taylor-float

taylor-float - Week 2 - Taylor series - Floating point -...

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Week 2 - Taylor series - Floating point - Nested Multiplication (maybe)
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Taylor series f(x) = f(0) + x f'(0) + x 2 f''(0) + x 3 f'''(0) + . .. 2! 3! f(x) = f(c) + (x-c)f'(c) + (x-c) 2 f''(c) + (x-c) 3 f'''(c) 2! 3! f(x+h) = f(x) + h f'(x) + h 2 f''(x) + h 3 f'''(x) + . .. 2! 3!
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Taylor series f(x) = cos 2 (x) f'(x) = -2 cos(x) sin(x) f''(x) = 2 sin 2 (x) - 2 cos 2 (x) f'''(x) = 4 sin(x) cos(x) +4 cos(x) sin(x) = 8 sin(x) cos(x)
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Taylor series f(x+h) = f(x) + h f'(x) + h 2 f''(x) + h 3 f'''(x) + . .. 2! 3! Expand around П/4 to find П/4+0.1 f(П/4+0.1) ≈ f(П/4) + 0.1 f'(П/4) + 0.1 2 f''( П/4 ) + 0.1 3 f'''( П/4 ) 2! 3!
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Taylor series f(П/4+0.1) ≈ 1/2 + 0.1 * (-1) + 0.01*0 + 0.001*4 2 6 f(П/4+0.1) ≈ 0.400666666. .. Real: f(П/4+0.1) ≈ 0.4006653346
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Error bounds: f(x+h) = f(x) + h f'(x) + h 2 f''(x) 2! Has error bound: h 3 f'''( ξ ) 3! where x < ξ < x+h Taylor series
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f'''(x) = 4 sin(x) cos(x) +4 cos(x) sin(x) = 8 sin(x) cos(x) f''''(x) = 8 [cos 2 (x) - sin 2 (x)] max h 4 f''''(z) П/4<z<П/4+0.1 4! = (0.1)
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taylor-float - Week 2 - Taylor series - Floating point -...

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