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### Course: Algebra (all content)>Unit 7

Lesson 12: Average rate of change

# Introduction to average rate of change

What's the average rate of change of a function over an interval?

## Want to join the conversation?

• While finding average of numbers,etc., we usually add up all those and divide by their count,but in here to find the average speed, we are actually taking up the slope formula.Would anyone please explain . Or am I thinking it in a wrong way?
• On a position-time graph, the slope at any particular point is the velocity at that point. This is because velocity is the rate of change of position, or change in position over time. Here, the average velocity is given as the total change in position over the time taken (in a given interval).

Using your idea of an average, to find the average velocity we'd want to measure the velocity at a bunch of (evenly spaced) points in that interval, and find the average of those. The question you might ask then would be: how many points should we take?

If we just took 2 points (the start and the end), we might get some idea of the average but this would likely be a bad representation of the true average. If the car started off stationary and ended stationary, its velocity is zero at those two points, which would suggest it's average velocity was zero - that can't be right! By taking just two points, we lost all the information about what happened between those points.

So we have to take some more points, and the more points we take, the more information we take into account, and so the closer our estimation should get to the actual answer. In fact, it seems like if we were able to take an infinite number of points we'd get the most accurate value possible. But since infinity is hard to do, let's just use a "large" number instead. So now we have two ways of finding an average velocity, Sal's way and your way. So you now may ask, what's the difference, what makes his way right and my way wrong? In fact, there is no difference, the two ways will give exactly the same answer!

An exact proof of this requires calculus or limits, but you could play around with this idea on paper or on a computer or even run some experiments to test this for yourself.
• Hi! I was wondering what the ∆ symbol means and where it can be used. Thank you!
• The symbol is the Greek letter called delta. It is commonly used as a abbreviation for "change in" something.
For example: ∆y means "change in y".
Hope this helps.
• Why that line is called secant line?
• A secant line is a line that intersects a curve of some sort, at two points. A secant line is what we use to find average rates of change.
• This video has a mistake at the end. The d(x) for 3 is 10, not 9, and that makes the drawing more logical.
• In past videos, Sal showed the slope-intercept form of an equation (y=mx+b). Could we use that to represent a function? f(x) = mx + b, m being the slope of the function?
And if so, in the function f(t) = t²+1, which is the same as saying t*t+1, why couldn't we say that the slope of that function is t?
• Yes, you could say m represents the slope in the linear function f(x)=mx+b.

The function f(t)=t^2 + 1 in your example is not linear (the graph isn’t a line). You can’t find the slope of a function that isn’t linear. There is a concept called a derivative that you’ll learn about in calculus, and it is like slope but for curves. The derivative of t^2+1 is 2t.
(Not just t, you’ll learn why in calc). This means that the slope changes depending on the values of t. For example, the slope of the curve at t=2 is 4 but at t=5 it is 10. The “slope” of lines is constant throughout the line, but the “slope” of a curve changes!
• I don't get this at all! Can anyone help?
• I don't understand this part, what does he mean by tangent line and secant line?
• A secant line intersects two points on a curve. The slope of a line secant to a curve gives the average rate of change between those points.

A tangent line just barely “kisses” a curve at a single point. The slope of the tangent line represents the instantaneous rate of change of the curve at that point. Don’t worry about this too much right now; you’ll learn more about it if you take a calculus class.
• Is the average rate of change really means"average"value of the slope?How can people just call it "average" rate of change?
• Here is my answer, I hope I have understood your question.
Slope = Rate of Change
For a straight line, the slope is the exact rate of change.
We are using the, by now familiar, concept of the slope of a function whose output is a straight line to introduce how we can think about the rate of change of a function that is not a straight line.

Using the Δy/Δx idea, we choose two points on our non-linear graph of some function `f`, and draw a straight line (a secant line) to calculate the slope of this straight line. Now this is not the exact value of the slope of the curved line, but it is a reasonable average of the rate of change of `f` between the two values we selected.

In time, you will learn how to calculate the instantaneous rate of change of a curvy graph of some function - that is, the exact slope (via a tangent line) at a point on the graph. We can do this by finding the derivative of `f` (this is calculus), and then plugging in the x value for which we we want to know the slope, and out pops the instantaneous rate of change of f at x.
Here is a sneak peek:

Example:
Let `f(x)=x²`, the derivative of `f` is `f'(x)=2x`, so the slope of the graph, when `x=3`, for our example is `f'(3)=(2)(3) = 6`. This is the instantaneous rate of change of `f` at `x=3`.

Don't worry about all this differentiation stuff right now, but do study algebra to be able to take a pre-calculus course to get into the calculus.