. We also give a derivation of the integration by parts formula.
Integrals Involving Trig Functions – In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.
Trig Substitutions – In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals.
Partial Fractions – In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions.
Integrals Involving Roots – In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots.
Integrals Involving Quadratics – In this section we are going to look at some integrals that involve quadratics for which the previous techniques won't work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. 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. Also note that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines.
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.e. not infinite) value. 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.e. if they have a finite value or not). 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.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.
Integrals Involving Trig Functions – In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.
Trig Substitutions – In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals.
Partial Fractions – In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions.
Integrals Involving Roots – In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots.
Integrals Involving Quadratics – In this section we are going to look at some integrals that involve quadratics for which the previous techniques won't work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. 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. Also note that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines.
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.e. not infinite) value. 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.e. if they have a finite value or not). 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.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.
Applications of Integrals - In this chapter we'll take a look at a few applications of integrals. We will look at determining the arc length of a curve, the surface area of a solid of revolution, the center of mass of a region bounded by two curves, the hydrostatic force/pressure on a plate submerged in water and a quick look at computing the mean of a probability density function. The applications given here tend to result in integrals that are typically covered in a Calculus II course.
Parametric Equations and Polar Coordinates - In this chapter we will introduce the ideas of parametric equations and polar coordinates. We will also look at many of the basic Calculus ideas (tangent lines, area, arc length and surface area) in terms of these two ideas.
Arc Length – In this section we'll determine the length of a curve over a given interval.
Surface Area – In this section we'll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis.
Center of Mass – In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the (x) or (y)-axis).
Hydrostatic Pressure and Force – In this section we'll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines.
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.
Surface Area – In this section we'll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis.
Center of Mass – In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the (x) or (y)-axis).
Hydrostatic Pressure and Force – In this section we'll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines.
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.
What Do You Know Ap Calculus College Board
Parametric Equations and Polar Coordinates - In this chapter we will introduce the ideas of parametric equations and polar coordinates. We will also look at many of the basic Calculus ideas (tangent lines, area, arc length and surface area) in terms of these two ideas.
Parametric Equations and Curves – In this section we will introduce parametric equations and parametric curves (i.e. graphs of parametric equations). 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.
Tangents with Parametric Equations – In this section we will discuss how to find the derivatives (frac{dy}{dx}) and (frac{d^{2}y}{dx^{2}}) for parametric curves. We will also discuss using these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down.
Area with Parametric Equations – In this section we will discuss how to find the area between a parametric curve and the (x)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus I techniques on the resulting algebraic equation).
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).
Surface Area with Parametric Equations – In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the (x) or (y)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Polar Coordinates – In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal' Cartesian/Rectangular coordinate system. 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.
Tangents with Polar Coordinates – In this section we will discuss how to find the derivative (frac{dy}{dx}) for polar curves. 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. The regions we look at in this section tend (although not always) to be shaped vaguely like a piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. 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).
Surface Area with Polar Coordinates – In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the (x) or (y)-axis 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.
Tangents with Parametric Equations – In this section we will discuss how to find the derivatives (frac{dy}{dx}) and (frac{d^{2}y}{dx^{2}}) for parametric curves. We will also discuss using these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down.
Area with Parametric Equations – In this section we will discuss how to find the area between a parametric curve and the (x)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus I techniques on the resulting algebraic equation).
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).
Surface Area with Parametric Equations – In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the (x) or (y)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Polar Coordinates – In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal' Cartesian/Rectangular coordinate system. 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.
Tangents with Polar Coordinates – In this section we will discuss how to find the derivative (frac{dy}{dx}) for polar curves. 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. The regions we look at in this section tend (although not always) to be shaped vaguely like a piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. 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).
Surface Area with Polar Coordinates – In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the (x) or (y)-axis 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.
- Credit card companies use calculus to set the minimum payments due on credit card statements at the exact time the statement is processed by considering multiple variables such as changing interest rates and a fluctuating available balance.
- Doctors and lawyers use calculus to help build the discipline necessary for solving complex problems, such as diagnosing patients or planning a prosecution case.
- The field of epidemiology — the study of the spread of infectious disease — relies heavily on calculus. It can be used to determine how far and fast a disease is spreading, where it may have originated from and how to best treat it.
For most professions, this topic is very important. So, this is why you can't dismiss it as just another subject. Despite its mystique as a more complex branch of mathematics, it touches our lives each day, in ways too numerous to calculate.
You can find tips from our experts here to become super at calculus.
More gamesunblocked games. Best of luck!
Show Mobile NoticeShow All NotesHide All NotesWhat Do You Know Ap Calculus Frq
You appear to be on a device with a 'narrow' screen width (i.e. you are probably on a mobile phone). Due to the nature of the mathematics on this site it is best views in landscape mode. If your device is not in landscape mode many of the equations will run off the side of your device (should be able to scroll to see them) and some of the menu items will be cut off due to the narrow screen width.
Here are my online notes for my Calculus II course that I teach here at Lamar University. Despite the fact that these are my 'class notes', they should be accessible to anyone wanting to learn Calculus II or needing a refresher in some of the topics from the class.
These notes do assume that the reader has a good working knowledge of Calculus I topics including limits, derivatives and basic integration and integration by substitution.
Calculus II tends to be a very difficult course for many students. There are many reasons for this.
The first reason is that this course does require that you have a very good working knowledge of Calculus I. The Calculus I portion of many of the problems tends to be skipped and left to the student to verify or fill in the details. If you don't have good Calculus I skills, and you are constantly getting stuck on the Calculus I portion of the problem, you will find this course very difficult to complete.
The second, and probably larger, reason many students have difficulty with Calculus II is that you will be asked to truly think in this class. That is not meant to insult anyone; it is simply an acknowledgment that you can't just memorize a bunch of formulas and expect to pass the course as you can do in many math classes. There are formulas in this class that you will need to know, but they tend to be fairly general. You will need to understand them, how they work, and more importantly whether they can be used or not. As an example, the first topic we will look at is Integration by Parts. The integration by parts formula is very easy to remember. However, just because you've got it memorized doesn't mean that you can use it. You'll need to be able to look at an integral and realize that integration by parts can be used (which isn't always obvious) and then decide which portions of the integral correspond to the parts in the formula (again, not always obvious).
Finally, many of the problems in this course will have multiple solution techniques and so you'll need to be able to identify all the possible techniques and then decide which will be the easiest technique to use.
Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
- Because I wanted to make this a fairly complete set of notes for anyone wanting to learn calculus II have included some material that I do not usually have time to cover in class and because this changes from semester to semester it is not noted here. You will need to find one of your fellow class mates to see if there is something in these notes that wasn't covered in class.
- Because I want these notes to provide some more examples for you to read through, I don't always work the same problems in class as those given in the notes. Likewise, even if I do work some of the problems in here I may work fewer problems in class than are presented here.
- Sometimes questions in class will lead down paths that are not covered here. I try to anticipate as many of the questions as possible when writing these up, but the reality is that I can't anticipate all the questions. Sometimes a very good question gets asked in class that leads to insights that I've not included here. You should always talk to someone who was in class on the day you missed and compare these notes to their notes and see what the differences are.
- This is somewhat related to the previous three items, but is important enough to merit its own item. THESE NOTES ARE NOT A SUBSTITUTE FOR ATTENDING CLASS!! Using these notes as a substitute for class is liable to get you in trouble. As already noted not everything in these notes is covered in class and often material or insights not in these notes is covered in class.
Here is a listing (and brief description) of the material that is in this set of notes.
Integration Techniques - In this chapter we will look at several integration techniques including Integration by Parts, Integrals Involving Trig Functions, Trig Substitutions and Partial Fractions. We will also look at Improper Integrals including using the Comparison Test for convergence/divergence of improper integrals.Integration by Parts – In this section we will be looking at Integration by Parts. Of all the techniques we'll be looking at in this class this is the technique that students are most likely to run into down the road in other classes. We also give a derivation of the integration by parts formula.
Integrals Involving Trig Functions – In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.
Trig Substitutions – In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals.
Partial Fractions – In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions.
Integrals Involving Roots – In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots.
Integrals Involving Quadratics – In this section we are going to look at some integrals that involve quadratics for which the previous techniques won't work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. 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. Also note that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines.
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.e. not infinite) value. 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.e. if they have a finite value or not). 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.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.
Integrals Involving Trig Functions – In this section we look at integrals that involve trig functions. In particular we concentrate integrating products of sines and cosines as well as products of secants and tangents. We will also briefly look at how to modify the work for products of these trig functions for some quotients of trig functions.
Trig Substitutions – In this section we will look at integrals (both indefinite and definite) that require the use of a substitutions involving trig functions and how they can be used to simplify certain integrals.
Partial Fractions – In this section we will use partial fractions to rewrite integrands into a form that will allow us to do integrals involving some rational functions.
Integrals Involving Roots – In this section we will take a look at a substitution that can, on occasion, be used with integrals involving roots.
Integrals Involving Quadratics – In this section we are going to look at some integrals that involve quadratics for which the previous techniques won't work right away. In some cases, manipulation of the quadratic needs to be done before we can do the integral. 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. Also note that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines.
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.e. not infinite) value. 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.e. if they have a finite value or not). 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.e. because it is not possible to do the indefinite integral) and yet we may need to know the value of the definite integral anyway. These methods allow us to at least get an approximate value which may be enough in a lot of cases.
Applications of Integrals - In this chapter we'll take a look at a few applications of integrals. We will look at determining the arc length of a curve, the surface area of a solid of revolution, the center of mass of a region bounded by two curves, the hydrostatic force/pressure on a plate submerged in water and a quick look at computing the mean of a probability density function. The applications given here tend to result in integrals that are typically covered in a Calculus II course.
Arc Length – In this section we'll determine the length of a curve over a given interval.
Surface Area – In this section we'll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis.
Center of Mass – In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the (x) or (y)-axis).
Hydrostatic Pressure and Force – In this section we'll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines.
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.
Surface Area – In this section we'll determine the surface area of a solid of revolution, i.e. a solid obtained by rotating a region bounded by two curves about a vertical or horizontal axis.
Center of Mass – In this section we will determine the center of mass or centroid of a thin plate where the plate can be described as a region bounded by two curves (one of which may the (x) or (y)-axis).
Hydrostatic Pressure and Force – In this section we'll determine the hydrostatic pressure and force on a vertical plate submerged in water. The plates used in the examples can all be described as regions bounded by one or more curves/lines.
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.
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Parametric Equations and Polar Coordinates - In this chapter we will introduce the ideas of parametric equations and polar coordinates. We will also look at many of the basic Calculus ideas (tangent lines, area, arc length and surface area) in terms of these two ideas.
Parametric Equations and Curves – In this section we will introduce parametric equations and parametric curves (i.e. graphs of parametric equations). 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.
Tangents with Parametric Equations – In this section we will discuss how to find the derivatives (frac{dy}{dx}) and (frac{d^{2}y}{dx^{2}}) for parametric curves. We will also discuss using these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down.
Area with Parametric Equations – In this section we will discuss how to find the area between a parametric curve and the (x)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus I techniques on the resulting algebraic equation).
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).
Surface Area with Parametric Equations – In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the (x) or (y)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Polar Coordinates – In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal' Cartesian/Rectangular coordinate system. 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.
Tangents with Polar Coordinates – In this section we will discuss how to find the derivative (frac{dy}{dx}) for polar curves. 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. The regions we look at in this section tend (although not always) to be shaped vaguely like a piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. 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).
Surface Area with Polar Coordinates – In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the (x) or (y)-axis 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.
Tangents with Parametric Equations – In this section we will discuss how to find the derivatives (frac{dy}{dx}) and (frac{d^{2}y}{dx^{2}}) for parametric curves. We will also discuss using these derivative formulas to find the tangent line for parametric curves as well as determining where a parametric curve in increasing/decreasing and concave up/concave down.
Area with Parametric Equations – In this section we will discuss how to find the area between a parametric curve and the (x)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus I techniques on the resulting algebraic equation).
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).
Surface Area with Parametric Equations – In this section we will discuss how to find the surface area of a solid obtained by rotating a parametric curve about the (x) or (y)-axis using only the parametric equations (rather than eliminating the parameter and using standard Calculus techniques on the resulting algebraic equation).
Polar Coordinates – In this section we will introduce polar coordinates an alternative coordinate system to the ‘normal' Cartesian/Rectangular coordinate system. 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.
Tangents with Polar Coordinates – In this section we will discuss how to find the derivative (frac{dy}{dx}) for polar curves. 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. The regions we look at in this section tend (although not always) to be shaped vaguely like a piece of pie or pizza and we are looking for the area of the region from the outer boundary (defined by the polar equation) and the origin/pole. 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).
Surface Area with Polar Coordinates – In this section we will discuss how to find the surface area of a solid obtained by rotating a polar curve about the (x) or (y)-axis 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.
Series and Sequences - In this chapter we introduce sequences and series. We discuss whether a sequence converges or diverges, is increasing or decreasing, or if the sequence is bounded. We will then define just what an infinite series is and discuss many of the basic concepts involved with series. We will discuss if a series will converge or diverge, including many of the tests that can be used to determine if a series converges or diverges. We will also discuss using either a power series or a Taylor series to represent a function and how to find the radius and interval of convergence for this series.
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. We will also give many of the basic facts and properties we'll need as we work with sequences.
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. We will also determine a sequence is bounded below, bounded above and/or bounded.
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).
Convergence/Divergence of Series – In this section we will discuss in greater detail the convergence and divergence of infinite series. 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. We will examine Geometric Series, Telescoping Series, and Harmonic Series.
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.
Comparison Test/Limit Comparison Test – In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. 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. Note as well that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. 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. We also discuss differentiation and integration of power series.
Taylor Series – In this section we will discuss how to find the Taylor/Maclaurin Series for a function. 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. We also derive some well known formulas for Taylor series of ({bf e}^{x}) , (cos(x)) and (sin(x)) around (x=0).
Applications of Series – In this section we will take a quick look at a couple of applications of series. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. We will also see how we can use the first few terms of a power series to approximate a function.
Binomial Series – In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form ( left(a+bright)^{n}) when (n) is an integer. In addition, when (n) is not an integer an extension to the Binomial Theorem can be used to give a power series representation of the term.
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. We will also determine a sequence is bounded below, bounded above and/or bounded.
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).
Convergence/Divergence of Series – In this section we will discuss in greater detail the convergence and divergence of infinite series. 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. We will examine Geometric Series, Telescoping Series, and Harmonic Series.
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.
Comparison Test/Limit Comparison Test – In this section we will discuss using the Comparison Test and Limit Comparison Tests to determine if an infinite series converges or diverges. 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. Note as well that there really isn't one set of guidelines that will always work and so you always need to be flexible in following this set of guidelines. 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. We also discuss differentiation and integration of power series.
Taylor Series – In this section we will discuss how to find the Taylor/Maclaurin Series for a function. 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. We also derive some well known formulas for Taylor series of ({bf e}^{x}) , (cos(x)) and (sin(x)) around (x=0).
Applications of Series – In this section we will take a quick look at a couple of applications of series. We will illustrate how we can find a series representation for indefinite integrals that cannot be evaluated by any other method. We will also see how we can use the first few terms of a power series to approximate a function.
Binomial Series – In this section we will give the Binomial Theorem and illustrate how it can be used to quickly expand terms in the form ( left(a+bright)^{n}) when (n) is an integer. In addition, when (n) is not an integer an extension to the Binomial Theorem can be used to give a power series representation of the term.
Vectors - In this (very brief) chapter we will take a look at the basics of vectors. Included are common notation for vectors, arithmetic of vectors, dot product of vectors (and applications) and cross product of vectors (and applications).
Basic Concepts – In this section we will introduce some common notation for vectors as well as some of the basic concepts about vectors such as the magnitude of a vector and unit vectors. We also illustrate how to find a vector from its starting and end points.
Vector Arithmetic – In this section we will discuss the mathematical and geometric interpretation of the sum and difference of two vectors. We also define and give a geometric interpretation for scalar multiplication. We also give some of the basic properties of vector arithmetic and introduce the common (i), (j), (k) notation for vectors.
Dot Product – In this section we will define the dot product of two vectors. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are orthogonal. We also discuss finding vector projections and direction cosines in this section.
Cross Product – In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products.
Vector Arithmetic – In this section we will discuss the mathematical and geometric interpretation of the sum and difference of two vectors. We also define and give a geometric interpretation for scalar multiplication. We also give some of the basic properties of vector arithmetic and introduce the common (i), (j), (k) notation for vectors.
Dot Product – In this section we will define the dot product of two vectors. We give some of the basic properties of dot products and define orthogonal vectors and show how to use the dot product to determine if two vectors are orthogonal. We also discuss finding vector projections and direction cosines in this section.
Cross Product – In this section we define the cross product of two vectors and give some of the basic facts and properties of cross products.
3-Dimensional Space - In this chapter we will start looking at three dimensional space. This chapter is generally prep work for Calculus III and so we will cover the standard 3D coordinate system as well as a couple of alternative coordinate systems. We will also discuss how to find the equations of lines and planes in three dimensional space. We will look at some standard 3D surfaces and their equations. In addition we will introduce vector functions and some of their applications (tangent and normal vectors, arc length, curvature and velocity and acceleration).
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The 3-D Coordinate System – In this section we will introduce the standard three dimensional coordinate system as well as some common notation and concepts needed to work in three dimensions.
Equations of Lines – In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.
Equations of Planes – In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane.
Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids.
Functions of Several Variables – In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.
Vector Functions – In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times.
Calculus with Vector Functions – In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions.
Tangent, Normal and Binormal Vectors – In this section we will define the tangent, normal and binormal vectors.
Arc Length with Vector Functions – In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we've already seen.
Curvature – In this section we give two formulas for computing the curvature (i.e. how fast the function is changing at a given point) of a vector function.
Velocity and Acceleration – In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function. For the acceleration we give formulas for both the normal acceleration and the tangential acceleration.
Cylindrical Coordinates – In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see cylindrical coordinates are really nothing more than a very natural extension of polar coordinates into a three dimensional setting.
Spherical Coordinates – In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the two).
Equations of Lines – In this section we will derive the vector form and parametric form for the equation of lines in three dimensional space. We will also give the symmetric equations of lines in three dimensional space. Note as well that while these forms can also be useful for lines in two dimensional space.
Equations of Planes – In this section we will derive the vector and scalar equation of a plane. We also show how to write the equation of a plane from three points that lie in the plane.
Quadric Surfaces – In this section we will be looking at some examples of quadric surfaces. Some examples of quadric surfaces are cones, cylinders, ellipsoids, and elliptic paraboloids.
Functions of Several Variables – In this section we will give a quick review of some important topics about functions of several variables. In particular we will discuss finding the domain of a function of several variables as well as level curves, level surfaces and traces.
Vector Functions – In this section we introduce the concept of vector functions concentrating primarily on curves in three dimensional space. We will however, touch briefly on surfaces as well. We will illustrate how to find the domain of a vector function and how to graph a vector function. We will also show a simple relationship between vector functions and parametric equations that will be very useful at times.
Calculus with Vector Functions – In this section here we discuss how to do basic calculus, i.e. limits, derivatives and integrals, with vector functions.
Tangent, Normal and Binormal Vectors – In this section we will define the tangent, normal and binormal vectors.
Arc Length with Vector Functions – In this section we will extend the arc length formula we used early in the material to include finding the arc length of a vector function. As we will see the new formula really is just an almost natural extension of one we've already seen.
Curvature – In this section we give two formulas for computing the curvature (i.e. how fast the function is changing at a given point) of a vector function.
Velocity and Acceleration – In this section we will revisit a standard application of derivatives, the velocity and acceleration of an object whose position function is given by a vector function. For the acceleration we give formulas for both the normal acceleration and the tangential acceleration.
Cylindrical Coordinates – In this section we will define the cylindrical coordinate system, an alternate coordinate system for the three dimensional coordinate system. As we will see cylindrical coordinates are really nothing more than a very natural extension of polar coordinates into a three dimensional setting.
Spherical Coordinates – In this section we will define the spherical coordinate system, yet another alternate coordinate system for the three dimensional coordinate system. This coordinates system is very useful for dealing with spherical objects. We will derive formulas to convert between cylindrical coordinates and spherical coordinates as well as between Cartesian and spherical coordinates (the more useful of the two).
