Let's talk about a concept that"s huge in calculus: the tangent line.
In this video we'll focus on three things: 1) the definition of the tangent line, 2)
how to find the tangent line for any curve, and 3) what it means for the tangent line
to be horizontal or vertical.
So what actually is the tangent line?
Well, it's a line that is tangent to a curve, meaning that it that touches the curve at
a single point and then continues in either direction with exactly the slope of the curve
at that point.
Notice that the tangent line isn't parallel to the curve itself; its slope just matches
the slope of the specific point on the curve.
One way to remember this is to think of an expression you've probably heard, "Going off
on a tangent."
That's where somebody in a conversation starts off by talking about something related to
the conversation, but then totally diverges from the topic at hand, like maybe one of
your relatives does over holiday dinners, no matter what the topic of conversation was
or what they're eating or...
Anyway, the tangent line is super helpful because it can tell you the rate of change
of a curve at a specific point on its graph, as long as you know the equation of the tangent
line.
You can find this equation a couple of different ways, depending on the information you've
been given.
The first way only requires you to know a point on the curve, and a point through which
the tangent line passes.
From there, you'll just need to use the point-slope formula to determine the equation for the
line.
This way of figuring out the tangent line doesn't require that you know the equation
of the curve or even what the curve looks like.
It could look like this, or like this.
But if you're given the equation for a function, you can use the derivative of the function
to find the slope and then the equation of the tangent line at a given point.
To do this, you'll first need to pick a value of x and determine f(x).
Let's use x=-4.
We can determine that f(-4)=2, so the point on the curve that the tangent line will touch
is (-4,2).
And now we just need to know the slope of the tangent line that'll pass through this
point.
Luckily, we know that the slope at any given point on a function is given by the derivative
of that function.
At x=-4, the derivative of the function is equal to -3.
Now we can use all this information to plug into the point-slope formula which will give
us the equation of the tangent line.
Sometimes in your calculus career, you may be asked to determine where, on a given curve,
the tangent line is either horizontal or vertical.
These special cases can give us some information about the behavior of a curve or function.
Here's a function that appears to have a horizontal tangent line pretty close to this point.
To be sure, we can look at the derivative of the function and determine where it's equal
to zero.
The derivative at a point will be equal to the slope of the tangent line at that point,
and a tangent line with slope zero, will be totally horizontal, and therefore parallel
to the x-axis.
So a function will have a horizontal tangent line anywhere where its derivative is equal
to zero.
A vertical tangent line, on the other hand aligns with a point on the curve at which
the slope is infinite, or exactly vertical.
This function, for example, appears to have a vertical tangent line here.
An infinitely steep slope means that the derivative along this line is undefined.
So to find out exactly what this line is, we need to see where the derivative of f(x)
is undefined.
We know that any function is undefined where its denominator is equal to zero, so we can
simply set the denominator of the derivative equal to zero and solve for x.
So the graph of f(x) has a vertical tangent line at x=2, which means that it's parallel
to the y-axis.
So a continuous function will have a vertical tangent line anywhere where its derivative
is undefined.
So to summarize, remember these key points:
The tangent line is the line that intersects a curve at only one point, that has the same
slope as the slope of the function at that point.
Horizontal tangent lines exist wherever the derivative is equal to zero.
And vertical tangent lines exist wherever the derivative is undefined.