[MUSIC] In this video, we discuss how secants in geometry can be used to approximate slopes of tangent lines. Slopes of lines are just fractions involving the vertical rise divided by the horizontal run. The fractions associated with lines formed by secants happen to be linked to fractions associated with tangent lines by a natural limiting process. It's an amazing fact, that under certain conditions we can understand and control the behavior of approximations that occur, producing very precise outcomes. This leads us towards formal definitions of the derivative in terms of limits of fractions, which is the main theme of this module. The underlying idea of calculus is to use tangent lines to approximate smooth curves, which arise as graphs of sophisticated functions that model interesting phenomena. Lines, of, course use simple arithmetic. So if we can achieve this then there's a substantial reduction in complexity. We have the possibility of creating very powerful and wide ranging tools. A tangent line, typically, just glances a curve at a single point of interest and acts as a good approximation to the curve near this point of intersection. There are cases where tangent lines actually cross the curve at the point of interest, but more about that later. Warning, not all curves are smooth, some might need special treatment. For example, this curve has a sharp point, this curve has a discontinuity where the curve breaks apart at some point, this curve also has a sharp point. In each case there's some ambiguity about choosing an appropriate tangent line. The third curve is an interesting and important example and the sharp point is known as a cusp. There is in fact a useful and meaningful tangent line for this cusp example. A vertical line that splits the cusp in the middle and even passes completely through the curve. Close to the cusp this vertical line is a very good approximation to the behavior of the curve on either side. In the previous video, we looked at a particular displacement curve associated with a cannonball fired directly upwards and eventually coming back down to Earth under the influence of gravity. The curve was in fact described by a quadratic equation. We looked at slopes of tangent lines in that example and a linear function was produced that represented the velocity. This looked rather special, but in fact turns out to be typical of behavior that occurs with all quadratic functions. To try to understand what's going on let's carefully analyze the simplest quadratic of all, y equals x squared. And see if we can extract some general principles. Note that we're reverting x again as the independent variable and y the dependent variable. A typical point on the curve has coordinates x on the horizontal axis and x squared on the vertical axis. And we can draw a tangent line to the curve passing through this point. We can perturb the input x slightly simply by adding h to get a new input x plus h. And then move up to the curve and across to the vertical axis to land at the value x plus h, all squared. Hence the new point on the curve has coordinates x plus h and x plus h all squared. We can join these two points on the green curve with a straight line segment, colored blue in the diagram. A line that joins two distinct points of a curve is called a secant in geometry. Because it's a line segment this secant has a slope in the usual sense, that is a vertical raise divided by an horizontal run. You can see that the horizontal run in the diagram is h, there's no harm throughout this discussion in assuming that h is positive. The vertical rise in this diagram is the difference in the y coordinates, namely, x plus h squared minus x squared. We'll look at the quotient of these carefully in a moment. Note that as h is made smaller and smaller approaching zero without actually reaching zero, the slope of the secant approaches the slope of the tangent line. And it's conventional to use these arrows as abbreviations for the word approaches. To see this visually imagine the second point slipping or sliding down the parabola towards the first point. Even though the secant gets shorter and shorter, the direction of the secant aligns more and more with the direction of the tangent line. So that their slopes look like they're going to coincide and we think of them as actually coinciding in some kind of limit. Note that even though h approaches zero, it doesn't actually reach zero as these diagrams require two distinct points in order to form a secant. If h were equal to zero then there would be only one point on the curve not two. So returning to the earlier diagram with horizontal run h and vertical rise x plus h squared minus x squared. Let's look at the quotient representing the slope of the secant. The numerator, representing the vertical rise, can be expanded and then there's some cancellation. So the fraction becomes 2xh plus h squared divided by h. And the numerator factorizes and the h's cancel in the numerator and denominator simplifying further to 2x plus h. As h gets closer and closer to zero this expression, 2x plus h, for the slope of the secant gets closer and closer to 2x. We regard 2x as the limit of this process of taking slopes of secants. So 2x represents the slope of the tangent line. We've derived this very important result, that the slope of the tangent line to the parabola y = x squared and the point of the coordinates x and x squared is just twice x. Now, if you go through the same process with y equals k x squared, where k is some fixed constant then the previous calculation simply gets scaled by a factor of k. And we'll get the following result. The slope of the tangent line to the parabola y=kx squared at the point with coordinates x and kx squared is just twice kx. We can now discover a surprising connection with areas of circles. Consider a circle of radius r, it's area, capital A, is given by the formula pi r squared. The perimeter, capital P, is given by the formula 2 pi r. That is, twice pi r, which is precisely the slope of the tangent line to the parabola A = pi r squared at the point with coordinates r and pi r squared. Where we interpret our previous result using r for x, capital A for y, and pi for k. This isn't a coincidence, it's an instance of something quite profound in mathematics. Forming slopes of tangent lines is called differentiation. Finding areas of regions in the plane, like finding the area of a circle, is called integration. There's a remarkable connection between them described in the last module of this course called the Fundamental Theorem of Calculus. Something as innocent looking as a simple relationship between slopes of tangent lines and areas of regions in the plane turns out to be one of the most transformative observations in the history of mathematics. And was first noticed by Newton and Leibniz in the 17th century. The idea behind the calculation we just did for y=x squared generalizes to any smooth curve y=f of x. Choose some point of interest with coordinates x on the horizontal axis and f of x on the vertical axis. Now, attach a tangent line to the curve that passes through this point. Now, perturb the input x slightly by adding h to get a new input, x plus h. The output corresponding to this new input is f of x plus h marked on the vertical axis. We now join the two points on the curve to form this blue line segment, called a secant as before. In moving from the first point to the second new point the horizontal run is h and the vertical rise is f of x plus h minus f of x. So the slope of the secant is the quotient f of x plus h minus f of x divided by h. The slope of the tangent line is the limiting slope of the secant as h tends towards zero. You can see this visually as a new point slides down the curve towards the original point, the secant gets shorter and shorter. But its slope gets more and more aligned with the slope of the tangent and we think of it as actually reaching the slope of the tangent in the limit. We capture this idea using this limit notation, lim is an abbreviation for limit. And together with h arrow zero beneath the full result reads, the slope of the tangent line is the limit as h approaches zero of f of x + h minus f x over h. Whatever this limit evaluates to representing the slope of the tangent line is called the derivative of f at x. Now, this technical expression involving limits has several equivalent formulations that we'll explore in later videos. It's quite astonishing, as you'll see, that we're able to calculate and control these limits precisely with so many important functions f. Today we considered in detail the process of taking the limiting slope of a secant to find the explicit formula for the slope of the tangent line to a parabola. The underlying method generalizes to any smooth function and is the basis for formal definitions of the derivative, which will be discussed and illustrated in detail in later videos. The idea has two main ingredients, forming slopes of line segments, which are special kinds of fractions, and then seeing what happens in the limit. Because of the importance of the notion of limits we'll discuss them in more detail in the next two videos. Please read the notes and when you're ready please attempt the exercises. Thank you very much for watching and I look forward to seeing you again soon. [MUSIC]
