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Calculus - Introductory Definitions - Functions, Limits, Continuity; Domain and Range , Left and Right Hand Limits; MCQ Quiz at the end

Outline of Contents in this tutorial 

(There is a PDF Document after this outline)


Target Audience: High School Students, College Freshmen and Sophomores, students preparing for the International Baccalaureate (IB), AP Calculus AB, AP Calculus BC, A Level, Singapore/GCE A-Level; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IIT-JEE/AIEEE Anyone else who needs this Tutorial as a reference!

 Differential Equations



Domain and Range of a Function: Let A and B be any two sets and let f denotes a rule which associates to each member of A a member of B. We say that f is a function from A into B. Also A is said

to be Domain of this function. If x denotes a member of the set A, then the member of the set B, which the function f associates to x ∈ A, is denoted by f (x) called the value of the function f for x or at x. The function may be described as x → f (x), or y = f (x) where x ∈ A and y ∈ B.
Range of f = {f (x) : x ∈ A}.

• Continuity: A function f is continuous at a point b in the domain of f if and only if for each positive real number ε such that for each x in the domain of f |x − b| < δ =⇒ |f (x) − f (b)| < ε.

A function f is continuous at a point b if and only if
1. b is in the domain of f
2. lim x→b f (x) exists
3. lim x→b f (x) = f (b)

• Limit: Let the function y = f (x) be defined in a certain neighbourhood of a point a or at certain points of this neighbourhood. The function y = f (x) approaches the limit b(y → b) as x approaches

a(x → a), if for every positive number ε, no matter how small, it is possible to indicate a positive number δ such that for all x, different from a and satisfying |x − a| < δ, we have |f (x) − b| < ε
If b is the limit of the function f (x) as x → a, we write
lim f (x) = b
x→a

• Left hand and Right Hand Limits: If f (x) approaches the limit b1 as x takes on only values less than a we write

lim f (x) = b1
x→a−
..then b1 is called the limit on the left at the point a of the function. If x takes on only values greater than a, we write lim f (x) = b2
x→a+
.. then b2 is called the limit on the right at the point a of the function. If the limit on the right and the limit on the left exist and equal, that is, b1 = b2 = b, then b will be the limit of f (x) at point a.

These are some of the rules which we will we introduce in this tutorial- Sum Rule, Product Rule, Quotient Rule, Constant Multiple Rule, Power Rule, Limit Of an Exponential Function, Limit of a Logarithm of a Function, L'Hospital Theorem, Common limits such as six/x, tan x/ x


This tutorial also introduces the Continuity Theorems, Extreme Value Theorem, Intermediate Value Theorem:


1. Let the function f (x) be continuous at x = a and let c be a constant. Then the function cf (x) is also continuous at x = a.
2. Let the functions f (x) and g(x) be continuous at x = a. Then the sum of the functions f (x) + g(x) is also continuous at x = a.
3. Let the functions f (x) and g(x) be continuous at x = a. Then the product of the functions f (x)g(x) is also continuous at x = a.
4. Let the functions f (x) and g(x) be continuous at x = a. Then the quotient of the functions f (x) is also continuous at x = a.
5. Let f (x) be differentiable at the point x = a. Then the function f (x) is continuous at that point. But the converse is not true.
6. Extreme Value Theorem: If f (x) is continuous on the closed, bounded interval [a, b], then it is bounded above and below in that interval. That is there exists numbers m and M such that m ≤ f (x) ≤ M , for every x in [a, b]
7. Intermediate Value Theorem: Let f (x) be continuous on the closed, bounded interval [a, b]. Then if c is any number between f (a) and f (b), there is a number x0 such that
f(c) =

Complete Tutorial Document 

(Also check out the MCQ Quiz below this, after understanding the basics)



MCQ Quiz on the basics of Functions, Limits, Continuity 

Companion MCQ Quiz for Functions, Limits, Continuity- test how much you know about the topic. Your score will be e-mailed to you at the address you provide.

MCQ Quiz- Functions, Limits, Continuity





 Our Set of Calculus Tutorials                          

Quick and introductory definitions related to Funtions, Limits and Continuity

Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions; L'Hospital's Rule

Functions, Limits and Continuity - Solved Problem Set II - Conditions for Continuity, More Limits, Approximations for ln (1+x) and sin x for infinitesimal values of x  

Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems

Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc

Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions 

Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives -

Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values, Rolle's Theorem

Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching

Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves 

Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  

Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves  

Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios  

Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsI 

Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations  

Integral Calculus - Solved Problems Set V- Integration as a summation of a series 

Introduction to Differential Equations and Solved Problems - Set I - Order and Degree, Linear and Non-Linear Differential Equations, Homogeneous Equations, Integrating Factor 

Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution 

Differential Equations - Solved Problems - Set III - More Differential Equations  

Differential Equations - Solved Problems - Set IV 


    


 


        



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