print barcode image c# Applications of the Derivative in Software

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CHAPTER 3 Applications of the Derivative
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8 A body is launched straight down at a velocity of 5 ft /sec from height 400 feet How long will it take this body to reach the ground 9 Sketch the graph of the function h(x) = x/(x 2 1) Exhibit maxima, minima, and concavity 10 A punctured balloon, in the shape of a sphere, is losing air at the rate of 2 in3 /sec At the moment that the balloon has volume 36 cubic inches, how is the radius changing 11 A ten-pound stone and a twenty-pound stone are each dropped from height 100 feet at the same moment Which will strike the ground rst 12 A man wants to determine how far below the surface of the earth is the water in a well How can he use the theory of falling bodies to do so 13 A rectangle is to be placed in the rst quadrant, with one side on the x-axis and one side on the y-axis, so that the rectangle lies below the line 3x +5y = 15 What dimensions of the rectangle will give greatest area 14 A rectangular box with square base is to be constructed to hold 100 cubic inches The material for the base and the top costs 10 cents per square inch and the material for the sides costs 20 cents per square inch What dimensions will give the most economical box 15 Sketch the graph of the function f (x) = [x 2 1]/[x 2 +1] Exhibit maxima, minima, and concavity 16 On the planet Zork, the acceleration due to gravity of a falling body near the surface of the planet is 20 ft /sec A body is dropped from height 100 feet How long will it take that body to hit the surface of Zork
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The Integral
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40 Introduction
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Many processes, both in mathematics and in nature, involve addition You are familiar with the discrete process of addition, in which you add nitely many numbers to obtain a sum or aggregate But there are important instances in which we wish to add in nitely many terms One important example is in the calculation of area especially the area of an unusual (non-rectilinear) shape A standard strategy is to approximate the desired area by the sum of small, thin rectangular regions (whose areas are easy to calculate) A second example is the calculation of work, in which we think of the work performed over an interval or curve as the aggregate of small increments of work performed over very short intervals We need a mathematical formalism for making such summation processes natural and comfortable Thus we will develop the concept of the integral
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Antiderivatives and Indefinite Integrals
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THE CONCEPT OF ANTIDERIVATIVE
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Let f be a given function We have already seen in the theory of falling bodies (Section 34) that it can be useful to nd a function F such that F = f We call
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The Integral
such a function F an antiderivative of f In fact we often want to nd the most general function F , or a family of functions, whose derivative equals f We can sometimes achieve this goal by a process of organized guessing Suppose that f (x) = cos x If we want to guess an antiderivative, then we are certainly not going to try a polynomial For if we differentiate a polynomial then we get another polynomial So that will not do the job For similar reasons we are not going to guess a logarithm or an exponential In fact the way that we get a trigonometric function through differentiation is by differentiating another trigonometric function What trigonometric function, when differentiated, gives cos x There are only six functions to try, and a moment s thought reveals that F (x) = sin x does the trick In fact an even better answer is F (x) = sin x + C The constant differentiates to 0, so F (x) = f (x) = cos x We have seen in our study of falling bodies that the additive constant gives us a certain amount of exibility in solving problems Now suppose that f (x) = x 2 We have already noted that the way to get a polynomial through differentiation is to differentiate another polynomial Since differentiation reduces the degree of the polynomial by 1, it is natural to guess that the F we seek is a polynomial of degree 3 What about F (x) = x 3 We calculate that F (x) = 3x 2 That does not quite work We seek x 2 for our derivative, but we got 3x 2 This result suggests adjusting our guess We instead try F (x) = x 3 /3 Then, indeed, F (x) = 3x 2 /3 = x 2 , as desired We will write F (x) = x 3 /3 + C for our antiderivative More generally, suppose that f (x) = ax 3 + bx 2 + cx + d Using the reasoning in the last paragraph, we may nd fairly easily that F (x) = ax 4 /4 + bx 3 /3 + cx 2 /2 + dx + e Notice that, once again, we have thrown in an additive constant You Try It: Find a family of antiderivatives for the function f (x) = sin 2x x 4 + ex
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