what is calculus?. origin of calculus the word calculus comes from the greek name for pebbles...
TRANSCRIPT
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
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
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 both go to 0
variation in F(x) variation in x
= lim
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.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 writelimx + 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 writelimx + f(x) = 0
Example of limits at infinity
• The function can diverge
The function doesn’tconverge to a single value but keeps growing.
It diverges. We can writelimx + f(x) = +
Example of limits at infinity
• The function can diverge
The function doesn’tconverge to a single value but its amplitudekeeps 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 writelimx + 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 writelimx + 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)
x0 2