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## AP®︎/College Calculus AB

### Course: AP®︎/College Calculus AB>Unit 2

Lesson 1: Defining average and instantaneous rates of change at a point

# Secant lines & average rate of change

Learn how to calculate the average rate of change for a function and its connection to the slope of a secant line. Grasp the concept of instantaneous rate of change and its significance in calculus, leading to the idea of the derivative.

## Want to join the conversation?

• At , if the secant line isn't the exact instantaneous rate of change, what's its purpose? It seems like it's just a crude version of the tangent line. If the tangent line is what's actually important in calculating derivatives, etc., why do we bother with the secant? I understand that the concept of the tangent line was probably derived from the secant line, but why not just use the tangent line and not bother with the secant?
• Calculating average change using secant lines is actually an important intermediate step to finding instantaneous change via tangent lines (and thus also derivatives). As you say, the process of finding the slope of the tangent line descends from the process of finding the slope of a secant line (the only difference is that a certain limit is taken of the difference quotient which is the expression for the slope of a general secant). As for "not bothering with the secant", there is no way to explain the process of finding instantaneous change without explaining how to find average change for the reason you just identified: the principles involved in finding average rate of change are part of finding instantaneous rate of change.

Also, average rates of change have advantages in their own right. In particular, in physics, there are a lot of phenomena that occur that have to deal with average rates of change instead of instantaneous rates of change. Furthermore, if you are looking at discrete data (as is the case in every real world observation), there is no way to get an instantaneous rate of change from that data because it is not continuous.

Lastly, "not having a purpose" (which is not the case with secant lines and average rates of change) is a poor argument for neglecting to study anything – especially in mathematics. There are plenty of things in mathematics that have no real world application, though they are studied nonetheless.

Comment if you have questions!
• What's the best way to memorize the difference between a tangent and a secant line?
• Secant is intersection at 2 points of a curve. Tangent is at 1 ponit.
• I love the way you build the idea and the general concept over multiple videos
• because the secant line is basically the hypotenuse of a triangle, could a^2 + b^2 = c^2 work? I tried it myself and got the answer 8.24.. which does not match sal's slope of 4. Am I missing something?
• The Pythagorean theorem involves the length of the hypotenuse, not the slope. We're approximating the slope of the function, so we don't care about the length of the secant line.
• What is the difference between a secant line and tangent line?
• A secant line represents the rate of change over a 'longer' interval.(The average rate of change) A tangent line represents the rate of change over an infinitesimally 'short' interval.(The instantaneous rate of change)
• Can anyone explain how the terms 'tangent' and 'secant' lines used here are connected to those which I learnt in trigonometry?
[In short; Why are they called tangent and secant lines?]
• Secant line is a line that touches a curve at two points, pretty much the average rate of change because it is the rate of change between two points on a curve (x1,y1), (x2,y2) the average rate of change is = (y2-y1)/(x2-x1) which is the slope of the secant line between the two points on the curve.
The tangent line is a line that touches a curve at one point, this line's slope at a point is the derivative in a sense the limit as the change in x between two points of a secant line approach 0. its slope is the derivative of the curve at the point.
Hope that helps
• can you explain what is the difference between scant line and slope??
• Say you have a curve, and this curve has two points on it. The line between those two points is called a secant line.
The slope is the m in the equation for any line, y=mx+b. The slope describes whether the line is going down or up on the graph, and how quickly it is doing so.
While a secant line has a slope, the two are otherwise rather unrelated.
• what is the difference between the slope of secant line and the slope of tangent line ?
• Basically, both are slopes, except secant and tangent lines are totally different. However, the method of finding the slop for each line remains the same. Secant line is one that intersects two points in a line, whereas tangent line intersects exactly one point on the curve.
• The slope of the secant line was the same as the derivative at 2. Are there other functions that have this trait where the average rate of change is the same as the rate of change in the center of the interval?
• For any function continuous on [a, b] and differentiable on (a, b), there is some point inside the interval where the derivative is equal to the slope of the secant line between a and b.

This is the Mean Value Theorem, which is discussed elsewhere on Khan Academy. This point only occurs at the midpoint of the interval in certain specific cases, and this is largely coincidental.
• Find the slope of the secant line between x=−2 and x=3 on the graph of the function f(x)=−2x2−5x+1.

how do you do problems like this?