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Daisy, our favorite track star, is learning how to throw discus via Youtube videos. She wants to know exactly when to release the discus so that it flies off in the right direction.
She learns that, at the point where she releases, the discus flies off tangent to the same point. It flies off in a straight line relative to the point that she releases it. She's not only awesome at track, but she rocks at calculus. She dreams in tangent lines, and knows she can master this event.
We aren't all track stars, but everyone can appreciate tangents.
The tangent line to f at a is the line approached by the secant lines between a and a + h as h approaches 0.
Tangent lines and secant lines are (usually) different things.
A secant line must hit two points on a graph. A secant line between the points on the graph where x = a and x = b will hit the graph at x = a and x = b.
A tangent line occurs at one single point on a graph and has the same slope as the graph at that point.
Visually, the tangent line to f at a bounces off the graph at x = a:
A tangent line may also cross through the graph somewhere else. The important thing that makes a tangent line a tangent line is that it grazes the graph at that one special point. This line is tangent to f at x = a because it bounces off there:
However, this line is also a secant line between x = a and x = b:
A tangent line usually doesn't "cross over" the graph from one side to the other. However, it may cross over the graph at x = a in cases like this:
Here, the tangent line isn't so much "bouncing off" as it is "laying along" the graph of f.
This phenomenon has to do with something called concavity.
When looking to see if a line is tangent to f at a, we're looking to see if the line "bounces off" or "lays along" the graph of f at a.
This function has a tangent line with infinite slope at x = a:
Since the slope here is infinite, f ' (a) doesn't exist.
If we have a function f that's already a line, the tangent line to f at any point a will be f again:
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So remember, tangent lines usually bounce off the graph at a single point without crossing it. There are a few exceptions to this, but we've covered them in detail so that you won't be shocked when one those other freak tangent lines gets thrown your way.
Tangent lines don't always exist. Since the slope of the tangent line to f at a is the same thing as the derivative of f at a, if f ' (a) doesn't exist then we can't draw a tangent line to f at a.
Let f(x) = |x|. We saw earlier that f ' (0) doesn't exist because of a disagreement between one-sided limits. If we try to draw a tangent line to f at 0, we run into difficulties. Should it look like this?
Or should it look like this?
Since there's no way to decide which it should be, we can't draw the tangent line to f at 0 at all.
We can't draw a tangent line to f at a for the function f shown below either:
Again, it's because of a disagreement between the one-sided limits. If we approach a from the right, we think the tangent line should look like this:
But if we approach a from the left, we think the tangent line should look like this:
In general, we can't draw tangent lines at parts of a graph that look like "points" or "corners":
We also can't draw tangent lines at places where the function doesn't exist.
This makes sense if we think about the limit definition of the derivative. We need to use f(a) to calculate f ' (a), so if f(a) doesn't exist, we're out of luck.
The following phrases all mean the same thing:
Since f ' (a) (the derivative of f at a) is equal to the slope of the tangent line to f at a, we can determine the sign of f ' (a) by looking at the tangent line to f at a.
Consider this function:
The tangent line to f at a is sloping upwards. This means the slope of the tangent line to f at a is positive. Since the slope of the tangent line to f at a equals f ' (a), we know f ' (a) is also positive.
We can also tell the sign of f ' (a) by looking at the function f and not thinking about tangent lines.
We can also use tangent lines to compare values of f ' at different points.
Remember these pictures. We'll use them again when we discuss concavity and second derivatives.
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