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Slopes of Tangent & Secant Lines

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  • 00:00 Average versus Instantaneous
  • 00:52 The Secant Line
  • 1:42 Finding the Slope of…
  • 2:24 The Tangent Line
  • 4:23 Lesson Summary
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Lesson Transcript
Instructor: Kevin Newton

Kevin has edited encyclopedias, taught middle and high school history, and has a master's degree in Islamic law.

Being able to find the slope of both tangent lines and secant lines allows us to calculate the rate of change of a curve. In this lesson, we learn how to do both, as well as learning that secant lines can only provide an average whereas tangent lines can be exact.

Average versus Instantaneous

Let's say you were going to take a trip from New York City to Washington, D.C. Ignoring the eighth wonder of the world that is the gridlock on the New Jersey Turnpike, the distance is 225 miles. Typically, people schedule about four hours for the trip. However, how fast do you have to go to make that happen? I just asked a bit of a trick question. If you were to find the average rate, that would come out to about 56 miles per hour. However, I seriously doubt that you will pull out of your driveway at 56 mph. Instead, there will be times when you go a little bit faster, and there will be times when you go a little bit slower. That 56 mph was only your average speed, whereas the speed you are going in an exact moment is the instantaneous speed. Not surprisingly, we have a way to graph for both of these using secant lines and tangent lines.

The Secant Line

A secant line is defined as a straight line that touches a curve at two points. Put in a different way, the secant line is a line that represents the average rate of change. If you were to graph the average rate of change of our speed as we drove between New York and Washington, what we would end up with is the secant line. It only takes the total time and total distance into account. It does not, on the other hand, factor in delays through waiting for toll plazas in New Jersey, but it also doesn't factor in a loose approach to speeding, taken in certain other parts of the route.

Finding the Slope of the Secant Line

Chances are, you're pretty familiar with the slope formula that allows you to take the slope of a line between two points. If not, it's the difference in change of y over the difference in change of x. In other words, it's rise over run. The slope of a line created by a curve between two points is the secant line. Now, if we were to take the secant line from New York to Washington, it would indicate that for every hour in change, we moved 56 mph. Let's say we got rid of the traffic jams and only focused on the fastest hour of that drive. Our secant line is changing, but we're getting closer to an instantaneous point. A secant line can't give us the exact value, but it can get really close. That is the weakness of both the secant line and the slope formula. It can only be used over a range of points. If the curve in question is actually a straight line that's okay. After all, it comes out to a constant slope. However, your drive from New York to Washington was not at a constant speed. After all, you didn't back out of your driveway at 56 mph. So, how can we check the exact speed at an exact moment?

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