IC ::={Σandk=1ak3k: a. € {0,2}}Let C be the set defined in Problem 1 above. Define the functions φ : C → [0, 1]and ψ : [0, 1] → [0,1] as follows(Σ)-Σαk=1ak22kψ(x) = sup{φ(y) : y € C and y Sa}.What is the range of the function & from last problem?

The given functions are ϕ:C→[0,1] defined as ϕ∑k=1∞ak3k=∑k=1∞ak22k and ψ:[0,1]→[0,1] defined as…

Answered: α c={ {Σ k=1 ak 3k and : ακ € {0,2}}…

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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1. Find curl(v) of the following in terms of (x1, x2, x3) and (e1, e2, e3). If curl(v) = 0, find the function f such that Vf = v. a. v(x) = (x² + x + x3)-'(x1e1 + x2€2 + X3@3). b. v(x) = (xỉ + X2X3)e1 + (x3 + X1X3)e2 + (xž + x2X1)e3 c. v(x) = e"1(sin(x2) cos(x3)e1 + sin(x2) sin(x3)e2 + cos(x2)e3)
q7
9 c and d
Theorem 11: Let f(t) and g(t) be two functions of class A and let L[f(p);1]= f(t) and L"[g(p}:t]= g(1). Then L"[7(p) . g(p);t]= [, S(u)g(t-u)du = f*g.
2. Let P denote the path that goes in straight lines from (1,1,1) to (-1,-1,1) to (-1,-1,-1) to (1,1,-1) and then back to (1,1, 1). 2xyz+z el +2 x²z+sin(z) Compute fp F. dr, where F(x, y, z) = P
6. Compute fryx (1,–3) and fyxy (1, -3) for f(x,y) = (x + y)*.
2 2 3. Evaluate f(x²+y²-ixy )dz from 1-i to 2+i2 along the following paths (a) horizontally from 1-i to 2-i then vertically to 2+i2 (b) directly on the line from 1-i to 2+i2
3. (.) Denote the number of derangements of {1, 2,..., n} by Dn. Using the formula for the number of the derangement that we found in class, provide a proof of the relation D, = (n – 1)(D-2 + Dn-1); (n = 3,4, 5, ...).
5) Suppose a path C is defined by r(t), 2 and r(8)= . If f is a differentiable function satisfying f(3,9) = 6 and f(6, 10) = 14, what is the value of Vf dr?

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α c={ {Σ k=1 ak 3k and : ακ € {0,2}} Let C be the set defined in Problem 1 above. Define the functions φ : C → [0,1] and ψ : [0, 1] → [0, 1] as follows (Σ)-Σ. Φ k=1 ψ(x) = sup{φ(y) : y € C and y < x}. = What is the range of the function & from last problem?