The monotonicity property of the integral implies that if the functions g,h : 0,00) → R are continuous and g(x) S h(x) for all x 2 0, then ghfor all r 2 0. 0 0 Use FTC1 to show that each one of the f...
s h) for all z c l e Sub-problem 3. Recall monotonicity of integration: If h() S [-1, 1], then This just says that integruls preserve inequalities 1. Explain why this is true graphically 2·Let g be continuous on [0,1]. Use the previous item, and the fact that to show that 3. Use the first two items to show that if g is bounded, say Ig(r)l s M for z [0, 1], then first two derivatives are continos on is...
2. Let DCR, and suppose the functions f:D + R and g: D R are continuous at to ED. Use the - definition of continuity to prove that f+g and fg are both continuous at ro e D: that is, prove that for every e > 0 there exists 8 >0 such that (+9)(2)-(+9)(20) <and (9)(x) - (49)(20) << whenever re D and r-rol < 8. (Hint. Use inequalities similar to those we used to prove the cor- responding results...
(a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g? (a) Can there be differentiable functions f,g (on R) with g(0)-f(0) 0 and f()g(x) for all z E R? What about if we ask (only) for continuous functions f,g?
3. Let f, g : a, b] → R be functions such that f is integrable, g is continuous. and g(x) 〉 0 for all x є a,b]. Since both f, g are bounded, let K 〉 0 be such that |f(x) K and g(x) < K for all x E [a,b (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that for all i 2. (b) Let P be a...
3. Let f, g : [a,b] → R be functions such that f is integrable, g is continuous, and g(x) >0 for all r E [a, b] Since both f,g are bounded, let K >0 be such that lf(z)| K and g(x) K for all x E [a3] (a) Let n > 0 be given. Prove that there is a partition P of [a, b such that U (P. f) _ L(P./) < η and Mi(P4)-mi(P4) < η for all...
3. Let f, g : a, bl → R be functions such that f is integrable, g is continuous. and g(x) >0 for al x E [a, b]. Since both f,g are bounded, let K> 0 be such that f(x)| 〈 K and g(x)-K for all x E la,b] (a) Let η 〉 0 be given. Prove that there is a partition P of a,b] such that for all i (b) Let P be a partition as in (a). Prove...
2. Consider the vector space C([0, 1]) consisting of all continuous functions f: [0,1]-R with the weighted inner product, (f.g)-f(x) g(x) x dr. (a) Let Po(z) = 1, Pi(z) = x-2, and P2(x) = x2-6r + 흡 Show that {Po, pi,r) are orthogonal with respect to this inner product b) Use Gram-Schmidt on f(x)3 to find a polynomial pa(r) which is orthogonal to each of po P1 P2 You may use your favorite web site or software to calculate the...
Let f,g be continuous functions on [a,b] with for all (a) show that there are such that (b) using (a) prove that there is a strictly between x1 and x2 such that f(x) 0 rE a, b a, 1 ( f(xgf(x) < g[x2}f{x)) We were unable to transcribe this imagef(r)g()da g(e) f(x)da f(x) 0 rE a, b a, 1 ( f(xgf(x)
S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9) Prove that if the function f is continuous on a, b, then there is c E [a, b such that f(x)dax a Ja f(e) S f(r)da= g(x)dz. Prove a,bsuch that (8) Suppose f and g are continuous functions on that there is ro e (a, b) such that f(ro) = g(xo). (9)...
Problem 2. Let C[0, 1] be the set of all continuous functions from [0, 1] to R. For any f, g є Cl0, 11 define - max f(x) - g(z) and di(f,g)-If(x) - g(x)d. a) Prove that for any n 2 1, one can find n points in C[O, 1 such that, in daup metric, the distance between any two points is equai to 1. b) Can one find 100 points in C[0, 1] such that, in di metric, the...