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Calculus - Introducing Differentiable functions and Differentiation - Basic Differentiation Formulas, Successive Differentiation; MCQ Quizzes




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Calculus - Introducing Differentiable functions and Differentiation

       

 
differentiation

Calculus - Introducing Differentiable functions and Differentiation - Outline of Contents (Also check out the MCQ Quizzes at the end):

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!


Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. 



Here's a very quick outline or summary of the concepts introduced in this tutorial.  Future tutorials, which appear further in this series of calculus tutorials, will focus on how these ideas are applied and we will solve interesting examples and problems where these concepts will be applied.


• Differentiability Of Functions &  Basic Differentiation Formulas: 

The condition for differentiability of functions will be introduced in the tutorial. Addition, Subtraction, Linearity, Product, Quotient, Chain and Power Rules, Exponential and Logarithmic Rules - these will be introduced in the tutorial document.


• Successive Differentiation: 

The derivative f' (x) of a derivable function f (x) is itself a function of x. We suppose that it also possesses a derivative, which is denoted by f'' (x) and called the second derivative of f (x). The third derivative of f (x) which is the derivative of f'' (x) is denoted by f '''(x) and so on. Thus the successive derivatives of f (x) are represented by the symbols, f (x), f; (x), . . . , f n (x), . . .
where each term is the derivative of the previous one. Sometimes y1 , y2 , y3 , . . . , yn , . . . are used to denote the successive derivatives of y.

• Leibnitz’s Theorem

The nth derivative of the product of two functions: If u, v be the two functions possessing derivatives of the nth order, then  (uv)n = un v +n C1 un−1 v1 +n C2 un−2 v2 + . . . +n Cr un−r vr + . . . + uvn .


• Rolle’s Theorem: 

If a function f (x) is derivable in an interval [a, b], and also f (a) = f (b), then there exists atleast one value c of x lying within [a, b] such that f (c) = 0.

• Lagrange’s Mean Value Theorem: 

If a function f (x) is derivable in an interval [a, b], then there exists atleast one value c of x lying within [a, b] such that
(f (b) − f (a)) / (b-a) = f(c)

Increasing and Decreasing Functions:

A function whose derivative is positive for every value of x in an interval is a monotonically increasing function of x in that interval, i.e,
If f (x) > 0 for every value of x in [a, b], then f (x) is an increasing function of x in that interval.
A function whose derivative is negative for every value of x in an interval is a monotonically decreasing function of x in that interval, i.e,
If f (x) < 0 for every value of x in [a, b], then f (x) is an decreasing function of x in that interval.

What Maxima and Minima mean :

Maximum Value of a function: f (c) is said to be a maximum value of f (x), if it is the greatest of all its values of x lying in some neighbourhood of c, i.e, f (c) is a maximum value of x if there exists a positive δ such that f (c) > f (c + h) or f (c) − f (c + h) > 0 for values of h lying between −δ and δ.
Minimum Value of a function: f (c) is said to be a minimum value of f (x), if it is the least of all its values of x lying in some neighbourhood of c, i.e, f (c) is a minimum value of x if there exists a positive δ such that f (c) < f (c + h) or f (c + h) − f (c) > 0 for values of h lying between −δ and δ.

Greatest and least values of a function in any interval

The greatest and least values of f (x) in any interval [a, b] are either f (a) and f (b), or are given by the values of x for which f (x) = 0.


Change of Sign

A function is said to change sign from positive to negative as x passes through a number c, if there exists some left-handed neighbourhood (c − h, c) of c for every point of which the function is positive, and also there exists some right-handed neighbourhood (c, c + h) of c for every point of which the function is negative.


Sufficient Criteria for extreme values

Prove that f (c) is an extreme value of f (x) if and only if f (x) changes sign as x passes through c, and to show that f (c) is a maximum value if the sign changes from positive to negative and a minimum value if the sign changes from negative to positive.

Minimum and Maximum Values :

Minimum Value: f (c) is a minimum value of f (x), if f '(c) = 0 and f'' (c) > 0.
Maximum Value: f (c) is a maximum value of f (x), if f '(c) = 0 and f ''(c) < 0.

Implicit Differentiation: 

A relation F (x, y) = 0 is said to define the function y = f (x) implicitly if, for x in the domain of f , F (x, f (x)) = 0. Given a differentiable relation F (x, y) = 0 which defines the differential function y = f (x), it is usually possible to find the derivative f even in the case when you cannot symbolically find f . The method of finding the derivative is called implicit differentiation.


Criteria for concavity, convexity and inflexion: 

Criteria to determine whether a curve y = f (x) is concave upwards, concave downwards, or has a point of inflexion at P [c, f (c)] are:
(i) the curve is concave upwards at P if f'' (c) > 0.
(ii) the curve is concave downwards at P if f'' (c) < 0.
(iii) the curve has inflexion at P if f'' (c) = 0 and f''' (c) = 0.



Complete Tutorial with Definitions and Formulas (Also try out the MCQ Quizzes below this) :




MCQ Quiz #1

Companion MCQ Quiz #1 - test how much you know about the topic. Your score will be e-mailed to you at the address you provide.



MCQ- Calculus- Functions, Differentiation, etc.


MCQ Quiz #2 

Companion MCQ Quiz #2 - 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



In Case you'd like to take a look at some of our other tutorials related to Single Variable Calculus :


 Quick and introductory definitions related to Funtions, Limits and Continuity - Defining the domain and range of a function, the meaning of continuity, limits, left and right hand limits, properties of limits and the "lim" operator; some common limits;  defining the L'Hospital rule, intermediate and extreme value theorems.   Functions, Limits and Continuity - Solved Problem Set I - The Domain, Range, Plots and Graphs of Functions;  L'Hospital's Rule- -  Solved problems demonstrating how to compute the domain and range of functions, drawing the graphs of functions, the mod function, deciding if a function is invertible or not; calculating limits for some elementary examples, solving 0/0 forms, applying L'Hospital 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   - More advanced cases of evaluating limits, conditions for continuity of functions, common approximations used while evaluating limits for ln ( 1 + x ), sin (x); continuity related problems for more advanced functions than the ones in the first group of problems (in the last tutorial).  Functions, Limits and Continuity - Solved Problem Set III - Continuity and Intermediate Value Theorems - Problems related to Continuity, intermediate value theorem.
Introductory concepts and definitions related to Differentiation - Basic formulas, Successive Differentiation, Leibnitz, Rolle and Lagrange Theorems, Maxima , Minima, Convexity, Concavity, etc - Theory and definitions introducing differentiability, basic differentiation formulas of common algebraic and trigonometric functions , successive differentiation, Leibnitz Theorem, Rolle's Theorem,  Lagrange's Mean Value Theorem, Increasing and decreasing functions, Maxima and Minima; Concavity, convexity and inflexion, implicit differentiation. 
Differential Calculus - Solved Problem Set I - Common Exponential, Log , trigonometric and polynomial functions - Examples and solved problems - differentiation of common algebraic, exponential, logarithmic, trigonometric and polynomial functions and terms; problems related to differentiability .   Differential Calculus - Solved Problem Set II - Derivability and continuity of functins - Change of Indepndent Variables - Finding N-th Derivatives -
 Examples and solved problems - related to derivability and continuity of functions; changing the independent variable in a differential equation; finding the N-th derivative of functions
 Differential Calculus - Solved Problems Set III- Maximia, Minima, Extreme Values,  Rolle's Theorem - 
Examples and solved problems - related to increasing and decreasing functions; maxima, minima and extreme values; Rolle's Theorem 
Differential Calculus - Solved Problems Set IV - Points of Inflexion, Radius of Curvature, Curve Sketching -  
Examples and solved problems - Slope of tangents to a curve, points of inflexion, convexity and concavity of curves, radius of curvature and asymptotes of curves, sketching curves 
Differential Calculus - Solved Problems Set V - Curve Sketching, Parametric Curves - More examples of investigating and sketching curves, parametric representation of curves
Introducing Integral Calculus - Definite and Indefinite Integrals - using Substitution , Integration By Parts, ILATE rule  - Theory and definitions. What integration means, the integral and the integrand. Indefinite integrals, integrals of common functions.  Definite integration and properties of definite integrals; Integration by  substitution, integration by parts, the LIATE rule, Integral as the limit of a sum. Important forms encountered in integration.  Integral Calculus - Solved Problems Set I - Basic examples of polynomials and trigonometric functions, area under curves - Examples and solved problems - elementary examples of integration involving trigonometric functions, polynomials; integration by parts; area under curves.
 Integral Calculus - Solved Problems Set II - More integrals, functions involving trigonometric and inverse trigonometric ratios - Examples and solved problems - integration by substitution, definite integrals, integration involving trigonometric and inverse trigonometric ratios. Integral Calculus - Solved Problems Set III - Reduction Formulas, Using Partial FractionsIExamples and solved problems - Reduction formulas, reducing the integrand to partial fractions, more of definite integrals
Integral Calculus - Solved Problems Set IV - More of integration using partial fractions, more complex substitutions and transformations - Examples and solved problems - More of integrals involving partial fractions, more complex substitutions and transformations Integral Calculus - Solved Problems Set V- Integration as a summation of a series - Examples and solved problems - More complex examples of integration, examples of integration as the limit of 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 - 
 Theory and definitions. What a differential equation is; ordinary and partial differential equations; order and degree of a differential equation; linear and non linear differential equations; General, particular and singular solutions; Initial and boundary value problems; Linear independence and dependence; Homogeneous equations; First order differential equations; Characteristic and auxiliary equations. Introductory problems demonstrating these concepts. Introducing the concept of Integrating Factor (IF). 
Differential Equations - Solved Problems - Set II - D operator, auxillary equation, General Solution - Examples and solved problems - Solving linear differential equations, the D operator, auxiliary equations. Finding the general solution ( CF + PI )

Differential Equations - Solved Problems - Set III - More Differential Equations - More complex cases of differential equations.  Differential Equations - Solved Problems - Set IV -
 Still more differential equations. 

 









Our 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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