MATH 1141 CALCULUS NOTES
CHAPTER 1 (TOPICS):
SETS
INEQUALITIES
ABSOLUTE VALUES
FUNCTIONS
POLYNOMIALS AND RATIONAL FUNCTIONS
THE TRIGONOMETRIC FUNCTIONS
THE ELEMENTARY FUNCTIONS
DEFINED FUNCTIONS
CONTINUOUS FUNCTIONS
SETS:
- The set N (natural numbers) given by:
N={0,1, 2, 3, 4 , … }
- The set Z (integers ) given by:
N={… ,−3 ,−2,−1,0, 1,2, 3, 4 , … }
- The set Q ( rationalnumbers ) is the collection of all numbers of the form p/q, where
p and q are integers and p ≠ 0
- If A is a set of numbers and x is within that set, then:
x∈ A
- A subset is when a set exists within another. For example {0,1,2} is a subset of
{0,1,2,3}
ABSOLUTE VALUES:
- Definition of an absolute value:
- Additional rules:
TRIANGLE INEQUALITY
,FUNCTIONS:
- Dom(f): set that contains all the input values i.e. Domain
- Codom(f): set that contains all the output values i.e. Range
- Additional rules for notation of functions:
- Composite functions:
POLYNOMIALS AND RATIONAL FUNCTIONS:
- Common Rule:
TRIGONOMETRIC FUNCTIONS:
- Complementary identities:
- Pythagorean identities:
ELEMENTARY FUNCTIONS:
- Functions that are comprised of a finite amount of varying functions
IMPLICITLY DEFINED FUNCTIONS:
- Those that over certain domains and ranges are varying curves
- E.g., piecewise function
CONTINUOUS FUNCTIONS:
,- Continuous functions are those that exist without splitting or being undefined at some
value in the domain
CHAPTER 2 (TOPICS):
LIMITS OF FUNCTIONS AT INFINITY
THE DEFINITION OF nlim
→∞
f (x )
PROVING LIMITS USING THE LIMIT DEFINITION
PROOFS OF BASIC LIMIT RESULTS
LIMITS OF FUNCTIONS AT A POINT
LIMITS OF FUNCTIONS AT INFINITY:
- Rules for limits:
- Pinching Theorem
Example:
, - Limits of the form √ f ( x )−√ g(x ) :
These limits involve multiplying the numerator and denominator by the same
equation with the opposite sign in the middle
For example:
THE DEFINITION OF A FUNCTION nlim →∞
f ( x ):
- The limit involves saying that for every small positive number ϵ , there is a real
number M, such that: the distance between the function f (x) and 0 is smaller than
ϵ , for x>M
- We then replace the limit 0 with L and we say the distance between the function and
its limit is |f ( x )−L|
- This allows for a general limit as such:
“For every positive number ϵ , there is a real number M such that if x> M then
|f ( x )−L|< ϵ ”
PROVING THE LIMIT USING THE LIMIT DEFINITION:
- Using the limit definition, we can now prove limits
- A step by step guide:
You first find the distance between the function and its limit -> |f ( x )−L|
This then requires that |f ( x )−L|< ϵ
This is then rearranged in terms of x, providing the values for x which satisfy
the previous statement
By showing this domain of x values in terms of ϵ , this means that the M
value is equal to the rearranged term
Therefore if M = “ …”, then |f ( x )−L|< ϵ , whenever x >M
This completes the proof
CHAPTER 1 (TOPICS):
SETS
INEQUALITIES
ABSOLUTE VALUES
FUNCTIONS
POLYNOMIALS AND RATIONAL FUNCTIONS
THE TRIGONOMETRIC FUNCTIONS
THE ELEMENTARY FUNCTIONS
DEFINED FUNCTIONS
CONTINUOUS FUNCTIONS
SETS:
- The set N (natural numbers) given by:
N={0,1, 2, 3, 4 , … }
- The set Z (integers ) given by:
N={… ,−3 ,−2,−1,0, 1,2, 3, 4 , … }
- The set Q ( rationalnumbers ) is the collection of all numbers of the form p/q, where
p and q are integers and p ≠ 0
- If A is a set of numbers and x is within that set, then:
x∈ A
- A subset is when a set exists within another. For example {0,1,2} is a subset of
{0,1,2,3}
ABSOLUTE VALUES:
- Definition of an absolute value:
- Additional rules:
TRIANGLE INEQUALITY
,FUNCTIONS:
- Dom(f): set that contains all the input values i.e. Domain
- Codom(f): set that contains all the output values i.e. Range
- Additional rules for notation of functions:
- Composite functions:
POLYNOMIALS AND RATIONAL FUNCTIONS:
- Common Rule:
TRIGONOMETRIC FUNCTIONS:
- Complementary identities:
- Pythagorean identities:
ELEMENTARY FUNCTIONS:
- Functions that are comprised of a finite amount of varying functions
IMPLICITLY DEFINED FUNCTIONS:
- Those that over certain domains and ranges are varying curves
- E.g., piecewise function
CONTINUOUS FUNCTIONS:
,- Continuous functions are those that exist without splitting or being undefined at some
value in the domain
CHAPTER 2 (TOPICS):
LIMITS OF FUNCTIONS AT INFINITY
THE DEFINITION OF nlim
→∞
f (x )
PROVING LIMITS USING THE LIMIT DEFINITION
PROOFS OF BASIC LIMIT RESULTS
LIMITS OF FUNCTIONS AT A POINT
LIMITS OF FUNCTIONS AT INFINITY:
- Rules for limits:
- Pinching Theorem
Example:
, - Limits of the form √ f ( x )−√ g(x ) :
These limits involve multiplying the numerator and denominator by the same
equation with the opposite sign in the middle
For example:
THE DEFINITION OF A FUNCTION nlim →∞
f ( x ):
- The limit involves saying that for every small positive number ϵ , there is a real
number M, such that: the distance between the function f (x) and 0 is smaller than
ϵ , for x>M
- We then replace the limit 0 with L and we say the distance between the function and
its limit is |f ( x )−L|
- This allows for a general limit as such:
“For every positive number ϵ , there is a real number M such that if x> M then
|f ( x )−L|< ϵ ”
PROVING THE LIMIT USING THE LIMIT DEFINITION:
- Using the limit definition, we can now prove limits
- A step by step guide:
You first find the distance between the function and its limit -> |f ( x )−L|
This then requires that |f ( x )−L|< ϵ
This is then rearranged in terms of x, providing the values for x which satisfy
the previous statement
By showing this domain of x values in terms of ϵ , this means that the M
value is equal to the rearranged term
Therefore if M = “ …”, then |f ( x )−L|< ϵ , whenever x >M
This completes the proof
