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MA1S11 Tutorial Sheet 91 8-11 December 2015 Useful facts: • Extremum points: If f 0 (x) > 0 a function f is increasing at x, if it is negative it is decreasing, if f 0 (x) = 0 then x is a critical point. If f 00 (x) > 0 at a critical point it is a minimum, if f 00 (x) < 0 at a critical point it is a maximum, if f 00 (x) = 0 it is undecided, it could be a point of inflection, a point joining a concave up f 00 (x) > 0 interval from a concave down, f 00 (x) < 0, interval. • Extrema: A point x = a is an relative or local maximum if there is an open interval I containing a such that f (a) ≥ f (x) for all x in I, a point x = a is an relative or local minimum if there is an open interval I containing a such that f (a) ≤ f (x) for all x in I. • Absolute extrema: A point x = a is an absolute or global maximum if f (a) ≥ f (x) for all x in the domain, it is an absolute or global minimum if f (a) ≤ f (x) for all x in the domain. • Newton’s method to find f (x) = 0 start with an initial guess x1 and iterate using xn+1 = xn − f (xn ) f 0 (xn ) (1) until it converges or is obviously not working. • The anti-derivative and the indefinite integral: For a function f (x) the function F (x) is the anti-derivative if dF (x) = f (x) (2) dx The indefinite integral is the family of all anti-derivatives Z f (x)dx = F (x) + C (3) where C is the arbitrary constant of integration. • Integration table 1 Stefan Sint, sint@maths.tcd.ie, see also http://www.maths.tcd.ie/~sint/MA1S11.html 1 R f (x) f (x)dx n+1 xn (n 6= −1) xn+1 + C sin x − cos x + C cos x sin x + C • Linearity: if Z Z f (x)dx = F (x) + C, g(x)dx = G(x) + C (4) then Z Z λf (x)dx = λF (x) + C, [f (x) + g(x)]dx = F (x) + G(x) + C (5) where λ is a constant. Questions 1. (2) Find the absolute maximum of 1 f (x) = − x4 + x3 + 2x2 + 1 4 2. (2) Use Newton’s method to find as a starting value. √ 5 (6) 3 to within 0.01 by solving x5 = 3, with x1 = 2 3. (4) Find the indefinite integral of x3 − 4x, √ √ √ x + 1/ x, x3 − 3 x and x(1 + x4 ). Extra Questions The questions are extra; you don’t need to do them in the tutorial class. 1. Evaluate Z 1 dx (7) 1 − sin x Hint: first multiply by 1 + sin x above and below the line, then use the identity cos2 x + sin2 x = 1. 2. Find the points of inflection of f (x) = x4 − 6x2 + 12x + 24 2 (8)