Calculus  Functions, Limits and Continuity  Problem Set III
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 ALevel; Class 11/12 students in India preparing for ISC/CBSE and Entrance Examinations like the IITJEE/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) =
A^{2} x − A, if x ≥ 3
Determine the value of constants A and B so that the following
function is continuous for all values of x.
2x^{2} + 3Ax + B, if −1 < 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) = 2x^{3} − 5x^{2} − 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 :
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 Nth 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 NonLinear 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

