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### Course: Get ready for AP® Calculus>Unit 2

Lesson 4: 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!
• Is it possible to find the rate of change (as a formula, linear or polynomial) for literally every changing thing in the universe? What do you guys think?
• Not sure about 'everything', but differentiation in Calculus is about getting the exact rate of change of a function at any given point!

The average rate of change takes two points and calculates `(y2 - y1)/(x2 - x1)`. It's not really accurate for the actual rate of change at the first point, though. So by bringing the second point closer and closer to the first point, we get closer and closer to the actual rate of change at that first point. There's more to it, but this is the basic idea that underlies differential calculus.
- in the video, Sal shows this visually, where the slope of the tangent line to any point on the graph is the 'instantaneous rate of change'. A tangent line touches the graph locally at only `1` point, not `2`, which reflects on this idea of moving the 2nd point closer and closer to the 1st point, to make it closer to a tangent line.

With differentiation, you can start with a function `f` and then obtain a function which describes the exact rate of change of `f` at any point on its graph, called the derivative of `f`.

It is possible to find the derivative of many types of elementary functions, including polynomials, exponentials, trigonometric functions, and so on... It's all in the Calculus course! https://www.khanacademy.org/math/ap-calculus-ab
• for d(t) = t^2+1, if t is 1, d(t) is 1+1 so its 2, not 1/1, isnt it?
• Sal is not finding d(t) when t=1. He is finding the average rate of change on the graph d(t) using the 2 points (0,1) and (1,2). The average rate of change is finding the slope between those 2 points.
Slope = Change of Y/Change in X = (y2-y1)/(x2-x1). In this case, it becomes Change of D/Change of t. Sal didn't write out the work, but he walks you thru it verbally.
• I don't get this at all! Can anyone help?