Reading: Tangent Lines
Example
The graph in figure 1 is the graph of y = f(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? 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). 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: [latex] m = \frac {\Delta y}{\Delta x} = \frac{(9 - 4)}{(3 - 2)} = 5 [/latex]. 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): [latex] m = \frac {\Delta y}{\Delta x} = \frac{(6.25 - 4)}{(2.5 - 2)} = \frac{2.25}{.5} = 4.5 [/latex] , 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 | y = x2 | slope of line through (x, y) 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 | y = x2 | slope of line through (x, y) and (2,4) |
3 | 9 | 5 |
2.5 | 6.25 | 4.5 |
2.01 | 4.0401 | 4.01 |
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?Solution
The tangent lines to the graph of g are horizontal (slope = 0) when x ≈ –1, 1, 2.5, and 5.Licenses & Attributions
CC licensed content, Shared previously
- Business Calculus. Provided by: Washington State Colleges Authored by: Dale Hoffman and Shana Calaway. Located at: https://docs.google.com/file/d/0B1lkHWwO61QEM0gwOFhES2N5Tlk/edit. License: CC BY: Attribution.