If we have a function \(f(x)\) and we pick two points on the curve of the function, what does the slope of a straight line connecting the two points tell us? What kind of behavior about \(f(x)\) does this slope describe?
We’re going to calculate and make some conjectures about slopes of lines between points, where the points are on the graph of a function. Let’s define the following function:
Now imagine a line that is tangent to the graph of \(f(x)\) at \(x=0\text{.}\) We are thinking of a line that touches the graph at \(x=0\text{,}\) but runs alongside of the curve at that point instead of through it.
Can you represent the slope you’re thinking of in the table above with a limit? What limit are we approximating in the slope calculations above? Set up the limit and evaluate it, confirming your conjecture.
Ok, we are going to think about this function at this point, so let’s find the coordinates of the point first. What’s the \(y\)-value on our curve at \(x=1\text{?}\)
We can investigate this definition visually. Consider the function \(f(x)\) plotted below, where we will look at the point \((-1, f(-1))\text{.}\) In the definition of the limit, we’ll let \(a=-1\text{,}\) and so consider:
Notice how repetitive this is: on one hand, we have to set up a completely different limit each time (since we’re looking at a different point on the function each time). On the other hand, you might have noticed that the work is all the same: you factor and cancel over and over. These limits are all ones that we covered in Section 1.3 First Indeterminate Forms, and so it’s no surprise that we keep using the same algebra manipulations over and over again to evaluate these limits.
Do you notice any patterns, any connections between the \(x\)-value you used for each point and the slope you calculated at that point? You might need to go back and do some more.
We’re going to try to think about the derivative as something that can be calculated in general, as well as something that can be calculated at a point. We’ll define a new way of calculating it, still a limit of slopes, that will be a bit more general.
This definition feels pretty different, but we hopefully can notice that this is really just calculating a slope. Notice in the following plot there is a significant difference. In the visualization of the Derivative at a Point, the first point was fixed into place and the second point was the one that we moved and changed. It was the one with the variable \(x\)-value.
Notice in the following visualization that the first point is the one that is moveable while the second point is defined based on the first one (and the horizontal difference between the points, \(\Delta x\)). This means that we don’t need to define one specific point, and can find the slope of the line tangent to \(f(x)\) at some changing \(x\)-value.
We define a derivative of\(f(x)\) at \(x=a\) as\(f'(x) = \displaystyle \lim_{x\to a} \left(\dfrac{f(x)-f(a)}{x-a}\right)\text{.}\) What is this limit? What does it represent in terms of\(f(x)\text{?}\) Feel free to sketch some pictures in order to discuss it visually.
For each of the following functions, use the definition of the derivative at a point to find the slope of the line tangent to the function’s curve at the specified point.
For each of the functions below, find the derivative function \(f'(x)\text{,}\) and evaluate that function at \(x=1\text{,}\)\(x=2\text{,}\)\(x=3\text{,}\) and \(x=4\text{.}\) Interpret these values.
