Limits and Continuity Unit Problem Sets
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 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. 2x2 + 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) = 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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