What is Calculus?

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

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.”

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”

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.

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

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

Examples Instantaneous velocity

Examples Instantaneous velocity

Examples Instantaneous velocity

Examples Instantaneous velocity both go to 0 distance time = lim

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

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

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!

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!)

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

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

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)

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

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

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) = +

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.

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

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

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

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…

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.

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

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

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

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

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

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