Estimating the Slope of Functions

Instructor: Laura Pennington

Laura received her Master's degree in Pure Mathematics from Michigan State University. She has 15 years of experience teaching collegiate mathematics at various institutions.

This lesson will review the definition of the slope of a line, and then go on to talk about how to estimate the slope of a linear function. Once we are comfortable with this process, we will look at using tangent lines to estimate the slopes of non-linear functions at a given point.

Estimating the Slope of a Linear Function

Let's say you got a kitten named Euler four years ago, and you've graphed Euler's weight since his birth. Let's consider the first three months.


estslopfunc1


Notice that because Euler's weight grew at a constant rate over the first three months, the graph of his weight with respect to time is a graph of a linear function (or a line). Furthermore, the rate at which the weight has grown is called the slope of a line.

In general, the slope of a line is the rate at which y is changing with respect to x. In this scenario, the slope is the rate at which Euler's weight is changing with respect to time in months.

We have a nice formula for finding the slope of a line if we have two points on the line, (x1,y1) and (x2,y2). That is as follows:

  • Slope = (y2 - y1) / (x2 - x1)

Therefore, when we have a graph of a line, we can estimate its slope in two easy steps:

  1. Find two points that appear to fall on the line.
  2. Use our slope formula.

For instance, take a look at the graph of Euler's weight again. It appears as though the two points (1,2) and (2,4) fall on the line (these don't have to be exact as this is an estimate), so we can estimate the slope by using our formula with these two points.


estslopfunc2


We see that the slope of the line through the estimated points (1,2) and (2,4) is 2. This tells us that in the first three months, Euler's weight grew at a constant rate of about 2 pounds per month.

Estimating the Slope of a Non-Linear Function

Now suppose we extend the graph of Euler's weight out to the present.


estslopfunc3


Notice that the graph is no longer the graph of a line, so it is a non-linear function that now takes on the pattern of growing quickly at first, but then much more slowly as time goes on. This graph represents a logarithmic function, and the slope isn't as simple as the slope of a line. However, we can still estimate the slope of this logarithmic function (or any function, for that matter) at a given point by using tangent lines.

The tangent line of a graph, at a given point, is the line that touches the line exactly at that point and does not cross over the graph. For example, consider the line that is tangent to Euler's graph at x = 12.


estslopfunc4


You may already be putting this together in your mind, but notice that the tangent line has a slope that is equal to the slope of the graph at the point where the tangent line touches the graph. Therefore, to estimate the slope of a non-linear function at a given point, we just need to estimate the slope of the tangent line at that point. This leads to the following steps to estimating the slope of a non-linear function at a given point, (x,y):

  1. Sketch in the tangent line to the graph at the point (x,y).
  2. Estimate the slope of the tangent line. This is the estimated slope of your function at the point (x,y).

Let's give this a try. Suppose we want to know how Euler's weight was changing at one year, or 12 months. To do this, we first sketch in the tangent line of the graph where x = 12. This is done for us in our graph of a tangent line.

Now, we just need to estimate the slope of the tangent line. It looks as though the line passes through points (8,9) and (12,10). All we have to do is plug this into our slope formula!


estslopfunc5


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