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

ball is thrown off a cliff. Its height (in feet) after 𝑡 seconds is given by the function ℎ(𝑡) = 200 + 100𝑡―16𝑡2Graph this function in Desmos. Use the Graph Settings button (the wrench) to set the X-axis to go from ―2 to 10 and the Y-axis from 50 to 500. Use the graph to answer the following questions. Give the values as they appear on the graph. Include units.

A coconut is fired from a cannon at the edge of a cliff over the water. The height of the coconut above the water is

hx=-32x2502+x+250

where x is the horizontal distance of the coconut from the face of the cliff. Height and horizontal distance are both measured in feet.

Graph the function in Desmos. Move/zoom as needed to see the graph.

Display the coordinates of all intercepts, maximum points, and minimum points by clicking on each point. Take a screenshot. Paste the screenshot below, then answer the questions about this scenario.

Scenario: A reclamation facility uses raw sewage as a source of water.

• Calculate microbial concentrations that meet daily acceptable risk. Assume that people drink 2 L of water daily. The daily intake dose based on probability of infection [D(Pi)] is 4.62 x10-7 for rotavirus, 5.61 x10-5 for Shigella, and 6.52 x10-5 for Cryptosporidium.

• How many log10 reductions are required to protect public health? Assume that the sewage contains the following concentrations of pathogens: rotavirus (enteric virus) 104 /100mL, Shigella 103 /100mL, and Cryptosporidium 105 /100mL

3. (6 pts) Determine whether the function

f(t) = 8/(1+t)

satisfies the hypotheses of the Mean Value Theorem on the interval [0, 3]. If so, find a value e in [0, 3] that satisfies the conclusion of the Mean Value Theorem