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Calculus - Functions, Limits, Continuity Problem Set III - Examples, Problems related to Continuity, Intermediate Value Theorems :


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                Calculus - Functions, Limits and Continuity - Problem Set III 

    

Domain and Range

Calculus - Functions, Limits and Continuity - Problem Set III - Outline of Contents:

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!
 

Here's a quick look at the kind of problems we will go through, in this tutorial - Problems related to Continuity, intermediate value theorem :


For what values of x is the following function continuous?
(x−1) /  √x - 1 if  x>1
5 − 3x, if  −2 ≤ x ≤ 1
6/x−4 ,  if  x < −2

Determine the values of the constant A so that the following 
function is continuous for all values of x.
f(x) =  
A2 x − A, if x ≥ 3
4, 
if x < 3

Determine the value of constants A and B so that the following 
function is continuous for all values of x.
Ax − B, if  
x ≤ −1
2x2 + 3Ax + B, if −1 < x ≤ 1
4, 
if 
x>1
 
Show that the following function is continuous for all values of x.
f(x) = 
e−1/x2 , if x != 0
0  if x = 0

Show that p(x) = 2x3 − 5x2 − 10x + 5 has a root somewhere in 
the interval [−1, 2].

Let f be a function on [0, 1]. Show that if −1 ≤ f (x) ≤ 1 for all x ∈ [0, 1] then there is c ∈ [0, 1] such that [f (c)]2 = c.


Complete Tutorial with Examples, Problems and Solutions :
 



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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