By Alan L. T. Paterson
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Post yr word: First released March 2d 1979
Introducing calculus on the easy point, this article covers hyperreal numbers and hyperreal line, non-stop capabilities, critical and differential calculus, basic theorem, endless sequences and sequence, limitless polynomials, topology of the genuine line, and traditional calculus and sequences of services. merely highschool arithmetic wanted.
From 1979 version
This old account begins with the Greek, Hindu, and Arabic assets that constituted the framework for the advance of infinitesimal tools within the seventeenth century. next chapters speak about the arithmetization of integration equipment, the position of research of distinct curves, techniques of tangent and arc, the composition of motions, extra.
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Additional info for Amenability
Hence the slope of the tangent is decreasing. This means that the second derivative must be negative. Chapter 3 : Introduction to Derivatives page 8 of 20 c First Year Calculus W W L Chen, 1982, 2008 The above heuristics can be summarized by the following result on stationary points and second derivatives which we shall establish formally in Chapter 8. PROPOSITION 3E. Suppose that I is an open interval containing a real number a. Suppose further that the function f (x) is differentiable at every x ∈ I, and that f (a) = 0.
Consider next the function f (x) = x3 . At any given point x, let us consider a small increment ∆x and the behaviour of the function as x changes to x + ∆x. Clearly the value f (x) changes to f (x + ∆x), giving rise to the error ∆f = f (x + ∆x) − f (x) = (x + ∆x)3 − x3 = 3x2 ∆x + 3x(∆x)2 + (∆x)3 , and the relative error f (x + ∆x) − f (x) ∆f = = 3x2 + 3x∆x + (∆x)2 . ∆x ∆x As ∆x is taken to be very small, we have respectively the approximations ∆f ≈ 3x2 ∆x and ∆f ≈ 3x2 . ∆x Note again 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.
STATIONARY POINTS. We now use our knowledge on derivatives to further our understanding of the functions. 2. 13. Let us continue our investigation of the function f (x) = 5 + x−3 . Simple calculation gives f (x) = −3x−4 . It follows that there is no stationary point. Next, note that f (x) < 0 whenever x = 0. It follows that the function is decreasing in the open intervals (−∞, 0) and (0, +∞). We now supplement our earlier effort with this extra information. 14. Let us continue our investigation of the function f (x) = 1/(x − 1)(x − 2).