2Origin of calculusThe word Calculus comes from the Greek name for pebblesPebbles were used for counting and doing simple algebra…
3Google 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.”
4Google 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”
5My definitionThe branch of mathematics that attempts to “do things” with very large numbers and very small numbersFormalising the concept of veryDeveloping tools to work with very large/small numbersSolving interesting problems with these tools.
11ExamplesInstantaneous velocityboth go to 0distancetime= lim
12(the study of what happens when things get very very small) ExamplesInstantaneous velocityTHAT’S CALCULUS TOO!(the study of what happens when things get very very small)both go to 0distancetime= lim
13Examples Local slope = lim variation in F(x) variation in x both go to 0
14Important 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!
15Important 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 advancementsThe 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!)
16Plan Keep working with functions Understand limits (for very small and very large numbers)Understand the concept of continuityLearn how to find local slopes of functions (derivatives)= differential calculusLearn how to use them in many applications
17Chapter V: Limits and continuity V Chapter V: Limits and continuity V.1: An informal introduction to limits
18V.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)
19Example of limits at infinity The function can convergeThe functionconverges to a single value (1), called the limit of f.We writelimx + f(x) = 1
20Example of limits at infinity The function can convergeThe functionconverges to a single value (0), called the limit of f.We writelimx + f(x) = 0
21Example of limits at infinity The function can divergeThe function doesn’tconverge to a single value but keeps growing.It diverges.We can writelimx + f(x) = +
22Example of limits at infinity The function can divergeThe function doesn’tconverge to a single value butits amplitudekeeps growing.It diverges.
23Example of limits at infinity The function may neither converge nor diverge!
24Example of limits at infinity The function can do all this either at + infinity or - infinityThe function converges at - and diverges at + .We can writelimx + f(x) = +limx - f(x) = 0
25Example of limits at infinity The function can do all this either at + infinity or - infinityThe function converges at + and diverges at -.We can writelimx + f(x) = 0
26Calculus…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…
28V.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.
29Examples of limits at x=0 (x becomes very small!) The function can have asymptotes (it diverges). The limit at 0 doesn’t exist…
30Examples of limits at x=0 The function can have a gap! The limit at 0 doesn’t exist…
31Examples of limits at x=0 The function can behave in a complicated (exciting) way.. (the limit at 0 doesn’t exist)
32Examples 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 writelimx 0 f(x) = 0
33Limits at a pointAll these behaviours also exist when x tends to another numberRemember: 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
34Limits at a pointf(x) = 1/xg(x) = f(x-2) = 1/(x-2)2x