Tangent Lines, Velocities, and Growth
Read this section for an introduction to connecting derivatives to quantities we can see in the real world. Work through practice problems 1-4.
The Slope of a Line Tangent to a Function at a Point
Our goal is to find a way of exactly determining the slope of the line which is tangent to a function (to the graph of the function) at a point in a way which does not require us to have the graph of the function.
Let's start with the problem of finding the slope of the line (Fig. 1) which is tangent to at the point We could estimate the slope of from the graph, but we won't. Instead, we can see that the line through
and on the graph of is an approximation of the slope of the tangent line, and we can calculate that slope exactly: . But is only
an estimate of the slope of the tangent line and not a very good estimate. It's too big. We can get a better estimate by picking a second point on the graph of f which is closer to the point is fixed and it must be one of the
points we use. From Fig. 2, we can see that the slope of the line through the points and is a better approximation of the slope of the tangent line at , a better estimate, but still an approximation. We can continue picking points closer and closer to on the graph of , and then calculating the slopes of the lines through each of these points and the point :
Points to the left of (2,4) | Points to the right of (2,4) | |
The only thing special about the x-values we picked is that they are numbers which are close, and very close, to . Someone else might have picked other nearby values for As the points we pick get closer and closer to the point on the graph of , the slopes of the lines through the points and are better approximations of the slope of the tangent line, and these slopes are getting closer and closer to
Practice 1: What is the slope of the line through and for and ?
We can bypass much of the calculating by not picking the points one at a time: let's look at a general point near . Define so is the increment from 2 to (Fig. 3). If is small, then is close to 2 and the point is close to . The slope of the line through the points and is a good approximation of the slope of the tangent line at the point :
If is very small, then is a very good approximation to the slope of the tangent line, and is very close to the value The value is called the slope of the secant line through the two points and The limiting value 4 of as gets smaller and smaller is called the slope of the tangent line to the graph of at .
Example 1: Find the slope of the line tangent to at the point by evaluating the slope of the secant line through and and then determining what happens as gets very small (Fig. 4).
Solution: The slope of the secant line through the points and is
As gets very small, the value of approaches the value 2, the slope of tangent line at the point .
Practice 2: Find the slope of the line tangent to the graph of at the point by finding the slope of the secant line, , through the points and and then determining what happens to as gets very small.