1.) T(n) = T(n/3) + c f(n) = versus nlog, a = Which is growing faster: f(n) or nlos, a? Which case of the Master Theorem does that potentially put us in? If you're potentially in case one or three, is it possible to find an epsilon which makes either f(n) = O(nlogs a-) (if you're in case one) or f(n) = N(nlossa+e) (if you're in case three) true? Choose one or show an inequality. If you're potentially in case three and there is an e, try to find a constant d < 1 such that af (n/b) ≤df (n) for large enough n's. What can you conclude?
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- 1- Let f: R⟶R be a function defined as f(x)=〖(x+2)〗^2. Determine whether the function is one to one. 2- Verify that f(x)=5x-1 and g(x)=(x+1)/5 are inverse functions.Use the master theorem to determine the run-time complexity of the following function. Clearly indicate the values of a, b, and c that you are using.Q1 a)Determine the Lagrange form of the interpolating polynomial P(x) that interpolates a function f(x) at x = 0, h and 3h, where h > 0. (Multiply the linear factors together, but leave P(x) as a sum of 3 quadratics in the variable x.) DELIVERABLES: All your work in constructing the polynomial. b) Derive the quadrature formula of the form a0f(0) + a1f(h) + a2f(3h) for approximating I = R 3h 0 f(x)dx that results from approximating the integral I by I ≈ R 3h 0 P(x)dx. Note: if you know only 3 function values of f(x) and they are at 3 unequally-spaced points 0, h and 3h, then this kind of quadrature formula can be used to approximate I. DELIVERABLES: All your work in deriving the quadrature formula.
- 5. (a) Write a function that solves the general linear least-squares problem. The inputs to your function should be a vector of a values, a vector of measured y values, and an anonymous function that calculates a single row of the Z matrix. Inside your function, use the inputted anonymous function to create Z row-by-row, then use Z to create the normal equations. Solve these normal equations to obtain the computed coefficients that define the best-fit of your model. Your function should output these calculated coefficients. Again, include an error check that makes sure the input vectors are the same size. You may not use any built-in MATLAB functions to solve the normal equation. You can use any functions you have developed solves Ax = b via LU-decomposition is one option. e.g. a function you have that (b) Test your function in (a) to fit the following model to the given dataset (see belo y = a + bx (c) Test your function in (a) to fit the following model to the given dataset…Arrange the following function in increasing order of their asymptotic growth. The domain of the functions belong to the natural numbers: 2^m, m^4, m^4/5, m^5/6,lg (m!), Ig^6m, m! Furthermore, justify your answers.Consider the function f(x) = 1 3x+1 . We approximate f(x) by the Lagrange interpolating polynomial P₂ (x) at the points xo = 1, x₁ = 1.5 and.x₂ = 2. A bound of the theoretical error of this approximation at x = 1.8 is:
- Let f: A→B and g: B→C be two functions. If g◦f is a one-to-one function, prove that f is one-to-oneUse the Quine McCluskey algorithm to simplify the function:F(w, x, y, z) = Σ(2, 6, 9, 10, 11, 13, 14, 15)Show all work – no credit will be given for an answer without accompanying workI'm having trouble understanding what exactly makes a linear ordinary differential equation "linear". I know that the major advantage of the linear ODEs is that a linear combination of particular solutions gives another particular solution. Characterized by additivity and homogeneity, this means that the output for a sum of inputs is equal to the sum of outputs for each individual input (e.g., f(3) + f(5) = f(3+5)= f(8)) and scaling the input by a factor scales the output by the same factor (e.g., f(6x) =6f(x)...but how would this look? Like say if I had 4e^5(t-1) satisfying the differential equation dy/dt=5y where y(1)=4, what would that "linear combination" look like?
- 4. Simplify the following Boolean function F, together with the don’t-care conditions d, and then express the simplified function in sum-of-minterms form: a. F(x,y,z) = Σ(0,1,4,5,6) d(x,y,z) = Σ(2,3,7) b. F(A,B,C,D) = Σ(0,6,8,13,14) d(A,B,C,D) = Σ(2,4,10)Electromagnetic Pulse propagating at oblique angle to a dielectric interface Consider a gaussian wave pulse propagating along the z-axis from region 1 with refractive index n1 and onto a dielectric interface y = m z (for all x). To the left of this dielectric interface, the refractive index is n2. Devise an initial value computer algorithm to determine the time evolution of the reflected and transmitted electromagnetic fields for this pulse. e.g., n1 = 1 , n2 = 2 initial profile (t = 0, with z0 < 0) Ex = E0 exp[-a (z-z0)^2] By = n1 * Ex Choose parameters so that the pulse width is at least a fact of 8 less than the z- domain of integration ( -L < z < L). For the slope of the interface, one could choose m = 1.I have these functions that are define on N* (Natural numbers >0), and I have to use the Big-O notation to find the asymptotic behavior (asymptotic expansion) of each function. I want the details solutions for each.