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
n 
THAT’S CALCULUS!
(the study of what happens when n gets very
very large)
Examples
• Instantaneous velocity
Examples
• Instantaneous velocity
Examples
• Instantaneous velocity
Examples
• Instantaneous velocity = lim
distance
both go to 0 time
Examples
• Instantaneous velocity = lim
distance
both go to 0 time
THAT’S CALCULUS TOO!
(the study of what happens when things
get very very small)
Examples
• Local slope = lim
variation in F(x)
both go to 0 variation in x
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 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)
0
2
x