Lecture 1 - Concavity, Points of Inflexion, Curve Sketching | Calculus - Engineering Mathematics PDF Download

 Page 1


Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Concavity, Points of Inflexion, Curve 
Sketching 
Course Developer: Dr. Sada Nand Prasad 
College/Department: A.N.D. College (D.U.) 
 
 
 
 
 
 
 
Page 2


Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Concavity, Points of Inflexion, Curve 
Sketching 
Course Developer: Dr. Sada Nand Prasad 
College/Department: A.N.D. College (D.U.) 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Table of Contents: 
 Chapter : Concavity, Points of Inflexion, Curve Sketching 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Concavity 
• 4. Points of Inflexion 
• 5. Curve Sketching 
• Summary 
• Exercises 
• Glossary 
• References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Concavity, Points of Inflexion, Curve 
Sketching 
Course Developer: Dr. Sada Nand Prasad 
College/Department: A.N.D. College (D.U.) 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Table of Contents: 
 Chapter : Concavity, Points of Inflexion, Curve Sketching 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Concavity 
• 4. Points of Inflexion 
• 5. Curve Sketching 
• Summary 
• Exercises 
• Glossary 
• References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
1. Learning Outcomes: 
After studying this chapter, you should be able to 
? apply second derivative to determine the behaviour of any curve in 
a given domain. 
? compute the range for which any function is convex or concave or 
having points of inflexion; 
? determine whether the curve is concave up or concave down; 
? obtain the points of inflexion of a curve; 
? identify and draw the graphs of some significant curves; 
 
2. Introduction: 
Application of differentiation is very useful in determining the solution of 
problems, we often face in almost all branches of science, like how to get 
accurate values of any function corresponding to any given values, how to 
find the maximum and minimum values of any function in a certain 
domain, how to determine the behaviour of any curve in a given domain 
etc. One of the ways of determining the behaviour of a curve is finding 
concavity and points of inflexion of the curve.  
In this chapter we shall explain the process of finding the concavity, 
convexity and points of inflexion of a function. We shall explain the 
methods of tracing a given curve. To start with, we have talked about the 
problem of finding concavity, convexity and points of inflexion, which are 
geometrical applications of differentiation. 
 
3. Concavity:  
Let P be a given point on a curve. Draw the tangent to the curve at the 
point P. Let L be a given straight line and let ? be the acute angle formed 
by the tangent at P with the line L.  
 
Page 4


Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Concavity, Points of Inflexion, Curve 
Sketching 
Course Developer: Dr. Sada Nand Prasad 
College/Department: A.N.D. College (D.U.) 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Table of Contents: 
 Chapter : Concavity, Points of Inflexion, Curve Sketching 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Concavity 
• 4. Points of Inflexion 
• 5. Curve Sketching 
• Summary 
• Exercises 
• Glossary 
• References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
1. Learning Outcomes: 
After studying this chapter, you should be able to 
? apply second derivative to determine the behaviour of any curve in 
a given domain. 
? compute the range for which any function is convex or concave or 
having points of inflexion; 
? determine whether the curve is concave up or concave down; 
? obtain the points of inflexion of a curve; 
? identify and draw the graphs of some significant curves; 
 
2. Introduction: 
Application of differentiation is very useful in determining the solution of 
problems, we often face in almost all branches of science, like how to get 
accurate values of any function corresponding to any given values, how to 
find the maximum and minimum values of any function in a certain 
domain, how to determine the behaviour of any curve in a given domain 
etc. One of the ways of determining the behaviour of a curve is finding 
concavity and points of inflexion of the curve.  
In this chapter we shall explain the process of finding the concavity, 
convexity and points of inflexion of a function. We shall explain the 
methods of tracing a given curve. To start with, we have talked about the 
problem of finding concavity, convexity and points of inflexion, which are 
geometrical applications of differentiation. 
 
3. Concavity:  
Let P be a given point on a curve. Draw the tangent to the curve at the 
point P. Let L be a given straight line and let ? be the acute angle formed 
by the tangent at P with the line L.  
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Definition (Concavity at a Point): The curve is said to be concave at P 
with respect to line L if a sufficiently small arc containing P, on extending 
to both sides of P lies entirely within the angle of ?.(Fig 1) 
 
O A 
L 
Fig 1: A curve concave or concave downwards at P 
Definition (Convexity at a Point): The curve is said to be convex at P 
with respect to line L if a sufficiently small arc containing P, on extending 
to both sides of P lies entirely outside the angle of ?.(Fig 2). 
O A 
                                                L 
Fig 2: A curve convex or concave upwards at P 
? 
P 
T 
T 
? 
P 
Page 5


Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
 
 
 
 
 
 
 
Subject: Mathematics 
Lesson: Concavity, Points of Inflexion, Curve 
Sketching 
Course Developer: Dr. Sada Nand Prasad 
College/Department: A.N.D. College (D.U.) 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Table of Contents: 
 Chapter : Concavity, Points of Inflexion, Curve Sketching 
• 1. Learning Outcomes 
• 2. Introduction 
• 3. Concavity 
• 4. Points of Inflexion 
• 5. Curve Sketching 
• Summary 
• Exercises 
• Glossary 
• References/ Further Reading 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
1. Learning Outcomes: 
After studying this chapter, you should be able to 
? apply second derivative to determine the behaviour of any curve in 
a given domain. 
? compute the range for which any function is convex or concave or 
having points of inflexion; 
? determine whether the curve is concave up or concave down; 
? obtain the points of inflexion of a curve; 
? identify and draw the graphs of some significant curves; 
 
2. Introduction: 
Application of differentiation is very useful in determining the solution of 
problems, we often face in almost all branches of science, like how to get 
accurate values of any function corresponding to any given values, how to 
find the maximum and minimum values of any function in a certain 
domain, how to determine the behaviour of any curve in a given domain 
etc. One of the ways of determining the behaviour of a curve is finding 
concavity and points of inflexion of the curve.  
In this chapter we shall explain the process of finding the concavity, 
convexity and points of inflexion of a function. We shall explain the 
methods of tracing a given curve. To start with, we have talked about the 
problem of finding concavity, convexity and points of inflexion, which are 
geometrical applications of differentiation. 
 
3. Concavity:  
Let P be a given point on a curve. Draw the tangent to the curve at the 
point P. Let L be a given straight line and let ? be the acute angle formed 
by the tangent at P with the line L.  
 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
Definition (Concavity at a Point): The curve is said to be concave at P 
with respect to line L if a sufficiently small arc containing P, on extending 
to both sides of P lies entirely within the angle of ?.(Fig 1) 
 
O A 
L 
Fig 1: A curve concave or concave downwards at P 
Definition (Convexity at a Point): The curve is said to be convex at P 
with respect to line L if a sufficiently small arc containing P, on extending 
to both sides of P lies entirely outside the angle of ?.(Fig 2). 
O A 
                                                L 
Fig 2: A curve convex or concave upwards at P 
? 
P 
T 
T 
? 
P 
Concavity, Points of Inflexion, Curve Sketching            
Institute of Lifelong Learning, University of Delhi                                                 
 
  
Value Additions: 
(1) If ( )
''
0, fx > at every point of the arc, then the arc is concave up. For 
( ) { }
'
0,
d
fx
dx
> gradient of the curve is an increasing function. 
(2) If ( )
''
0, fx < at every point of the arc, then the arc is concave down or 
convex up since the gradient of the curve is a decreasing function. 
(3) The curve is convex or concave at a point P with respect to the x – 
axis according as 
2
2
dy
y
dx
 is positive or negative at P. 
 
Example 1: Show that the curve 
x
ye = is convex everywhere.  
Solution: We have, 
  ,
x
ye =   
Differentiating the equation w.r.t. x we get 
 ,
x
dy
e
dx
= 
Again differentiating w.r.t. x we get 
 
2
2
,
x
dy
e
dx
= 
? 
( )
2
2
2
.0
x x x
dy
y e e e
dx
= = > 
Since 
2
2
dy
y
dx
is positive for all values of x, the curve is at every point 
convex to the foot of the corresponding ordinate. 
 
Example 2: Find the range of values of x for which the curve 
43 2
6 12 5 9 yx x x x = - + +- is concave upwards or downwards. 
Solution: Given curve is 
 
43 2
6 12 5 9 yx x x x = - + +- 
Differentiating twice with respect to x, we get