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### Course: Calculus, all content (2017 edition)>Unit 2

Lesson 3: Derivative as instantaneous rate of change

# Approximating instantaneous rate of change with average rate of change

Sal approximates the instantaneous velocity of a motorcyclist. Created by Sal Khan.

## Want to join the conversation?

• When does creating a tangent line from the smooth curve become applicable in life?
• Consider this . . . .
When does skipping a rope become applicable to a boxer's life?
They never jump rope in the ring.

When does running through tires become applicable to a football player's life?
There are no tires on a football field.

When does a c major scale become applicable in a musician's life?
They never give a concert featuring the c major scale.

The answer to the purpose of what you are asking is the same answer for the others, it is training for what is to come, and a skill that will yield benefits later even though you may not be able to appreciate (or even in some cases, understand) it at the moment.

Most of the math things you will do in "life" will not be as easy as graphing a tangent line, you will need much more sophisticated math, but to understand that math, some of it depends on building a foundation of what a tangent line actually means.

In the end, it is possible you may have a career that, from your point of view as a student now, seems to be math free. But learning math is more than just numbers and tangent lines; it is an ordered system of logic that helps you be a better thinker, no matter what you choose to do. Being a better thinker will help you in all areas of your life, from managing your income and expenses, to being able to see logical holes in ideas so that people can't take advantage of you; both very important life skills.
• At Sal says we can get the slope of the tangent line @ t=2 seconds﻿ by averaging the "slopes of the tangent lines between 1.5 and 2 and 2 and 2.5" ... Aren't they secant lines, i.e. the﻿ slope of a line between two points on a curve?
• We don't actually have a curve in this case, just a list of data points, but conceptually they're points on a curve and of course there's no such thing as a tangent between two points on a curve, so Sal misspoke when he referred to them as tangents. He meant to say secant lines, or just lines.
• Why does simply taking away 18.7 from 46.1 give the same answer of 27.4?
• Its not a coincidence. Think about all the steps that went in to finding that 27.4. First was finding the average velocity between those two intervals. Then, you averaged those two to get 27.4. Plugging everything in ...

27.4 = (((30.2 - 18.7) / 0.5) + ((46.1 - 30.2) / 0.5))/2.

The two terms that are divided by 0.5 are the first part of calculating the average velocity. Then you add these together and divide by 2 to average them.
The 2 cancels with the 0.5's, so you have 27.4 = 30.2 - 18.7 + 46.1 - 30.2,
or, 27.4 = 46.1 - 18.7

(Its helpful if you write it out on paper as its hard to format things here)
• I s it possible to further simplify the final equation? While working the problem before I finished the video I ended at (s=27.4t-24.6) Since "s" is the f(t) isn't the equation now in slope-intercept form, or have I made some mathematical gaff?
• Simplifying the final answer is not as important in calculus the way it is in algebra. We are more interested in having the answer in a useful form.

But, yes, I think that is the correct way to put that tangent line in slope-intercept form.
• What did the question mean by saying t is an independent varible? How di it affect the answer?
(1 vote)
• t changes on its own, what else does it change? It changes S as well. Because S is dependent on t. In other words, a function of t. So when we differentiate it with respect to t, it would be how S changes when t changes.
• The general question is why the 'secant' line rather than the 'tangent' line?
(1 vote)
• A secant line is just the line between two points on a graph. A tangent line is a line that touches exactly one point on the graph. We can express tangent lines in calculus by saying that they are secant lines that have two points that get infinitely close together, such that they look like exactly one point.
• So what form is y = mx+c called?
(1 vote)
• That is slope intercept form. Depending on the country you are from, the letters used for slope and intercept can change - different letters, but the concept is the same.
• at Sal says to take the Tagent we use the average of [1.5, 2] and [2, 2.5]. why don't we take the average of the [1.5, 2] and [1, 1.5]?
(1 vote)
• I believe it is because we want the line tangent to t=2, if you notice t=2 is between [1.5,2] and [2,2.5]. If you took the average of [1, 1.5] and [1.5, 2] you would get the line tangent to t= 1.5.