What is Calculus?

1. What is Calculus?

2. Origin of calculus • The word Calculus comes from the Greek name for pebbles • Pebbles were used for counting and doing simple algebra…

3. Google answer • “A method of computation or calculation in a special notation (like logic or symbolic logic). (You'll see this at the end of high school or in college.)” • “The hard deposit of mineralized plaque that forms on the crown and/or root of the tooth. Also referred to as tartar.”

4. Google answers • “The branch of mathematics involving derivatives and integrals.” • “The branch of mathematics that is concerned with limits and with the differentiation and integration of functions”

5. My definition • The branch of mathematics that attempts to “do things” with very large numbers and very small numbers • Formalising the concept of very • Developing tools to work with very large/small numbers • Solving interesting problems with these tools.

6. Examples • Limits of sequences: lim an = a n 

7. Examples • Limits of sequences: lim an = a THAT’S CALCULUS! (the study of what happens when n gets very very large) n 

8. Examples • Instantaneous velocity

9. Examples • Instantaneous velocity

10. Examples • Instantaneous velocity

11. distance time = lim both go to 0 Examples • Instantaneous velocity

12. distance time = lim both go to 0 Examples • Instantaneous velocity THAT’S CALCULUS TOO! (the study of what happens when things get very very small)

13. Examples • Local slope = lim variation in F(x) variation in x both go to 0

14. Important new concepts! • So far, we have always dealt with actual numbers (variables) • Example: f(x) = x2 + 1 is a rule for taking actual values of x, and getting out actual values f(x). • Now we want to create a mathematical formalism to manipulate functions when x is no longer a number, but a concept of something very large, or very small!

15. Important new concepts! • Leibnitz, followed by Newton (end of 17th century), created calculus to do that and much much more. • Mathematical revolution! New notations and new tools facilitated further mathematical developments enormously. • Similar advancements • The invention of the “0” (India, sometimes in 7th century) • The invention of negative numbers (same, invented for banking purposes) • The invention of arithmetic symbols (+, -, x, = …) is very recent (from 16th century!)

16. Plan • Keep working with functions • Understand limits (for very small and very large numbers) • Understand the concept of continuity • Learn how to find local slopes of functions (derivatives) = differential calculus • Learn how to use them in many applications

17. Chapter V: Limits and continuityV.1: An informal introduction to limits

18. V.1.1: Introduction to limits at infinity. • Similar concept to limits of sequences at infinity: what happens to a function f(x) when x becomes very large. • This time, x can be either positive or negative so the limit is at both + infinity and - infinity: • lim x  + f(x) • limx  - f(x)

19. Example of limits at infinity • The function can converge The function converges to a single value (1), called the limit of f. We write limx + f(x) = 1

20. Example of limits at infinity • The function can converge The function converges to a single value (0), called the limit of f. We write limx + f(x) = 0

21. Example of limits at infinity • The function can diverge The function doesn’t converge to a single value but keeps growing. It diverges. We can write limx + f(x) = +

22. Example of limits at infinity • The function can diverge The function doesn’t converge to a single value but its amplitude keeps growing. It diverges.

23. Example of limits at infinity • The function may neither converge nor diverge!

24. Example of limits at infinity • The function can do all this either at + infinity or - infinity The function converges at - and diverges at + . We can write limx + f(x) = + limx - f(x) = 0

25. Example of limits at infinity • The function can do all this either at + infinity or - infinity The function converges at + and diverges at -. We can write limx + f(x) = 0

26. Calculus… • Helps us understand what happens to a function when x is very large (either positive or negative) • Will give us tools to study this without having to plot the function f(x) for all x! • So we don’t fall into traps…

28. V.1.2: Introduction to limits at a point • Limit of a function at a point: New concept! • What happens to a function f(x) when x tends to a specific value. • Be careful! A specific value can be approached from both sides so we have a limit from the left, and a limit from the right.

29. Examples of limits at x=0 (x becomes very small!) • The function can have asymptotes (it diverges). The limit at 0 doesn’t exist…

30. Examples of limits at x=0 • The function can have a gap! The limit at 0 doesn’t exist…

31. Examples of limits at x=0 • The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)

32. Examples of limits at x=0 • But most functions at most points behave in a simple (boring) way. The function has a limit when x tends to 0 and that limit is 0. We write limx  0 f(x) = 0

33. Limits at a point • All these behaviours also exist when x tends to another number • Remember: if g(x) = f(x-c) then the graph of g is the same as the graph of f but shifted right by an amount c

34. Limits at a point f(x) = 1/x g(x) = f(x-2) = 1/(x-2) 0 2 x