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Question 1: What is the derivative u'(x) of the function u(x) = ao, where ao is a real number? When working directly with limits we often faced the complication that limits do not, always exist.

Question 1: What is the derivative u'(x) of the function u(x) = ao, where ao is a real number? When working directly with limits we often faced the complication that limits do not, always exist.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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In this course, we have been exploring the elementary foundations of applied mathematics. We have been studying limits, and have found that they allow us to calculate rates of change of functions, find maximum and minimum values of functions, and also to approximate function values. In practical applications, one first collects data and finds a function which fits the data well. This function can then be analysed using the methods we have studied. This kind of mathematics is known as the calculus. It was developed in the late 17th. century in England (Isaac Newton) and Germany (Gottfried Wilhelm Leibniz), but in quite different ways. Knowing this will help you to understand the fact that many different notations exist for what are mathematically the same ideas. Each notation has advantages and disadvantages, and they continue to be used where they are appropriate. As a result, a standard notation never emerged. The slope of the graph of a function f(x) at a point x = a, which is commonly called the derivative of the function f at a, can be written in the following equivalent ways: f'(a) j(a) d f(x)| f(x) dx d f(a + h) – f(a) lim dx h→0 h |x=a x=a or, if we are looking for a general answer which works for any value of x, f' (x) f (x) d f(x) d (x) f(x +h) – f(x) lim dx dx h→0 h We already know that d (k x") = nk x"-1 dx for n a positive natural number. What about n = 0° is not defined, so we cannot simply write x° and calculate its slope. Instead, we can be sure that xº = 1 for x > 0 so we usually work with the constant function that has the value 1 for all x. 0? This is a special case, because the value of
Transcribed Image Text:In this course, we have been exploring the elementary foundations of applied mathematics. We have been studying limits, and have found that they allow us to calculate rates of change of functions, find maximum and minimum values of functions, and also to approximate function values. In practical applications, one first collects data and finds a function which fits the data well. This function can then be analysed using the methods we have studied. This kind of mathematics is known as the calculus. It was developed in the late 17th. century in England (Isaac Newton) and Germany (Gottfried Wilhelm Leibniz), but in quite different ways. Knowing this will help you to understand the fact that many different notations exist for what are mathematically the same ideas. Each notation has advantages and disadvantages, and they continue to be used where they are appropriate. As a result, a standard notation never emerged. The slope of the graph of a function f(x) at a point x = a, which is commonly called the derivative of the function f at a, can be written in the following equivalent ways: f'(a) j(a) d f(x)| f(x) dx d f(a + h) – f(a) lim dx h→0 h |x=a x=a or, if we are looking for a general answer which works for any value of x, f' (x) f (x) d f(x) d (x) f(x +h) – f(x) lim dx dx h→0 h We already know that d (k x") = nk x"-1 dx for n a positive natural number. What about n = 0° is not defined, so we cannot simply write x° and calculate its slope. Instead, we can be sure that xº = 1 for x > 0 so we usually work with the constant function that has the value 1 for all x. 0? This is a special case, because the value of
Question 1: What is the derivative u'(x) of the function u(x) = ao, where ao is a real number? When working directly with limits, we often faced the complication that limits do not always exist, so it was difficult to establish general rules. However, if we limit ourselves to functions which can be analysed (these are called analytic functions) then we can be sure that certain limits we need do exist. One class of functions which are particularly useful in practical problems (in engineering, economics and science, for example) are the polynomials. These can be written in the general form P(x) = ao + a1 x + a2 x² + a3 x³ + ... + an x" , where the natural number n is called the degree of the polynomial. For polynomials P and Q, we can write ( e) + («) - #(ra) + Qla) d d P(3) ) + (Q«) ) = ( P(x) + Q(x) dx dx dx
Transcribed Image Text:Question 1: What is the derivative u'(x) of the function u(x) = ao, where ao is a real number? When working directly with limits, we often faced the complication that limits do not always exist, so it was difficult to establish general rules. However, if we limit ourselves to functions which can be analysed (these are called analytic functions) then we can be sure that certain limits we need do exist. One class of functions which are particularly useful in practical problems (in engineering, economics and science, for example) are the polynomials. These can be written in the general form P(x) = ao + a1 x + a2 x² + a3 x³ + ... + an x" , where the natural number n is called the degree of the polynomial. For polynomials P and Q, we can write ( e) + («) - #(ra) + Qla) d d P(3) ) + (Q«) ) = ( P(x) + Q(x) dx dx dx
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