Lecture notes
CALCULUS_NOTES_FINAL _COMPLETE CHAPTER 2024
- Module
- Maths
- Institution
- PEARSON (PEARSON)
Calculus Complete Notes
[Show more]
1 review
By: Bruce5759wayne • 4 months ago
Content preview
MATH 1141 CALCULUS NOTESCHAPTER 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
The benefits of buying summaries with Stuvia:
Guaranteed quality through customer reviews
Stuvia customers have reviewed more than 700,000 summaries. This how you know that you are buying the best documents.
Quick and easy check-out
You can quickly pay through credit card for the summaries. There is no membership needed.
Focus on what matters
Your fellow students write the study notes themselves, which is why the documents are always reliable and up-to-date. This ensures you quickly get to the core!
Frequently asked questions
What do I get when I buy this document?
You get a PDF, available immediately after your purchase. The purchased document is accessible anytime, anywhere and indefinitely through your profile.
Satisfaction guarantee: how does it work?
Our satisfaction guarantee ensures that you always find a study document that suits you well. You fill out a form, and our customer service team takes care of the rest.
Who am I buying these notes from?
Stuvia is a marketplace, so you are not buying this document from us, but from seller APlusAchiever. Stuvia facilitates payment to the seller.
Will I be stuck with a subscription?
No, you only buy these notes for £6.57. You're not tied to anything after your purchase.
Can Stuvia be trusted?
4.6 stars on Google & Trustpilot (+1000 reviews)
53068 documents were sold in the last 30 days
Founded in 2010, the go-to place to buy revision notes and other study material for 14 years now