\(Sinx+cosx= sin^2x+cos^2x\)

Find the value of X. 

n is positive integer，f（x）is continues in R，when x →∞，f（x）=x^（2n）+o（x^（2n）），prove that f（x）has the minimun value

Differentiation with respect  to x. \(Sinx+cos2x+y\)

Find  derivative\(Y = {-sinx+ \sqrt{sinx^2-4sinxcosx} \over 2sinx}\)

Find derivative

\(Y= {-sinx +cos 2x\over cosx+sin2x}\)

Find the graph of \(Sin \theta\)

Find the value of asymtotes.

\(X^2\)

A 144 lb weight stretches a spring 18 feet. The weight hangs vertically from the spring and a damping force numerically equal to 12 times the instantaneous velocity acts on the system. The weight is released from 9 feet above the equilibrium position with a downward velocity of 38 ft/s.

(a) Determine the time (in seconds) at which the mass passes through the equilibrium position.

(b) Find the time (in seconds) at which the mass attains its extreme displacement from the equilibrium position.

Question Four Suppose we have estimated the following logit model with 3 explanatory variables using maximum likelihood. 1 Pi = 1+ e-(0.1+0.3X21-0.6X3i+0.9X4i) Suppose also that X2 = 1.6 tz = 0.2 x4 = 0.1

a) Find the estimate of F(zi)

b) Find the Marginal effects

c) Give an interpretation of your results in (b

(a) Matlab problem Write a MATLAB function z = spline3.coeff (t, y) that takes as input the knot vector t = [0,₁.....] and the function values y = [yo. Y...] and returns the vector z = [0= 20, 21, 22, 2-1,2n = 0]. The z vector is computed by setting up and solving a tridiagonal system as dis cussed during the lecture. You can use the MATLAB backslash for solving the tridiagonal system.

(b) Matlab problem Compute the natural cubic spline, S(r), using the function you just wrote for t = [-1 = 0, 1 = 1] and y = [yo = f(ro), y₁ = f(₁),...Un f(x)] for n = 11 and plot S(r) as well as f(x) on [-1.1]. How does the natural cubic spline behave relative to the polynomial interpolation from the previous homework?

You may use the function eval_spine.m, provided with this assignment, to evaluate S(x).

During the decay of a radioactive substance, the rate of decay(the rate at which the mass is lost) is proportional to the mass present at that instant. Let M denote the mass in grams of radioactive Thorium 229 at time t in years after it has been introduced, where the initial amount of Thorium-229 was Mo = 112

g. Now M satisfies the differential equation