Continuity, Differentiability, and Integrability
The derivative of a function is the rate of change of that function. It shows how fast or how slow the function is changing. This can be useful in determining things such as instantaneous rates of change, velocity, acceleration and maximum profits. A good way to explain the concept of a derivative is to do it graphically. To illustrate, think of a drag car race. The track is only ¼ of a mile long, or 1320 feet. The dragster crosses the finish line in six seconds. How fast was the dragster going when it crossed the finish line? The dragster traveled 1320 feet in 6 seconds, so the average speed of the dragster is 1320 divided by 6 which equals 220 feet per second, or 150 miles per hour. The following graph represents the dragster’s position function as the red curve. The position function for the dragster is 36 2/3 x^2. The green line is the secant line connecting the dragster’s starting point and end point. The slope of this secant line is the average speed of the dragster, 220 feet per second, or 150 miles per hour.
This only tells us the average rate of speed. To find the exact speed the car was going when it crossed the finished line, approximate it by finding the slope of the line between the point we want, in this case, (6, 1320), and a point closer to it along the curve. As we move the secant line closer and closer, the slope changes and gets closer and closer to the instantaneous speed. By doing this, mathematicians discovered that the slope of the line tangent to the point is the instantaneous rate of change. So to find the speed of the dragster as it crossed the finished line, we can use the slope of the line tangent to the point of the finish line. The blue line in the following graph represents this tangent line.
The derivative of our position function will give us the slope of this line. The position function is 36 2/3 x^2 so the derivative is73 1/3 x. The slope of the blue tangent line is 440 feet per second, or 300 miles per hour. This is the exact speed that the dragster was traveling at as it crossed the finished line.
For a function to be differentiable, its graph needs to be smooth and continuous. How is a function continuous? The mathematical definition of continuity at a point as defined in Thomas' Calculus is: "Interior point: A function y=f(x) is continuous at an interior point c of its domain if lim┬(x→c)〖f(x)=f(c).〗 Endpoint: A function y=f(x) is continuous at a left endpoint a or is continuous at a right endpoint b of its domain if lim┬(x→a^+ )〖f(x)=f(a)〗 or lim┬(x→b^- )〖f(x)=f(b),〗 respectively (Weir, Hass, & Giordano, 2008, p. 120)." Thomas’ Calculus goes further to define a test for continuity: “Continuity Test: A function f(x) is continuous at an interior point of its domain x=c if and only if it meets the following three conditions:
f(c) exists (c lies in the domain of f)
lim┬(x→c)〖f(x)〗 exists (f...