Theorem 20.8 (The Mean Value Theorem for Integral Calculus). Let f a, bR be continuous, and g a, ...
Answer C 6. Let f be a continuous function on [0, oo) such that 0 f(z) Cl- for some C,e> 0, and let a = fo° f(x) da. (The estimate on f implies the convergence of this integral.) Let fk(x) = kf(ka) a. Show that lim00 fk(x) = 0 for all r > 0 and that the convergence is uniform on [8, oo) for any 6> 0. b. Show that limk00 So ()dz = a. c. Show that lim00 So...
Problem 4: Let f: [0, 1] → R be an integrable function that is continuous at 0. Prove that lim f(") dx = f(0). n+Jo [ Hint: there are several approaches. It might help to first show that for a fixed 0 <b< 1, we have limn700 Sº f(x) dx = b. f(0). ]
please answer this with work. this is discrete math. 4. In Calculus, the Mean Value Theorem states: If /(x) is defined and continuous on (a, b) and differentiable on (a,b), then there is at least one number con (a,b) such that f'(c) = f(b)-f(a) b-a Hint: Let S = lſis defined and continuous on (a,b) and differentiable on (a,b)} Translate the sentence into a symbolic sentence with quantifiers. a. b. What is the simplified negation of your symbolic sentence?
2a) Let a, b e R with a < b and let g [a, bR be continuous. Show that g(x) cos(nx) dx→ 0 n →oo. as Hint: Let ε > 0, By uniform continuity of g, there exists δ > 0 such that 2(b - a Choose points a = xo < x1 < . . . < Xm such that Irh-1-2k| < δ. Then we may write rb g (z) cos(nx) dx = An + Bn where 7m (g(x)...
Question Details SCalcET8 5.3.504.XP.MI 12. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the function. cos(V5t) dt G(x) G'(x) = Show My Work (Optional) Question Details SCalcET8 5.2.074 6. Express the limit as a definite integral. n 9 lim 1 1 (i/n) nco n j = 1 1 dx JO Show My Work (Optional) Question Details SCalcET8 5.3.504.XP.MI 12. Use Part 1 of the Fundamental Theorem of Calculus to find the derivative of the...
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...
8. Let f and g be scalar functions with continuous partial derivatives, and let C and S satisfy the conditions of Stokes's Theorem. Verify each identity. (a) dr = Vg) N ds X (b) dr 0 (e) 8. Let f and g be scalar functions with continuous partial derivatives, and let C and S satisfy the conditions of Stokes's Theorem. Verify each identity. (a) dr = Vg) N ds X (b) dr 0 (e)
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...
Problem 6. (Mean Value Property) Let f : RR be a function with continuous second derivative. (a) Suppose f"( to f( ). 0 for all r E IR. P al rove that the average value of f on the interval a, bs equ f, onla b is equal tore !) Prove intervals la, b, the average。 (b) (Braus) Supposeerall Hint: To prove the second part, try to use the fundamental theorem of calculus or Jensen's inequality. Problem 6. (Mean Value...
Problem 6. (Mean Value Property) Let f : RR be a function with continuous second derivative. (a) Suppose f"( to f( ). 0 for all r E IR. P al rove that the average value of f on the interval a, bs equ f, onla b is equal tore !) Prove intervals la, b, the average。 (b) (Braus) Supposeerall Hint: To prove the second part, try to use the fundamental theorem of calculus or Jensen's inequality. Problem 6. (Mean Value...