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Differential Differential Calculus- Solved Problem Set IV: Curve Sketching, tangents , points of inflexion, convexity, concavity, radius of curvature, symptotes



 Differential Equations



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                                        Calculus - Differential Calculus - Problem Set IV

 
Differentiation- Curves

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!

 
 

Calculus - Differential Calculus - Problem Set IV - Outline of Contents:


Here's a quick outline of the kind of problems which will be solved in this tutorial


Curve Sketching and Curve Tracing

We will learn a lot about curves by taking a very detailed tour through problems such as the ones below :
  • Find all the points (x, y) on the graph of x 2/3 + y 2/3 = 8 where lines tangent to the graph at (x, y) have slope −1.
  • Find the points of inflexion and determine the intervals of convexity and concavity of the curve y = e− (x ^ 2)
  • Show that the curve (a2 + x2 )y = a2 x has three points of inflexion.
  • Find the points of inflexion on the curve x = (log y)3 .
  • Find the asymptotes of  curves like  y =  (x2 + 2x − 1) / x , y = e−x sin x + x

Apart from the above, you will also be able to get an idea about :

  • How to change the independent variable from x to t in a given differential equation .
  • How to transform the formula for the radius of curvature, into polar co-ordinates .
  • How to determine the mutual positions of a curve and an asymptote ,
  • Investigate functions, finding their critical points and and constructing their graphs .
  • Determining the domains of increase and decrease of the function .
  • Determining the domains of convexity and concavity of the curve and the points of inflexion .
  • Determining the maximum and minimum values of the function .
  • Finding the domains of functions .


Complete Tutorial with Examples and Solved Problems :





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


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 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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