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124 integration par t 4 integration 125 vedic

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Unformatted text preview: zero (tan θ = 0 ) then that point must be a maxim um maximum um, minimum or inflexion point. We mentioned earlier that pol ynomials ar e to functions w ha t r a tionals polynomials are wha ra hat p ol ar e to r eal n umber s . We can do all our calculations of reals to any degree are real number umbers of accuracy using the rationals. Likewise we may approximate any single-valued function by a suitable polynomial. In fact, thinking more deeply on what we just covered in the last few chapters, we may be more specific. We may piece-wise approximate any single-valued function by a suitable choice of polynomials of maximum degree 3 or cubic. We just saw how a function is either increasing, decreasing or changing direction. Polynomials of degree 0,1,2, and 3 are sufficient to cover all these variations. We only need to identify the maxim um minim um and maximum minimum um, um, infle lexion inf lexion points of the function thru which the approximating polynomials must interpolate. Note : Later the student will learn ROLLE’S THEOREM, which is a special case of the MEAN VALUE THEOREM. In the special case when f(a) = f(b) then there exists _ _ some MEAN point x in the interval [a, b] such that f ’( x ) = 0. Is it possible to find _ the point x now ? (see note page 64) 120 2 4. Po Po i n t s o f I N F L E X I O N A point at which a curve changes its shape in known as a point of inf lexion. infle lexion. infle lexion inf lexion = a bending in the curve, a change in curvature or shape, a change in direction, modulation of sound. (flexible, inflexible) We have seen in the previous sections that at a maxim um or minim um maximum minimum point a cur ve changes its shape. maximum : concave down and going left to right from rising to falling. minimum : concave up and going left to right from falling to rising. There is another type of change in shape possible: concave up to concave down or vice versa concave down to concave up. Consider the function f(x) = x 3 + 1 . Please see the graph. f ’(x) = 3x 2 a n...
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