f(x) = Ω(g(x)) means there are positive constants c and x, such that f(x) >= cg(x) for all x ≥ x0 x^4 - 50x^3 + 1 = Ω(x^4) => x^4 - 50x^3 + 1 = Ω(x^4) Let's assume c = 0.5 => x^4 - 50x^3 + 1 = Ω(x^4) => x^4 - 50x^3 + 1 = 0.5(x^4) => 0.5x^4 >= 50x^3 - 1 => x^4 >= 100x^3 - 2 This above equation is true, for all x >= 100 so, x^4 - 50x^3 + 1 = Ω(x^4) for c = 0.5 and x0 = 100, using the definition of Big-Ω
2. (10 points) Given the following functions: f()= r* - 50x3 +1, g(x) = 24 Show...
3. Show that (a) the function g: R” → R, given by g(x) = ||2||2, is convex. (b) if f : RM → R is convex, then g:R" + R given by g(x) = f(Ax – b) is also convex. A here is an m x n matrix, and b ERM is a vector. You may use any of the results we covered in class (but the definition of convexity may be an easy way to do this, and gives...
*9. For each of the following pairs of functions, determine the highest order of contact between the two functions at the indicated point xo: (e) f,g : R-R given by f(x)and g(x) 1+2r ro0 (f) f, g : (0, oo) → R given by f(r) = In(2) and g(z) = (z-1)3 + In(z): zo = 1. (g) f.g: (0, oo) -R given by f(x)-In(x) and g(x)-(x 1)200 +ln(x); ro 1 x-1)200
*9. For each of the following pairs of functions,...
3. (10 points) Let F denote the vector space of functions f: R + R over the field R. Consider the functions fi, f2. f3 E F given by f1(x) = 24/3, f2() = 2x In(9), f() = 37*+42 Determine whether {f1, f2, f3} is linearly dependent or linearly independent, and provide a proof of your answer.
Question 1 (10 points) Which of the following functions is not an onto function? f: R → R, where f(x) = 2x + 7 f: R – R, where f(x) = 6x - 1 Of: Z – Z, where f(n) = n + 3 f: Z - Z, where f(n) = 3n + 1
1-> X- Let f :S → R and g:S → R be functions and c be a cluster point. Assume lim f (x), lim g(x) exists. Using the definition of the limit prove the following lim(af (x) + Bg(x)) =a lim f(x) + Blim g(x) for any a,ßeR xc XC X-> b. lim( f(x))} = (lim f(x)) f(x) lim f (x) c. If (Vxe S)g(x) # () and lim g(x)() then prove lim X-C XC 10 g(x) lim g(x) X-C
It would be very useful to have a theory about computability of functions R" -> R. Given that there Q.2 are an uncountable number of real numbers, we would need to start with a definition of which numbers are themselves computable. A natural definition would be that a real number x is computable if it is the limit of computable sequence of rational numbers (so that we can compute it to whatever accuracy we like). More carefully V a Definition:...
8. Suppose that we are given the following information about the functions f, g, h and k and their derivatives; • f(1) = 3 • f'(1) = 2 • g(1) = 4 • g'(1) = -2 • h(1) = 9 . h'(1) = -1 k(1) = 10 • k'(1) = -3 (e) (5 points) Set F(x) = log2[f(x) + g(x)]. Compute F'(1). (f) (5 points) Set F(T) = log: [f(r)g(r)h(r)k(r)]. Compute F'(1).
Question 2 (20 points): Consider the functions f(x, y)-xe y sin y and g(x, y)-ys 1. Show f is differentiable in its domain 2. Compute the partial derivatives of g at (0,0) 3. Show that g is not differentiable at (0,0) 4. You are told that there is a function F : R2 → R with partial derivatives F(x,y) = x2 +4y and Fy(x, y 3x - y. Should you believe it? Explain why. (Hint: use Clairaut's theorem)
Question 2...
15 points LarCAApCalc2 2.7.006. Consider the functions given below. Find the following. f(x) = 4x 7, g(x) = 1 -x (a) (fg)(x) (f g)(x) (b) (c) (fg)(x) (d) (flg)(x) What is the domain of f/g? (Enter your answer using interval notation.) A Your work in question(s) 1, 2, 3, 4, 5, 6 will also be submitted or saved. Submit Assignment Save Assignment Progress -15 points LarCAApCalc2 2.7.010 Consider the functions given below. Find the following. f(x) = x-16, g(x) =...
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Question 3. E-6 Proof (Show Working) 10 points 249 Show that f:RR defined by f(x) is continuous at x = 7 using only r +3 cosa the epsilon-delta definition of continuity. Note that we want you to do it the hard way: you are not allowed to use the limit laws or the combination of continuous functions theorem or similar. You must give an 'e-δ style proof Solution: Let ε > 0 be given and choose δ =...