Next week my revised physics/calculus/chemistry monthly plan kicks in. The end of the year goal is to plug a leak in my calculus videos--this one and one more.
The Precise Definition of a Limit
Showing posts with label calculus. Show all posts
Showing posts with label calculus. Show all posts
Friday, December 29, 2017
Friday, December 23, 2016
Friday Gen Eds MS10: The Basics of Calculus
This is the tenth post in the math/science part of my "Gen Eds in a Nutshell" series. The Gen Ed series consists of ten subjects you might study in a general education or "liberal arts" core at a university or college. I've already done the subject of philosophy, and I'm over half way through the world history subject on Wednesdays. I'm combining the last two on math and science into one series on Fridays.
Thus far in the math/science subjects:
1. The calculus was invented in the 1600s. Isaac Newton (1643-1727) usually gets the credit, but Gottfried Leibniz (1646-1716) developed it independently at the same time. Calculus was invented to address two problems, namely, the tangent problem and the area problem.
2. We might introduce the tangent problem in this way. If you drive from location x to location y in 60 minutes and the roadway between is 60 miles, we know that you have driven an average of 1 mile per minute.
But what about the instantaneous speed? How might we know how fast you were driving 29 minutes into the trip?
We might plot your trip on a graph. We might put distance on the y-axis and time on the x-axis. Perhaps we could come up with an equation that expressed the graph.
The average speed between any two points is the difference in distance divided by the difference in time, or the "rise" (y2 - y1) divided by the "run" (x2 - x1). In other words, the average speed is the slope of the graph between those two points. And if we wanted to know the speed at any instant, it would be the slope of the tangent to the initial equation at that point.
3. What Newton and Leibniz invented was, first, a way to determine the equation of this tangent line, the instantaneous slope at a point on an equation. More broadly, they found a way to derive an equation for the tangent to an equation at every point along the equation.
Their idea is really very simple. The slope between two points, as we have said, is the rise (of y) divided by the run (of x). So the rise is the y value of the equation (or function) at x2 minus the y value of the equation at x1. Another way to put this is f(x + Δx) - f(x). The symbol Δ means the change in x, and f(x) means the value of the equation (or function) at x.
If we divide f(x + Δx) - f(x) by Δx (the change in x), then we have divided the change in y by the change in x, which is the slope.
Differential Calculus
4. Newton's idea was to take Δx to zero. When the change in x was infinitesimally small, the "limit" as the difference in x approached zero would approach the equation of the tangent line to the equation. We can write the process in this way:
The expression f'(x) is called the derivative of x. If we take an equation and plug in x and x + Δx and go through the process of simplifying it, the Δx part always cancels out and we are left with the equation for the tangents to the initial equation at every point. As we will see in the next post, this allows us to do all sorts of things in science, economics, and in any field that involves change.
5. The branch of calculus that has to do with finding the instantaneous rate of change in this way is called differential calculus. There are some basic patterns for the derivatives of certain kinds of equations. For example, the derivative of a polynomial (an equation with x raised to some power) is very predictable.
The expression dy/dx is a way of saying "the derivative of y." In this case, y = an + b. To find an equation for a line that is tangent to this equation, you multiply the power (n) times whatever number is in front of a. Then you lower the power of n down by 1. [1]
6. You can take the derivative of a derivative, called a second order derivative. In fact, you can keep taking derivatives of derivatives as long as you still have elements left of which to take the derivative.
For example, if you have an equation for the distance you've traveled in a certain amount of time, the derivative of that equation (the first order derivative) gives you an equation for the speed in relation to time. Then if you take the derivative of that equation, the second order derivative is an equation for the acceleration in relation to time. Acceleration is the change in speed in relation to time.
7. This all may sound complicated, but it can be used to do a lot of things. For example, when an equation reaches a maximum or a minimum, its slope will be zero at that point (because the graph changes directions at that point). So if you make the equation of the derivative equal to zero and solve for x, then you will have identified points where the initial equation hits maxima and minima.
If you look at whether the second derivative is positive or negative, you can decide whether it is a maximum or minimum. If the second derivative is negative, then its graph is concave down and the point where the slope equals 0 is a minimum. Otherwise, it's a maximum.
Integral Calculus
8. The opposite problem to the tangent problem might be called the area problem. Let's say you have an equation again that you have graphed. How do you find the area under the graph?
The answer is similar. You slice up the area under the graph into smaller and smaller rectangles and add up their areas. As the width of the rectangles becomes infinitesimal (approaches zero), the sum of all the rectangles approaches the area under the equation.
The way we might write this situation up is:
What this means is that we are going to slice up the area under the equation into "n" number of rectangles. As the number of them approaches infinity (the limit as n approaches infinity), the sum approaches the area. The big symbol is the summation sign. The Δx refers to the little x part of each rectangle and the f(xi) refers to the y part of each rectangle. i = 1 just refers to the fact that we start adding with the first rectangle and we keep adding till we get to the "nth" one.
We call this operation, "integrating" or "finding the integral" of an equation. An "indefinite integral" finds the equation for the area under the entire equation. A "definite integral" finds the area under a specific part of the equation.
9. As it turns out, differentiation and integration are opposite operations. In fact, this is the fundamental theorem of calculus. If you take the derivative of an equation, you can get back to the initial equation by integrating the derivative. Or if you take the integral of an equation, you can get back to the original equation by "differentiating" it.
For example, the derivative of a distance equation is a speed equation, and the derivative of a speed equation is an acceleration equation. But it works the other way too. The integral of an acceleration equation is a speed equation, and the integral of a speed equation is a distance equation.
10. The tools of calculus are immensely helpful in countless fields, especially physics, but also fields like economics, population growth--any topic that involves change.
Next Week: Math/Science 11: The Physics of Motion
[1] The b disappears because it was times an x to the 0 power (1). Zero times b is zero. When taking a derivative, all "constants" like b disappear.
Thus far in the math/science subjects:
- Math/Science Overview
- Basic Types of Numbers
- The Atom and Quantum Physics
- The Periodic Table
- Molecules and Ions
- Chemical Reactions
- The Basic Tools of Algebra
- Heat and Thermodynamics
- Basic Geometry and Trigonometry
1. The calculus was invented in the 1600s. Isaac Newton (1643-1727) usually gets the credit, but Gottfried Leibniz (1646-1716) developed it independently at the same time. Calculus was invented to address two problems, namely, the tangent problem and the area problem.
2. We might introduce the tangent problem in this way. If you drive from location x to location y in 60 minutes and the roadway between is 60 miles, we know that you have driven an average of 1 mile per minute.
But what about the instantaneous speed? How might we know how fast you were driving 29 minutes into the trip?
We might plot your trip on a graph. We might put distance on the y-axis and time on the x-axis. Perhaps we could come up with an equation that expressed the graph.
The average speed between any two points is the difference in distance divided by the difference in time, or the "rise" (y2 - y1) divided by the "run" (x2 - x1). In other words, the average speed is the slope of the graph between those two points. And if we wanted to know the speed at any instant, it would be the slope of the tangent to the initial equation at that point.3. What Newton and Leibniz invented was, first, a way to determine the equation of this tangent line, the instantaneous slope at a point on an equation. More broadly, they found a way to derive an equation for the tangent to an equation at every point along the equation.
Their idea is really very simple. The slope between two points, as we have said, is the rise (of y) divided by the run (of x). So the rise is the y value of the equation (or function) at x2 minus the y value of the equation at x1. Another way to put this is f(x + Δx) - f(x). The symbol Δ means the change in x, and f(x) means the value of the equation (or function) at x.
If we divide f(x + Δx) - f(x) by Δx (the change in x), then we have divided the change in y by the change in x, which is the slope.
Differential Calculus
4. Newton's idea was to take Δx to zero. When the change in x was infinitesimally small, the "limit" as the difference in x approached zero would approach the equation of the tangent line to the equation. We can write the process in this way:
5. The branch of calculus that has to do with finding the instantaneous rate of change in this way is called differential calculus. There are some basic patterns for the derivatives of certain kinds of equations. For example, the derivative of a polynomial (an equation with x raised to some power) is very predictable.
6. You can take the derivative of a derivative, called a second order derivative. In fact, you can keep taking derivatives of derivatives as long as you still have elements left of which to take the derivative.
For example, if you have an equation for the distance you've traveled in a certain amount of time, the derivative of that equation (the first order derivative) gives you an equation for the speed in relation to time. Then if you take the derivative of that equation, the second order derivative is an equation for the acceleration in relation to time. Acceleration is the change in speed in relation to time.
7. This all may sound complicated, but it can be used to do a lot of things. For example, when an equation reaches a maximum or a minimum, its slope will be zero at that point (because the graph changes directions at that point). So if you make the equation of the derivative equal to zero and solve for x, then you will have identified points where the initial equation hits maxima and minima.
If you look at whether the second derivative is positive or negative, you can decide whether it is a maximum or minimum. If the second derivative is negative, then its graph is concave down and the point where the slope equals 0 is a minimum. Otherwise, it's a maximum.
Integral Calculus
8. The opposite problem to the tangent problem might be called the area problem. Let's say you have an equation again that you have graphed. How do you find the area under the graph?The answer is similar. You slice up the area under the graph into smaller and smaller rectangles and add up their areas. As the width of the rectangles becomes infinitesimal (approaches zero), the sum of all the rectangles approaches the area under the equation.
The way we might write this situation up is:
We call this operation, "integrating" or "finding the integral" of an equation. An "indefinite integral" finds the equation for the area under the entire equation. A "definite integral" finds the area under a specific part of the equation.
9. As it turns out, differentiation and integration are opposite operations. In fact, this is the fundamental theorem of calculus. If you take the derivative of an equation, you can get back to the initial equation by integrating the derivative. Or if you take the integral of an equation, you can get back to the original equation by "differentiating" it.
For example, the derivative of a distance equation is a speed equation, and the derivative of a speed equation is an acceleration equation. But it works the other way too. The integral of an acceleration equation is a speed equation, and the integral of a speed equation is a distance equation.
10. The tools of calculus are immensely helpful in countless fields, especially physics, but also fields like economics, population growth--any topic that involves change.
Next Week: Math/Science 11: The Physics of Motion
[1] The b disappears because it was times an x to the 0 power (1). Zero times b is zero. When taking a derivative, all "constants" like b disappear.
Monday, May 18, 2015
Teaching math as needed for physics (for nerds only)
Although I haven't finished or published examples of this approach. I have used something like it to teach biblical languages and the Seminary uses an approah something like this in its integrative approach to the practice of ministry. I believe it is far more effective than the "building block" model if it is done correctly. It is something like problem-based learning.
But I can imagine putting together the beginnings of a science curriculum this way.
1. Instantaneous velocity and acceleration (while discussing motion)
This is a great place to introduce the basics of finding a derivative, and this is how the Marion/Hornyak (MH) book pictured began. Integration can also be introduced as the inverse process in brief. Both would be introduced simply, to be expanded upon later.
2. Vector addition (motion in more than one dimension)
Vector addition begins to come into play as soon as you hit motion in more than one dimension. I find Young and Freedman's (YF) introduction of all things vector in chapter 1 a teaching problem. The cross product in particular is, I think, a difficult concept to introduce at the very beginning. Wouldn't it be more helpful to introduce the various characteristics of vector addition and multiplication as they arise in specific topics? This is also a place to review some basic trig.
3. Summation (forces)
When you get into forces, you could introduce summation notation and extend integral calculus. MH review definite integrals when they get to the application of Newton's laws.
4. Vector dot product (work)
Young and Freedman introduce dot products in chapter 1, but students don't need it until chapter 6. Why not introduce it there? There are more integrals in the treatment of work. So an integrative approach could be solidifying and extending techniques of integration as the student went along.
5. Partial derivatives, nablus (energy)
It is amazing to me that I wasn't introduced to partial derivatives until my third semester of calculus and I don't remember hearing about the nablus/gradient in four semesters. Yet these are relatively easy concepts I could have learned in a first semester. This is frequently the case. Teaching the calculus in order, you don't get to simple and useful concepts until way down the line.
Yet chapter 7 of YF already introduces these fairly straightforward concepts in their treatment of energy.
6. Integration in two and three dimensions
MH have a calculus 5 section on this before a chapter on angular momentum. YF also have an advanced section involving integration in their chapter on angular velocity.
7. Radian measurement (angular velocity)
YF review radian measurement as they begin their chapter on angular velocity.
8. Cross product
It is not until chapter 10 and the treatment of torque that YF ever use the vector cross product. Why introduce it in chapter 1, when most students will have no idea what it means or what it is for?
9. Taylor series
MH review this third semester calculus topic in between their chapters on gravitation and periodic motion.
10. Differentials
I see some differentials by themselves in YF treatment of thermodynamics.
11. Surface integrals
When you get into Gauss' law regarding electric flux
12. The gradient
More partial differentials when you get to electric potential
13. More cross product
When you get into electromagnetic induction
14. Second order differential equations
MH finally review second order differential equations, a fourth semester calculus topic, as they are digging deeper into electromagnetic waves.
I picture, perhaps, two teachers tag teaming over the course of a year or summer intensive. Problems might circle back around to earlier physics topics after new mathematical concepts were introduced. Similarly, methods of application (e.g., in calculus) could be introduced in the process of doing problems.
Just some ideas for a Monday morning...
2. Vector addition (motion in more than one dimension)
Vector addition begins to come into play as soon as you hit motion in more than one dimension. I find Young and Freedman's (YF) introduction of all things vector in chapter 1 a teaching problem. The cross product in particular is, I think, a difficult concept to introduce at the very beginning. Wouldn't it be more helpful to introduce the various characteristics of vector addition and multiplication as they arise in specific topics? This is also a place to review some basic trig.
3. Summation (forces)
When you get into forces, you could introduce summation notation and extend integral calculus. MH review definite integrals when they get to the application of Newton's laws.
4. Vector dot product (work)
Young and Freedman introduce dot products in chapter 1, but students don't need it until chapter 6. Why not introduce it there? There are more integrals in the treatment of work. So an integrative approach could be solidifying and extending techniques of integration as the student went along.
5. Partial derivatives, nablus (energy)
It is amazing to me that I wasn't introduced to partial derivatives until my third semester of calculus and I don't remember hearing about the nablus/gradient in four semesters. Yet these are relatively easy concepts I could have learned in a first semester. This is frequently the case. Teaching the calculus in order, you don't get to simple and useful concepts until way down the line.
Yet chapter 7 of YF already introduces these fairly straightforward concepts in their treatment of energy.
6. Integration in two and three dimensions
MH have a calculus 5 section on this before a chapter on angular momentum. YF also have an advanced section involving integration in their chapter on angular velocity.
7. Radian measurement (angular velocity)
YF review radian measurement as they begin their chapter on angular velocity.
8. Cross product
It is not until chapter 10 and the treatment of torque that YF ever use the vector cross product. Why introduce it in chapter 1, when most students will have no idea what it means or what it is for?
9. Taylor series
MH review this third semester calculus topic in between their chapters on gravitation and periodic motion.
10. Differentials
I see some differentials by themselves in YF treatment of thermodynamics.
11. Surface integrals
When you get into Gauss' law regarding electric flux
12. The gradient
More partial differentials when you get to electric potential
13. More cross product
When you get into electromagnetic induction
14. Second order differential equations
MH finally review second order differential equations, a fourth semester calculus topic, as they are digging deeper into electromagnetic waves.
I picture, perhaps, two teachers tag teaming over the course of a year or summer intensive. Problems might circle back around to earlier physics topics after new mathematical concepts were introduced. Similarly, methods of application (e.g., in calculus) could be introduced in the process of doing problems.
Just some ideas for a Monday morning...
Wednesday, August 15, 2012
Calculus is a method, not a reality
Just sayin'.
In one of the more important breakthroughs of my life, it occurred to me the other day that calculus is not really a deeper understanding of reality. It's a method for approximating values so close that you can figure out what they actually are. It's a way of tricking the math into telling us exact figures no one was able to figure out until the 1600's.
So differential calculus tricks math into telling us the slope of a tangent line to some other function, its instantaneous rate of change. Integral calculus tricks math into telling us the area under a curve, the sum of all the infinitesimal rectangles under it.
But it's all a trick to tell us a reality we already knew about, we just didn't know what its value was exactly.
I thought you'd want to know.
In one of the more important breakthroughs of my life, it occurred to me the other day that calculus is not really a deeper understanding of reality. It's a method for approximating values so close that you can figure out what they actually are. It's a way of tricking the math into telling us exact figures no one was able to figure out until the 1600's.
So differential calculus tricks math into telling us the slope of a tangent line to some other function, its instantaneous rate of change. Integral calculus tricks math into telling us the area under a curve, the sum of all the infinitesimal rectangles under it.
But it's all a trick to tell us a reality we already knew about, we just didn't know what its value was exactly.
I thought you'd want to know.
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