Application of Derivatives Class 12 Notes | EduRev

 Page 1


 85
5. APPLICATIONS OF DERIVATIVES 
 
Derivatives are everywhere in engineering, physics, biology, economics, and 
much more. In this chapter we seek to elucidate a number of general ideas 
which cut across many disciplines.  
 
Linearization of a function is the process of approximating a function by a 
line near some point. The tangent line is the graph of the linearization. 
 
Given some algebraic relation that connects different dynamical quantities 
we can differentiate implicitly. This relates the rates of change for the 
various quantities involved. Such problems are called “related rates 
problems”. 
 
The shape of a graph  can be ciphered through analyzing how the 
first and second derivatives of the function behave. Rolle’s Theorem and the 
Mean Value Theorem are discussed as they provide foundational support for 
later technical arguments. Fermat’s Theorem tells us that local extrema 
happen at critical points.  
If a function is increasing on an interval then the derivative will be positive 
on that same interval. Likewise, a decreasing function will have a negative 
derivative.  These observation lead straight to the First Derivative Test 
which allows us to classify critical points as being local minimas, maximas or 
neither. Concavity is discussed and shown to be described by the second 
derivative of the function. If a function is concave up on an interval then the 
second derivative of the function will be positive on that interval. Likewise, 
the second derivative is negative when the function is concave down. 
Concavity’s connection to the second derivative gives us another test; the 
Second Derivative Test. Sometimes the second derivative test helps us 
determine what type of extrema reside at a particular critical point. However, 
the First Derivative Test has wider application. We also discuss the Closed 
Interval Method which is based on the same ideas plus the insight that when 
we restrict a function to a closed interval then the extreme values might 
occur at endpoints.  In total, precalculus and college algebra skill is 
supplemented with new calculus-based insight.  Calculus helps us graph with 
new found confidence. 
 
Optimization is the application of calculus-based graphical analysis to 
particular physical examples. We have to find critical points then 
characterize them as minima or maxima depending on the problem. As 
always word problems pose extra troubles as the interpretation of the 
problem and invention of needed variables are themselves conceptually 
challenging. This part of calculus allows for much creativity. Often drawing a 
picture is an essential step to organize your ideas to forge ahead.  
 
Page 2


 85
5. APPLICATIONS OF DERIVATIVES 
 
Derivatives are everywhere in engineering, physics, biology, economics, and 
much more. In this chapter we seek to elucidate a number of general ideas 
which cut across many disciplines.  
 
Linearization of a function is the process of approximating a function by a 
line near some point. The tangent line is the graph of the linearization. 
 
Given some algebraic relation that connects different dynamical quantities 
we can differentiate implicitly. This relates the rates of change for the 
various quantities involved. Such problems are called “related rates 
problems”. 
 
The shape of a graph  can be ciphered through analyzing how the 
first and second derivatives of the function behave. Rolle’s Theorem and the 
Mean Value Theorem are discussed as they provide foundational support for 
later technical arguments. Fermat’s Theorem tells us that local extrema 
happen at critical points.  
If a function is increasing on an interval then the derivative will be positive 
on that same interval. Likewise, a decreasing function will have a negative 
derivative.  These observation lead straight to the First Derivative Test 
which allows us to classify critical points as being local minimas, maximas or 
neither. Concavity is discussed and shown to be described by the second 
derivative of the function. If a function is concave up on an interval then the 
second derivative of the function will be positive on that interval. Likewise, 
the second derivative is negative when the function is concave down. 
Concavity’s connection to the second derivative gives us another test; the 
Second Derivative Test. Sometimes the second derivative test helps us 
determine what type of extrema reside at a particular critical point. However, 
the First Derivative Test has wider application. We also discuss the Closed 
Interval Method which is based on the same ideas plus the insight that when 
we restrict a function to a closed interval then the extreme values might 
occur at endpoints.  In total, precalculus and college algebra skill is 
supplemented with new calculus-based insight.  Calculus helps us graph with 
new found confidence. 
 
Optimization is the application of calculus-based graphical analysis to 
particular physical examples. We have to find critical points then 
characterize them as minima or maxima depending on the problem. As 
always word problems pose extra troubles as the interpretation of the 
problem and invention of needed variables are themselves conceptually 
challenging. This part of calculus allows for much creativity. Often drawing a 
picture is an essential step to organize your ideas to forge ahead.  
 
 86
Finally we discuss limits at infinity. Graphically these limits tell us about 
horizontal asymptotes. Generally there are many different types of 
asymptotic behavior, we focus on the basic types.  Again this helps us graph 
better. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
Page 3


 85
5. APPLICATIONS OF DERIVATIVES 
 
Derivatives are everywhere in engineering, physics, biology, economics, and 
much more. In this chapter we seek to elucidate a number of general ideas 
which cut across many disciplines.  
 
Linearization of a function is the process of approximating a function by a 
line near some point. The tangent line is the graph of the linearization. 
 
Given some algebraic relation that connects different dynamical quantities 
we can differentiate implicitly. This relates the rates of change for the 
various quantities involved. Such problems are called “related rates 
problems”. 
 
The shape of a graph  can be ciphered through analyzing how the 
first and second derivatives of the function behave. Rolle’s Theorem and the 
Mean Value Theorem are discussed as they provide foundational support for 
later technical arguments. Fermat’s Theorem tells us that local extrema 
happen at critical points.  
If a function is increasing on an interval then the derivative will be positive 
on that same interval. Likewise, a decreasing function will have a negative 
derivative.  These observation lead straight to the First Derivative Test 
which allows us to classify critical points as being local minimas, maximas or 
neither. Concavity is discussed and shown to be described by the second 
derivative of the function. If a function is concave up on an interval then the 
second derivative of the function will be positive on that interval. Likewise, 
the second derivative is negative when the function is concave down. 
Concavity’s connection to the second derivative gives us another test; the 
Second Derivative Test. Sometimes the second derivative test helps us 
determine what type of extrema reside at a particular critical point. However, 
the First Derivative Test has wider application. We also discuss the Closed 
Interval Method which is based on the same ideas plus the insight that when 
we restrict a function to a closed interval then the extreme values might 
occur at endpoints.  In total, precalculus and college algebra skill is 
supplemented with new calculus-based insight.  Calculus helps us graph with 
new found confidence. 
 
Optimization is the application of calculus-based graphical analysis to 
particular physical examples. We have to find critical points then 
characterize them as minima or maxima depending on the problem. As 
always word problems pose extra troubles as the interpretation of the 
problem and invention of needed variables are themselves conceptually 
challenging. This part of calculus allows for much creativity. Often drawing a 
picture is an essential step to organize your ideas to forge ahead.  
 
 86
Finally we discuss limits at infinity. Graphically these limits tell us about 
horizontal asymptotes. Generally there are many different types of 
asymptotic behavior, we focus on the basic types.  Again this helps us graph 
better. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 87
5.1. LINEARIZATIONS 
We have already found the linearization of a function a number of times. The 
idea is to replace the function by its tangent line at some point. This provides 
a fairly good approximation if we are near to the point. How near is near? 
Well, that depends on the example and what your idea of a “good 
approximation” should be. These are questions best left to a good numerical 
methods course. The linearization of a function  at a point  is 
denoted by  or simply  in this course, 
 
The graph of   is the tangent line to  at . 
 
Example 5.1.1:  (linearization can be used to calculate square roots) 
 
 
 
Page 4


 85
5. APPLICATIONS OF DERIVATIVES 
 
Derivatives are everywhere in engineering, physics, biology, economics, and 
much more. In this chapter we seek to elucidate a number of general ideas 
which cut across many disciplines.  
 
Linearization of a function is the process of approximating a function by a 
line near some point. The tangent line is the graph of the linearization. 
 
Given some algebraic relation that connects different dynamical quantities 
we can differentiate implicitly. This relates the rates of change for the 
various quantities involved. Such problems are called “related rates 
problems”. 
 
The shape of a graph  can be ciphered through analyzing how the 
first and second derivatives of the function behave. Rolle’s Theorem and the 
Mean Value Theorem are discussed as they provide foundational support for 
later technical arguments. Fermat’s Theorem tells us that local extrema 
happen at critical points.  
If a function is increasing on an interval then the derivative will be positive 
on that same interval. Likewise, a decreasing function will have a negative 
derivative.  These observation lead straight to the First Derivative Test 
which allows us to classify critical points as being local minimas, maximas or 
neither. Concavity is discussed and shown to be described by the second 
derivative of the function. If a function is concave up on an interval then the 
second derivative of the function will be positive on that interval. Likewise, 
the second derivative is negative when the function is concave down. 
Concavity’s connection to the second derivative gives us another test; the 
Second Derivative Test. Sometimes the second derivative test helps us 
determine what type of extrema reside at a particular critical point. However, 
the First Derivative Test has wider application. We also discuss the Closed 
Interval Method which is based on the same ideas plus the insight that when 
we restrict a function to a closed interval then the extreme values might 
occur at endpoints.  In total, precalculus and college algebra skill is 
supplemented with new calculus-based insight.  Calculus helps us graph with 
new found confidence. 
 
Optimization is the application of calculus-based graphical analysis to 
particular physical examples. We have to find critical points then 
characterize them as minima or maxima depending on the problem. As 
always word problems pose extra troubles as the interpretation of the 
problem and invention of needed variables are themselves conceptually 
challenging. This part of calculus allows for much creativity. Often drawing a 
picture is an essential step to organize your ideas to forge ahead.  
 
 86
Finally we discuss limits at infinity. Graphically these limits tell us about 
horizontal asymptotes. Generally there are many different types of 
asymptotic behavior, we focus on the basic types.  Again this helps us graph 
better. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 87
5.1. LINEARIZATIONS 
We have already found the linearization of a function a number of times. The 
idea is to replace the function by its tangent line at some point. This provides 
a fairly good approximation if we are near to the point. How near is near? 
Well, that depends on the example and what your idea of a “good 
approximation” should be. These are questions best left to a good numerical 
methods course. The linearization of a function  at a point  is 
denoted by  or simply  in this course, 
 
The graph of   is the tangent line to  at . 
 
Example 5.1.1:  (linearization can be used to calculate square roots) 
 
 
 
 88
This example shows that we can calculate good approximations to square roots, even 
when the computers and their robot slaves turn against us.  
 
Example 5.1.2 and 5.1.3: 
 
 
Page 5


 85
5. APPLICATIONS OF DERIVATIVES 
 
Derivatives are everywhere in engineering, physics, biology, economics, and 
much more. In this chapter we seek to elucidate a number of general ideas 
which cut across many disciplines.  
 
Linearization of a function is the process of approximating a function by a 
line near some point. The tangent line is the graph of the linearization. 
 
Given some algebraic relation that connects different dynamical quantities 
we can differentiate implicitly. This relates the rates of change for the 
various quantities involved. Such problems are called “related rates 
problems”. 
 
The shape of a graph  can be ciphered through analyzing how the 
first and second derivatives of the function behave. Rolle’s Theorem and the 
Mean Value Theorem are discussed as they provide foundational support for 
later technical arguments. Fermat’s Theorem tells us that local extrema 
happen at critical points.  
If a function is increasing on an interval then the derivative will be positive 
on that same interval. Likewise, a decreasing function will have a negative 
derivative.  These observation lead straight to the First Derivative Test 
which allows us to classify critical points as being local minimas, maximas or 
neither. Concavity is discussed and shown to be described by the second 
derivative of the function. If a function is concave up on an interval then the 
second derivative of the function will be positive on that interval. Likewise, 
the second derivative is negative when the function is concave down. 
Concavity’s connection to the second derivative gives us another test; the 
Second Derivative Test. Sometimes the second derivative test helps us 
determine what type of extrema reside at a particular critical point. However, 
the First Derivative Test has wider application. We also discuss the Closed 
Interval Method which is based on the same ideas plus the insight that when 
we restrict a function to a closed interval then the extreme values might 
occur at endpoints.  In total, precalculus and college algebra skill is 
supplemented with new calculus-based insight.  Calculus helps us graph with 
new found confidence. 
 
Optimization is the application of calculus-based graphical analysis to 
particular physical examples. We have to find critical points then 
characterize them as minima or maxima depending on the problem. As 
always word problems pose extra troubles as the interpretation of the 
problem and invention of needed variables are themselves conceptually 
challenging. This part of calculus allows for much creativity. Often drawing a 
picture is an essential step to organize your ideas to forge ahead.  
 
 86
Finally we discuss limits at infinity. Graphically these limits tell us about 
horizontal asymptotes. Generally there are many different types of 
asymptotic behavior, we focus on the basic types.  Again this helps us graph 
better. 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 
 87
5.1. LINEARIZATIONS 
We have already found the linearization of a function a number of times. The 
idea is to replace the function by its tangent line at some point. This provides 
a fairly good approximation if we are near to the point. How near is near? 
Well, that depends on the example and what your idea of a “good 
approximation” should be. These are questions best left to a good numerical 
methods course. The linearization of a function  at a point  is 
denoted by  or simply  in this course, 
 
The graph of   is the tangent line to  at . 
 
Example 5.1.1:  (linearization can be used to calculate square roots) 
 
 
 
 88
This example shows that we can calculate good approximations to square roots, even 
when the computers and their robot slaves turn against us.  
 
Example 5.1.2 and 5.1.3: 
 
 
 89
 
These examples just give you a small window into the utility of linearization. You should 
take our numerical methods course if you want to know more about how to perform these 
sorts of calculations with care. For applications, the true error in the approximation 
should be quantified.