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Calculus: Differential Calculus: Solved Problem Set II - Derivability / Differentiability, Continuity; Changing Independent Variables; N-th derivative


 Differential Equations
                                            
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Calculus: Differential Calculus: Solved Problem Set II

 Differentiation- Continuity; Changing Independent Variables

Calculus - Differential Calculus - Problem Set II - 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!


In this tutorial, we'll pick up somewhat challenging problems based on the concepts of differential calculus introduced so far. 
We'll examine the continuity and derivability of more complicated functions. We'll differentiate more complex functions; using substitutions and transformations. We'll also pick up examples of chain/successive differentiation to find the n-th derivatives. 

Here's a quick look at the functions which will be inspected in this tutorial.


1.  Show that the function f(x) is derivable for every value of x but the derivative is not continuous for x = 0.
f (x) = x2 sin (1/x) when x = 0
0 when x = 0

2.  Examine the continuity and derivability in the interval (−∞, ∞) for the following function f(x) =
1              in − ∞ < x < 0
1 + sin x    in 0 ≤ x < π
2+ (x− π)in π ≤ x < ∞

 
3.  Find the nth derivative of x / (x2 + a2)
This will be solved using a technique involving complex numbers.

4.  Find the nth derivatives of  
(i) cos4 x
(ii)eax cos2 x sin x
These will be solved using compound and multiple angle trigonometric identities. 

Complete tutorial with examples, solutions to problems :




Other 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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