If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

# Introduction to average rate of change

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

## Want to join the conversation?

• This is probably a silly question, but why do you need differential calculus to find the instantaneous slope of the line? Why couldn't you just look at it like:
y = mx+b
y = t^2+1 which is the same as y = t * t + 1
t = m = slope of the line at time = t? • It's impossible to determine the instantaneous rate of change without calculus. You can approach it, but you can't just pick the average value between two points no matter how close they are to the point of interest. If you zoom in you'd see that the curve before the point of interest is different from the curve after the point of interest.
That being said, a close approximation may provide a good enough answer for your work.
• 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.
• • Why that line is called secant line? • 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.
• What relationship does a tangent line in graphs have with the tangent of a circle?How about secant lines? • This video has a mistake at the end. The d(x) for 3 is 10, not 9, and that makes the drawing more logical. • Hi! I was wondering what the ∆ symbol means and where it can be used. Thank you!
(1 vote) • At , Sal talks about slope-intercept form. Can anyone give me an explanation of what that is and how it can be used to find average rate of change.  