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" (y- y1) divided by the "run" (x- 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.

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