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(5 Marks) ii. Let f:[0, ] → R be the function such that f(0) = cos...
5. Let f : [a, b] → R be bounded, a : [a, b] → R monotonically increasing, and P a partition of [a, b]. (a) Define upper and lower Riemann-Stieltjes sums of f with respect to P and a. (b) Let P' be the partition obtained from P by inserting one additional point x' into the subinterval (2k-1, xk] of P. Prove that for the lower and upper Riemann- Stieltjes sums of f we have L(P, f, a) <L(P',...
Please all thank you Exercise 25: Let f 0,R be defined by f(x)-1/n, m, with m,nENand n is the minimal n such that m/n a) Show that L(f, P)0 for all partitions P of [0, 1] b) Let mE N. Show that the cardinality of the set A bounded by m(m1)/2. e [0, 1]: f(x) > 1/m) is c) Given m E N construct a partition P such that U(f, Pm)2/m. d) Show that f is integrable and compute Jo...
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...
[EUM 114 1. Let f(x) be a function of period 2 (a) over the interval 0<x<2 such that f(x) = - f(x)pada selang Diberikan f(x) sebagai fungsi dengan tempoh 2t yang mana 0<x<2m Sketch a graph of f (x) in the interval of 0 <x< 4 (1 marks/markah) Demonstrate that the Fourier Series for f(x) in the interval 0<x< 2n is (ii) 1 2x+-sin 3x + 1 sin x + (6 marks/markah) Determine the half range cosine Fourier series expansion...
5 (10 pts) Let b 0 be a number and f)for (o.b, Lt artition of [O, b, wherefor0, 1,2, ns bea 72 ( 1) Find the upper sum U(f, P) . (2) Find lim Uf. P). 5 (10 pts) Let b 0 be a number and f)for (o.b, Lt artition of [O, b, wherefor0, 1,2, ns bea 72 ( 1) Find the upper sum U(f, P) . (2) Find lim Uf. P).
Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the largest integer less than or equal to r. (a) Evaluate the upper and lower sums U(f, P) and L(f, P) of f with respect to or if P is the partition {0、름, î,3.3.2) of [O, 2]. 4 42 (b) Explain why f є [0,2] and use results in part (a) to give a range of fda. Let f(x)-12.2-2x-11 and a(z) = x2 + 12n, where Ir] is the...
Q-5: [5x1 marks] Let f(x) = 10 + (x – 2)4 a) Find f'(x) and f'(x). b) Find the intervals on which f is increasing or decreasing. c) Find the local maximum and minimum of f, if any. d) Find the intervals on which the graph of f is concave up or concave down. e) Find the points of inflection, if any.
You will calculate L5 and Us for the quadratic function y4x+ 13 between x 0 andx3 Enter Ax NumberNumberx1 NumberNumber x3 Number x4 Number x5Number Enter the upper bounds on each interval: M1 Number M2 Number M3 Number M4 Number M5 Number Hence enter the upper sum U5 Number Enter the lower bounds on each interval: m2 Number ms Number m Number ,m3 Number m4 Number Hence enter the lower sum L5: Number
true or false The real valued function f : (1,7) + R defined by f(x) = 2is uniformly contin- uous on (0,7). Let an = 1 -1/n for all n € N. Then for all e > 0) and any N E N we have that Jan - am) < e for all n, m > N. Let f :(a,b) → R be a differentiable function, if f'() = 0 for some point Xo € (a, b) then X, is...
Consider the following function. f(x) = cos(x) - sin(x), (0, 2) (a) Find the critical numbers of f, if any. (Enter your answers as a comma-separated list.) (b) Find the open intervals on which the function is increasing or decreasing (Enter your answers using interval notation. If an answer does not exist, enter DNE.) increasing х decreasing X (c) Apply the First Derivative Test to identify all relative extrema. (If an answer does not exist, enter DNE.) relative minimum (X,Y)...