Notes, Lesson 2.6
Tangents, Velocities, and Other Rates of Change

In this lesson we take the slopes of tangent lines and secant lines which we studied in lesson 2.1, and our newly acquired limit skills and we develop the formal definition for the slope of a tangent line at any instance. This is then how to find a derivative.

 Once again, we examine a generic function, . We are trying to find the slope of a tangent line to the function at the point . We choose another point so that we can have a secant line (green) to begin with. This second point is h units to the right of . Slope of secant line: Using the slope formula and simplification. Slope of tangent line: To make a tangent line out of a secant line, we must shrink the h value, and take its limit as it approaches zero. This is called a Difference Quotient.

This taking of a limit as h approaches zero is illustrated in animated form below:

Let's now do an example problem.
We'll find the slope of the tangent line to the curve  at the exact instance when x=7.

 The point where we are to find the slope of the tangent line is (7, 31). This is found by substituting 7 into the function in place of x. In our problem, a = 7, so we substitute this into our tangent formula. According the the function defined, here is what we are to do. Using our algebra skills, we get this. We gather like terms and this is the result. Notice that we still cannot take the limit. Because we would get , we know that there is a limit, but that we will have to further algebraically simplify first. Finally, we can substitute and find the limit. We have found that the exact slope of the tangent line at x = 7 is 10!

Even better, let's find the exact slope of the tangent line to the function  at any point. Here we wil be finding our first derivative!

 Becasue we do not know any specific point a to find the slope of the tangent line at, we leave our independent variable, x in place of a. We follow the given function to define  and . Using our algebra skills, we get this. We gather like terms and this is the result. Notice that we still cannot take the limit. Because we would get , we know that there is a limit, but that we will have to further algebraically simplify first. We have found the formula for the exact slope of the tangent line at any point! We can now calculate the slope of the tangent line at any instant! This is a derivative! There are two notations that are commonly used for indicating a derivative:  (1) The first is the dy/dx notation. This can be interpreted as "the derivative of y with respect to x." (2) The second is the f prime notation. The single prime mark (') indicates the first derivative of the function f.

Let's check our work from above, when we were finding the slope of the tangent line to the curve  at the exact instance when x=7.

 Using our new derivative notation, we begin at our newly found derivative. Because this formula represents the slope of the tangent at any instant, we simply substitute the value of x that we want to find the derivative at (in this case 7). We find that the slope of the tangent line at x = 7 is 10. This agrees with our previous work.

 If you have the Journey Through Calculus CD, load and run Resources/Module 3/Derivative at a Point/The Falling Robot.. This module will help you learn about average and instantaneous velocity by comparing falling objects..

 Check Concepts

 #1: True or False: Without limits we could not find a derivative. Choose One True False #2: True or False: Derivatives give us a new tools to explore the shape of graphs. Choose One True False #3: When you find a derivative you have really found ________________. Choose One The slope of the curve at one instant A formula for the slope at any instant The slope of a secant line #4: If your function represents distance plotted agains time, then the derivative gives a formula for the ____________ at any instant. Choose One Distance Velocity DAcceleration Jerk #5 Another way to define a derivative is that it is .... Choose One Instantaneous rate of change Average rate of change Instantaneous amount of changeAverage amount of change