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

                                        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

A Quick look at some of the examples we cover in the tutorial document and first, some hints on how to handle such problems. 

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. 
First differentiate both sides to get the derivative. Then find the point of intersection of the tangent and the curve. 

  • Find the points of inflexion and determine the intervals of convexity and concavity of the curve y = e− (x ^ 2)
Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero. Find how the sign of the 2nd order derivatives changes while passing through specific points. FInd intervals of convexity and concavity and plot the curve.

  • Show that the curve (a2 + x2 )y = a2 x has three points of inflexion.
Again, a similar approach. Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero. Find how the sign of the 2nd order derivatives changes while passing through specific points.
 
  • Find the points of inflexion on the curve x = (log y)3 .
Again, a similar approach. Find the expression for the first and second order derivatives. Then find the points where the second order derivative is zero. 

  • Find the asymptotes of  curves like  y =  (x2 + 2x − 1) / x , y = e−x sin x + x
We discover both a vertical and an inclined asymptote. To determine the mutual positions of a curve and an asymptote, let us consider the difference of the ordinates of the curve and the asymptote for one and the same value of x. We find that his difference is negative for x > 0 and positive for x < 0; and so for x > 0, the curve lies below the asymptote, and for x < 0 it lies above. This is also something which helps us sketch the curve. 


  • Transforming the formula for radius of curvature of a curve, at a point, into the polar form (using the substitutions x = ρ cos θ and y = ρ sin θ)
We use parametric differentiation to replace (dy/dx) and (d2y/dx2) in terms of first and second order derivatives of ρ w.r.t θ

While going through these problems 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 .
  • Detecting the presence or absence of a vanishing point.
You will also gain an appreciation of why curve sketching is important

Curve sketching is important to help students visualize what they're doing. If they understand the basis of how a curve is sketch, it also set the ground for them to understand optimization and the concepts of maxima and minima. Free hand sketches of curves are not made for accuracy, they are made for the purpose of understanding general features of the curves such as zeroes, poles, points of discontinuity, inflexion points, aymptotes, tangents, loops, etc. You sketch curves in order to understand the properties of the function. Calculus helps you to determine those properties (such as. gradients and turning points)

Complete Tutorial with Examples and Solved 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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