Example

The graph in figure 1 is the graph of yf(x). We want to find the slope of the tangent line at the point (1, 2).

First, draw the secant line between (1, 2) and (2, −1) and compute its slope. Then draw the secant line between (1, 2) and (1.5, 1) and compute its slope. Compare the two lines you have drawn. Which would be a better approximation of the tangent line to the curve at (1, 2)?

Now draw the secant line between (1, 2) and (1.3, 1.5) and compute its slope. Is this line an even better approximation of the tangent line? Then draw your best guess for the tangent line and measure its slope. Do you see a pattern in the slopes?

Figure 1

You should have noticed that as the interval got smaller and smaller, the secant line got closer to the tangent line and its slope got closer to the slope of the tangent line. That’s good news—we know how to find the slope of a secant line.

Example

Now let’s look at the problem of finding the slope of the line L (Figure 2) which is tangent to f(x) = x2 at the point (2,4).

Figure 2

We could estimate the slope of L from the graph, but we won’t. Instead, we will use the idea that secant lines over tiny intervals approximate the tangent line.

We can see that the line through (2,4) and (3,9) on the graph of f is an approximation of the slope of the tangent line, and we can calculate that slope exactly: $m = \frac {\Delta y}{\Delta x} = \frac{(9 - 4)}{(3 - 2)} = 5$. But m = 5 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 (2,4)—the point (2,4) is fixed and it must be one of the points we use. From Figure 7, we can see that the slope of the line through the points (2,4) and (2.5,6.25) is a better approximation of the slope of the tangent line at (2,4): $m = \frac {\Delta y}{\Delta x} = \frac{(6.25 - 4)}{(2.5 - 2)} = \frac{2.25}{.5} = 4.5$ , a better estimate, but still an approximation. We can continue picking points closer and closer to (2,4) on the graph of f, and then calculating the slopes of the lines through each of these points and the point (2,4):

Points to the left of (2,4)
x yx2 slope of line through (xy) and (2,4)
1.5 2.25 3.5
1.9 3.61 3.9
1.99 3.9601 3.99
Points to the right of (2,4)
x yx2 slope of line through (xy) and (2,4)
3 9 5
2.5 6.25 4.5
2.01 4.0401 4.01

Figure 3

The only thing special about the x–values we picked is that they are numbers which are close, and very close, to x = 2. Someone else might have picked other nearby values for x. As the points we pick get closer and closer to the point (2,4) on the graph of y = x2 , the slopes of the lines through the points and (2,4) are better approximations of the slope of the tangent line, and these slopes are getting closer and closer to 4.

We can bypass much of the calculating by not picking the points one at a time: let’s look at a general point near (2,4). Define x = 2 + h so h is the increment from 2 to x (figure 3). If h is small, then x = 2 + h is close to 2 and the point (2 + h, f(2 + h)) = (2 + h, (2 + h)2) is close to (2,4). The slope m of the line through the points (2,4) and (2 + h, (2 + h)2) is a good approximation of the slope of the tangent line at the point (2,4):

$m = \frac{\Delta y}{\Delta x} = \frac{(2 + h)^2 - 4}{(2 + h) - 2} = \frac{(4 + 4h + h^2) - 4}{h} = \frac{4h + h^2}{h} = \frac{h(4+h)}{h} = 4 + h$

If h is very small, then m = 4 + h is a very good approximation to the slope of the tangent line, and m = 4 + h is very close to the value 4.

The value m = 4 + h is the slope of the secant line through the two points (2,4) and (2 + h, (2 + h)2). As h gets smaller and smaller, this slope approaches the slope of the tangent line to the graph of f at (2,4).

In some applications, we need to know where the graph of a function f(x) has horizontal tangent lines (slopes = 0). In figure 3, the slopes of the tangent lines to graph of y = f(x) are 0 when x = 2 or x ≈ 4.5 .

Example

Figure 4 is the graph of y = g(x). At what values of x does the graph of y = g(x) Does figure 4 have horizontal tangent lines?

Figure 4

Solution

The tangent lines to the graph of g are horizontal (slope = 0) when x ≈ –1, 1, 2.5, and 5.