Read e-book online Applied Exterior Calculus (1985) PDF

Calculus

By Dominic G. B. Edelen

ISBN-10: 0471807737

ISBN-13: 9780471807735

This ebook supplies an utilized creation to external calculus for higher department undergraduates and starting graduate scholars. improvement is operational with an emphasis on computation talent and trouble-free geometric notions. attention is restricted to neighborhood questions. The publication additionally positive factors absolutely labored out examples and issues of solutions.

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Example text

Clearly the value f (x) changes to f (x + ∆x), giving rise to the error ∆f = f (x + ∆x) − f (x) = (x + ∆x)2 − x2 = 2x∆x + (∆x)2 , and the relative error ∆f f (x + ∆x) − f (x) = = 2x + ∆x. ∆x ∆x Chapter 3 : Introduction to Derivatives page 12 of 20 c First Year Calculus W W L Chen, 1982, 2008 As ∆x is taken to be very small, we have respectively the approximations ∆f ≈ 2x∆x and ∆f ≈ 2x. ∆x Note that the first of these suggests that ∆f is essentially directly proportional to ∆x, and the second shows that the relative error is an approximation of the derivative.

Now  < 0 if x < −1,    2  = 0 if x = −1, 2 − 2x f (x) = 2 > 0 if −1 < x < 1, (x + 1)2     = 0 if x = 1, < 0 if x > 1. It follows that the function has a local minimum at x = 0. Also it has points of inflection at x = −1 and at x = 1. Furthermore, the slope of the curve is decreasing in the intervals (−∞, −1) and (1, ∞), and increasing in the interval (−1, 1). 3. Derivatives of the Inverse Trigonometric Functions The purpose of this section is to determine the derivatives of the inverse trigonometric functions by using implicit differentiation and our knowledge on the derivatives of the trigonometric functions.

If we write u = f (x) = x3 + 1 and y = g(u) = u2 , then (g ◦ f )(x) = g(f (x)) = g(x3 + 1) = (x3 + 1)2 . It follows that our original function is really a composition of two functions. As we vary x, the value u = f (x) changes at the rate of du/dx. This change in the value of u = f (x) in turn causes a change in the value of y = g(u) at the rate of dy/du. It is therefore not unreasonable to expect the change in x causes a change in y at the rate (dy/du)(du/dx). Indeed, this is the case, and the following result is known as the Chain rule for differentiation which we shall prove in Chapter 8.

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Applied Exterior Calculus (1985) by Dominic G. B. Edelen


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