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.Read More

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