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## Differential Calculus

### Course: Differential Calculus>Unit 2

Lesson 1: Average vs. instantaneous rate of change

# Secant lines & average rate of change

Understanding average rate of change and its relation to slope of a secant line.

## 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? • 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? • What is the difference between a secant line and tangent line? • can you explain what is the difference between scant line and slope?? • 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
• what is the difference between the slope of secant line and the slope of tangent line ? • 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? • av.speed is total distance covered/total time taken
but,(delta dis.)/(delta time)
is change in dis./t1-t2? • Av. speed is Δd/Δt (Δ means 'change in'), which is:
(final d. -initial d.)/(final t. -initial t.).

This is the same as saying (total d. traveled)/(total time passed)
For example, imagine you are waching a car race. If at some random instant after the race has started you start a timer (t=0) and at t=2s you see a car that's at 500m from the starting line. At t=7s you look again and the car is at 600m from the starting line.
The distance the car traveled over that interval is 600m-500m (change in distance), or 100m, and the change in time is 7s-2s, or 5s. This means over that interval the car's average speed was 100m/5s = 20m/s

Hope this helps!