What's the answer to the 2nd question? Please see the attached screenshots for the quesiton. In this course, we have been exploring the elementary foundations of applied mathematics. Wehave been studying limits, and have found that they allow us to calculate rates of change offunctions, find maximum and minimum values of functions, and also to approximate functionvalues. In practical applications, one first collects data and finds a function which fits the datawell. 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 inEngland (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 whatare mathematically the same ideas. Each notation has advantages and disadvantages, and theycontinue 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 derivativeof the function f at a, can be written in the following equivalent ways:f'(a)j(a)d f(x)|f(x)dxdf(a + h) – f(a)limdxh→0h|x=ax=aor, 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)limdxdxh→0hWe already know thatd(k x") = nk x"-1dxfor 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 surethat 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 canbe analysed (these are called analytic functions) then we can be sure that certain limits we needdo 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 formP(x) = ao + a1 x + a2 x² + az x³ ++ an x" ,where the natural number n is called the degree of the polynomial. For polynomials P and Q, wecan write)=dddP(x)dxP(x)+Q(x)dxdxQuestion 2: Use this property of polynomials to calculate the derivative of 3 + x³, by firstcalculating the derivatives of the constant function 3 and the function x³, and then addingthem. Show that you get the same answer using limits for the function 3+x³ by direct calculation.

Answered: Image /qna-images/answer/b255c8ae-b4d5-440b-802c-95cae23ba390.jpg

Question 2: Use this property of polynomials to calculate the derivative of 3 + x³, by first calculating the derivatives of the constant function 3 and the function x³, and then adding them. Show that you get the same answer using limits for the function 3+x³ by direct calculation.

Question 2: Use this property of polynomials to calculate the derivative of 3 + x³, by first calculating the derivatives of the constant function 3 and the function x³, and then adding them. Show that you get the same answer using limits for the function 3+x³ by direct calculation.

Calculus: Early Transcendentals

8th Edition

ISBN:9781285741550

Author:James Stewart

Publisher:James Stewart

Chapter1: Functions And Models

Section: Chapter Questions

Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...

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