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Differential and Integral Calculus by Feliciano and Uy: A Comprehensive and Rigorous Introduction



- Why is it important to learn? - What are the main topics covered in the book by Feliciano and Uy? H2: Limits - What are limits and how to find them? - What are the properties of limits? - What are the indeterminate forms and how to evaluate them? H2: Differentiation of Algebraic Functions - What is differentiation and how to find derivatives? - What are the rules of differentiation? - How to use derivatives to find slopes, tangents, normals, rates of change, etc.? H2: Some Applications of the Derivatives - How to use derivatives to find extrema, concavity, inflection points, etc.? - How to use derivatives to sketch graphs of functions? - How to use derivatives to solve optimization problems? H2: Differentiation of Transcendental Functions - How to find derivatives of exponential, logarithmic, trigonometric, inverse trigonometric, and hyperbolic functions? - How to use derivatives to solve related rates problems? - How to use derivatives to find linear approximations and differentials? H2: The Indeterminate Forms - How to use L'Hopital's rule to evaluate indeterminate forms? - How to use Taylor's theorem and Maclaurin's theorem to evaluate indeterminate forms? - How to use series expansions to evaluate indeterminate forms? H2: Integration - What is integration and how to find antiderivatives? - What are the methods of integration? - How to use integration to find areas, volumes, lengths, etc.? H2: Some Applications of the Integrals - How to use integration to find work, force, pressure, etc.? - How to use integration to find average value, mean value, etc.? - How to use integration to solve differential equations? H2: Improper Integrals - What are improper integrals and how to evaluate them? - What are the convergence tests for improper integrals? - How to use improper integrals to find arc length, surface area, etc.? H1: Conclusion - Summarize the main points of the article. - Provide some tips and resources for further learning. - End with a call-to-action for the readers. Table 2: Article with HTML formatting Differential and Integral Calculus by Feliciano and Uy




Differential and integral calculus are two branches of mathematics that deal with the study of rates of change, functions, curves, areas, volumes, and other concepts. They have many applications in science, engineering, economics, physics, biology, and more. In this article, we will explore the main topics covered in the book "Differential and Integral Calculus" by Florentino T. Feliciano and Fausto B. Uy. This book is a comprehensive and rigorous introduction to calculus for college students. It covers both theory and practice with numerous examples, exercises, and solutions.




Differential And Integral Calculus By Feliciano And Uy



Limits




A limit is a value that a function approaches as its input variable gets closer and closer to a certain point. For example, the limit of f(x) = x as x approaches 2 is 4. We write this as limx→2f(x) = 4. Limits help us understand the behavior of functions near points where they are not defined or where they have discontinuities.


To find limits, we can use various methods such as direct substitution, factoring, rationalizing, expanding, etc. We can also use some properties of limits such as sum rule, product rule, quotient rule, power rule, etc. For example,


  • limx→a(f(x) + g(x)) = limx→af(x) + limx→ag(x)



  • limx→a(f(x)g(x)) = limx→af(x)limx→ag(x)



  • limx→a(f(x)/g(x)) = limx→af(x)/limx→ag(x), if limx→ag(x) ≠ 0



  • limx→a(f(x)) = (limx→af(x)), if n is a positive integer



Sometimes, the limit of a function does not exist or is infinite. This happens when the function has different values from the left and right sides of the point, or when the function grows without bound as the input variable approaches the point. These are called indeterminate forms and they require special techniques to evaluate.


Differentiation of Algebraic Functions




Differentiation is the process of finding the derivative of a function. The derivative of a function is a measure of how fast the function changes with respect to its input variable. For example, the derivative of f(x) = x is f'(x) = 2x. This means that for every unit increase in x, the function f(x) increases by 2x units.


To find derivatives, we can use various rules such as constant rule, power rule, sum rule, difference rule, product rule, quotient rule, chain rule, etc. For example,


  • f'(x) = 0, if f(x) is a constant function



  • f'(x) = nx, if f(x) = x, where n is any constant



  • f'(x) = g'(x) + h'(x), if f(x) = g(x) + h(x)



  • f'(x) = g'(x) - h'(x), if f(x) = g(x) - h(x)



  • f'(x) = g'(x)h(x) + g(x)h'(x), if f(x) = g(x)h(x)



  • f'(x) = (g'(x)h(x) - g(x)h'(x))/(h(x)), if f(x) = g(x)/h(x), where h(x) ≠ 0



  • f'(x) = g'(u)u'(x), if f(x) = g(u), where u is a function of x (chain rule)



We can use derivatives to find various quantities such as slopes, tangents, normals, rates of change, etc. For example,


  • The slope of the curve y = f(x) at the point (a,f(a)) is f'(a)



  • The equation of the tangent line to the curve y = f(x) at the point (a,f(a)) is y - f(a) = f'(a)(x - a)



  • The equation of the normal line to the curve y = f(x) at the point (a,f(a)) is y - f(a) = -(1/f'(a))(x - a)



  • The rate of change of y with respect to x is dy/dx = f'(x)



Some Applications of the Derivatives




Derivatives have many applications in various fields such as physics, engineering, economics, biology, etc. Some of these applications are:


  • Finding extrema: We can use derivatives to find the maximum and minimum values of a function on a given interval or domain. To do this, we need to find the critical points of the function where the derivative is either zero or undefined, and then use the first or second derivative test to determine whether they are local maxima or minima. We also need to check the endpoints of the interval if any.



Finding concavity and inflection points: We can use derivatives to find the concavity and inflection points of a function on a given interval or domain. To do this, we need to find the second derivative of the function and determine its sign 71b2f0854b