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## AP®︎ Calculus AB (2017 edition)

### Course: AP®︎ Calculus AB (2017 edition)>Unit 2

Lesson 1: The derivative: An intuitive introduction

# Newton, Leibniz, and Usain Bolt

AP.CALC:
CHA‑2 (EU)
,
CHA‑2.A (LO)
,
CHA‑2.A.1 (EK)
,
CHA‑2.B (LO)
,
CHA‑2.B.1 (EK)
Why we study differential calculus. Created by Sal Khan.

## Want to join the conversation?

• why did they name this subject calculus? •   It's from the Latin meaning reckon or account (like calculate), which came from using pebbles (calx) to count with.
• People say that we see math in our everyday lives -- and while I understand how this concept applies to beginning math, pre-algebra, algebra, and trig, how does this apply to calculus? At , Sal says that differential calculus is all about finding instantaneous rate of change, but is that the only "everyday use"? Or is calculus simply a concept that is used in other subjects, or even professions, like engineering?

Thanks! •   I think the best way to find a good answer to this question is to just keep watching the videos! If you attend college for any engineering discipline, you have to learn calculus before you even begin learning the specifics of your discipline (whether it be mechanical, electrical, civil, computer, computer science, etc...). The best way to understand what "every day" things calculus will enable you to do is to learn calculus and start doing incredible things every day :-)
• Who invented calculus? •   Lots of people were involved in the development. Archimedes approached the idea in his iterative computation of the value of pi, but he was hampered by a poor numeric system and lack of algebraic notation. Ultimately, Leibniz and Newton should get the credit for actually formulating it, and we still use the Leibniz notation of dy/dx for most purposes.
• what exactly is a derivative? •  What a great question......A derivative calculates the slope of a line.
Well then,, why not just call it the slope?
But what if the line is curving? Crap. Now the slope keeps changing.
So draw a curve,,,any curve,,,,,,,,,,,,,now draw a tangent line at any point on that curve . THE DERIVATIVE FUNCTION WILL CALCULATE THE SLOPE AT ANY POINT ON THE CURVE.
• why is it called "differential" calculus •  The name comes from the fact that we're dealing with rates of change, which are calculated by dividing the difference in one quantity by the difference in another quantity (for example, speed is the difference in position divided by the difference in time).
• what is the secant and tangent line? • So calculus is all about finding instantaneous rate of change, or is there a better, simpler way of describing it. • That is essentially what differential calculus is all about. You will later learn that there are actually two primary types of calculus: differential and integral. Differential calculus focuses on, as you pointed out, the instantaneous rate of change of a function at any point on that function. This is necessary because, as Sal pointed out in the video, the rate of change of a function can vary wildly on any given interval. The other type, integral calculus, focuses on finding the area underneath a curve (for example, finding the area underneath a standard y=x^2 parabola on the interval x=1 and x=4). While these two may seem like completely separate topics, you would be surprised by how closely related the two are, and how you can apply each in a variety of situations.
• Was it Newton or Leibniz, who invented the form of notation that we use today in calculus? • We actually use a mixture of notations, though mostly Leibniz and Lagrange.

For differential calculus:
The notation with the lowercase letter d is from Leibniz.
The notation involving the primes as in f'(x), is from Lagrange.
And there are still some other notations by a variety of mathematicians, mostly for more advanced calculus.
Newton's notion uses dots placed over the variable. I've never seen anyone use that notation other than to say "this is Newton's notation", but I suppose there must be somebody, somewhere that uses it.

For Integral Calculus, the integral symbol was invented by Leibniz, but the notation for the definite integral, based on Leibniz's `∫` symbol was invented by Fourier.
• So, was calculus originally invented to find out the instantaneous speed of an object? • Calculus for the most part was rooted in the ancient obsession with trying to square the circle (reduce the area of a circle to a simple x*y formula) as well as computing areas and tangents to other curves. Since there are a lot of curves in position-speed-acceleration calculations, calculus proves an extremely valuable tool with which to analyze and predict behavior but it wasn't really the precipitating factor in its creation, for one main reason that there really wasn't the ability to precisely measure time on small scales.

For a brief introduction:

http://www-groups.dcs.st-and.ac.uk/~history/HistTopics/The_rise_of_calculus.html

And if you are further curious:

http://www.amazon.com/Story-Number-Princeton-Science-Library/dp/0691141347

Although this book is specifically centered about the evolution of "e", the author does a great job of tying the discovery of logarithm properties to modern calculus methods by building up the history of calculus from its earliest ancient dabbling to its explosion in the 17th and 18th centuries.

Here’s a great video series for the history of mathematics, from Pythagoras to calculus and beyond: 