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(2)Suppose that g(x) is a continuous function on an interval a;b such that g(x) >0 for all x Show that Z b a g(x)dx>0 Solution Since g(x) 6= 0 on a;b the function 1 g is de ned and continuous on a;b Hence there is M > 0 so that 1 g(x) < M for all x This means thatBy Gauss f(x) must have a linear factor in Z, f(x) = (ax b)g(x) Looking at the leading coe cients, we must have that adivides 1 So a= 1 Possibly replacing g(x) by g(x) we may assume that a= 1 If m= bthen f(x) = (x m)g(x) m2Z is a zero of f(x) Considering the constant coe cients mmust divide aThe Fundamental Theorem of Calculus, Part 1 If f is continuous on a,b, then the function g defined by g(x) = Z x a f(t)dt a ≤ x ≤ b is continuous on a,b and differentiable on

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F Bn B where f(x1,x2,,xn) is a Boolean expression in x1,x2,,xn Examples f(x,y,z)=xyx'z is a 3variable Boolean function The function g(x,y,z,w)=(xyz')(x'y'w)xyw' is also a Boolean function Definition Two Boolean expressions are said to be equivalent if their corresponding Boolean functions are the sameTheorem (713) If g is Riemann integrable on a,b and if f(x) = g(x) except for a finite number of points in a,b, then f is Riemann integrable and Z b a f = Z b a g Theorem (715) Suppose f,g 2 Ra,b Then (a) if k 2 R, kf 2 Ra,b and Z b a kf = k Z b a f (b) f g 2 Ra,b and Z b a (f g) = Z b a f Z b a g (c) If f(x) g(x) 8x 2Title INBZASDINRUpdf Author kkasprzak Created Date AM

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