PLEASE HELP ME. kindly show all your work 1. Prove that∀k ∈ N, 1k + 2k + · · · + nk ∈ Θ(nk+1). 2. Suppose that the functions f1, f2, g1, g2 : N → R≥0 are such that f1 ∈ Θ(g1) and f2 ∈ Θ(g2).Prove that (f1 + f2) ∈ Θ(max{g1, g2}). Here (f1 + f2)(n) = f1(n) + f2(n) and max{g1, g2}(n) = max{g1(n), g2(n)}. THANK YOU3OoyQaGwGPYnEumugVXv

Suppose that f(x, y) = -2z4+3 zy2 + 3y2, z=rcos (t) and y=rsin(t). Find af af and (write your answers as functions of the pair r and t). Ot Or af Or af ICH

6. Sketch: f(x) = [x+ 2] -1, on [-2,2]

Please Show all work so I can better understand ! , Thank You

Let x, y e R. Thenx^2+y^2 = 16 True False defines y as a function of x.

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1) In this problem, let x = 2, the value Ax = dx, dy = f'(x)dx, and Ay = f(x + Ax)-f(x) to complete the table f(x)=√x 1.0000 0.5000 0.1000 0.0100 0.0010 2) dx= Ax dy Ay Ay - dy dy Ay 3) 4) f(x) = 5x √2x+3 Find the first derivative given that dy Find X by implicit differentiation. √√x+y+√√x = 4 dx -лr² h A conical tank (with vertex down) is 10 feet across the top and 12 feet deep. If water is flowing into the tank at the rate of 10 cubic feet per minute, find the rate of change of the depth of the water the instant it is 8 feet deep? 1 V

Discuss the analyticity of f(z) = Re(z 3)   Find whether the given function is analytic or not. Do it satisfying Cauchy Reimann equations.

We are given the following functionsf : Z → Z × Z, f(x) = (x^2, x^3). (a) Find images f({1, 2, 3}) and f(N).(b) Find those preimages f^−1({1, 2, 3, 4}) and f^−1({(1, −1),(4, 8)}) that are welldefined.

See attached image please