Derivatives


Uploaded by TheIntegralCALC on 26.04.2012

Transcript:
Last time, in our second section, we talked about limits and continuity, which gave us
a great foundation for today’s topic, derivatives. We’ll be wrapping our heads around the first
of two fundamentally important questions in calculus: how to find the rate of change of
a function at a point.
Before we go further, let’s talk about what we mean by the rate of change of a function
at a point. We all know how to find the rate of change of a line; it’s just the slope
of the line. For example, picture the graph of 2x+2. We can tell just by looking at the
equation that the slope of the graph is 2. And it doesn’t matter where we are on the
graph of the function, the slope is 2 everywhere.
But what about this graph? What is the slope, or rate of change of this function at this
point, or this one? What if we need to know the slope at both of these points? How can
we find the slope of a curved surface? Well, we don’t, really. We’ll actually find
the equation of the line tangent to the curve, that passes through that point on the graph
because remember, we already know how to find the slope of a line.
Let me take a half a step back, and talk for a brief moment about something I just said:
the line tangent to the curve. What does it mean for a line to be tangent to a curve?
It means that the line just barely skims the graph, intersecting it at exactly one point.
Think of the tangent line in contrast to a secant line, which cuts through the graph
at one or more points.
Now imagine a secant line, intersecting the graph at two points. As I move this second
point closer to the tangent point, the distance between the points of intersection becomes
smaller and smaller and the secant and tangent lines become more and more similar. Eventually,
the distance between the points will become infinitely small, until eventually the two
points come together at a single point, and the line no longer intersects the function
at two points, but instead only at one, and this is the tangent line to the curve at that
point.
So in order to calculate the slope of the function at a point, we need to find the slope
of the tangent line, and therefore obviously the equation of the tangent line. How can
we do this? Well, recall from algebra that to find the equation of a line, we either
need two points, or a point and a slope. We know that the slope is the very thing we’re
trying to solve for, so we’re going to have to calculate the equation of the line using
two points. We already have one point; it’s the point where the function and the tangent
line intersect each other. So we just need one more point.
Which point can we pick? Well, since we currently only have one equation on the table, the equation
of the function itself, we can only pick points on the graph of the function. So, let’s
pick another point on the graph; one that’s relatively close to our tangent point. Using
these two points, we can find the equation of the secant line that connects the two,
and call that an estimate of the equation of the tangent line. But of course, an estimate
isn’t good enough. We want the exact equation of the tangent line.
To get a better estimate, we can move this second point closer and closer to the tangent
point. The closer we move the points together, the more identical the secant and tangent
lines become. You might be starting to see by now that, if we eventually decrease the
distance between these points all the way to zero, we’ll have only one point, and
that will be the tangent point.
To model this mathematically, we’ll have to borrow an equation we learned way back
in algebra, for the average rate of change of a function between two points. (a,f(a))
is our tangent point, and (b,f(b)) is our second point. Using these two points, we can
calculate the average rate of change of the function between them. Remember though that
we’re trying to find the instantaneous rate of change at the tangent point. If we start
moving (b,f(b)) closer and closer to (a,f(a)), and we continue calculating the average rate
of change between the closer and closer points, we’ll start to see the average rate of change
approach a specific value. The value it’s approaching is the instantaneous rate of change,
also known as the slope of the tangent line, which is the slope of the function at the
tangent point.
This brings us to the definition of the derivative. Let’s modify slightly the formula for average
rate of change. Instead of using a and f(a) we’ll use x and f(x), and we’ll call the
distance between the tangent point and the secant point h.
Now we can simplify the function by canceling the x’s in the denominator. Then, thinking
again about the tangent line and borrowing what we learned last time about limits, we
can take the limit as h goes to zero, and this will give us the slope of the tangent
line.
In other words, as we bring those two points closer and closer together, h, which is the
distance between the points, gets smaller and smaller, until eventually it becomes 0,
and then we have the slope of the tangent line, or the instantaneous rate of change
of the function at the tangent point.
Before we wrap up today, let’s take a look at one example. The problem says we need to
use the definition of the derivative to find the instantaneous rate of change of the function
f(x)=x/(1-2x) when x=1.
You can see that the definition has more or less two components: f(x+h) and f(x). We already
have f(x); it’s our original function. To calculate f(x+h), we’ll take all of the
x’s out of our original equation, and put an (x+h) in their place. Now we can take f(x)
and f(x+h) and put them both into the definition.
Now we need to start simplifying. The simplification process will be different depending on the
original function. In this case, we have to find a common denominator in the numerator
of our function. Eventually, once we’ve fully simplified, we’ll take the limit as
h goes to zero. In other words, we’ll plug in 0 for h, and any term involving h will
disappear from our derivative function. What we’re left with is the derivative of the
original function, which we denote as f prime of x.
To find the slope of the function when x=1, we just plug 1 into the derivative function,
and we see that the slope of the function is equal to 1, or put another way, the instantaneous
rate of change of the function at x=1 is 1.
Next time we’re going to expand on what we’ve learned about the concept and definition
of the derivative to talk about some better techniques we can use to calculate derivatives.
I’ll see you then.