What are some practical examples of the use of standard deviation in data use for business?

These instructions are written with the assumption that code will be done in matlab. You might find the following built in commands useful: length, plot, xlabel, ylabel, title, legend, fzero, plot, disp, axis, axes, min, max.

3. Root finding; Bisection, Secant, Newton. a) Write your own version of the bisection method to solve e 2x − 10x = 0. Assume the root is in the interval [0, 1]. Try to use the matlab command ”fzero” to accomplish the same thing. b) Write your own version of secant method to solve e −2x − 7x = 0. Assume the root is in the interval [1/9, 2/3]. Try to use the matlab command ”fzero” to accomplish the same thing. c) Write your own version of Newton’s method to solve x + sin(x) − cos(2x) = 0.

These instructions are written with the assumption that code will be done in matlab. You might find the following built in commands useful: length, plot, xlabel, ylabel, title, legend, fzero, plot, disp, axis, axes, min, max.

2. Numerical Integration (Quadrature). Write FOUR of your own numerical integration routines. One should use left-end, right-end, or mid-points, another should use the trapezoid method, another should use Simpson’s method, and the fourth should use either Guassian Quadrature or Romberg’s method. Use your four routines to approximate two integrals numerically so that they are accurate to 1 part in 1 billion (if possible, if this is not possible explain why). First use your routines to approximate f(x) = x 3 − x cos(x) on the interval [0, 2π], you can and should check that your answer is correct. Then use your routines to approximate f(x) = 4e −2x 2 on the interval [−1, 1]. Use one of matlab’s built in integration functions to approximate both integrals also. Each integral should be: numerically approximated FOUR ways, checked vs one of matlab’s built-in numerical integration methods, and the polynomial/trig function should be checked exactly.

Find all the steady states of the system

? ′ = ?(10 − ? − ?) ? ′ = ?(30 − 2? − ?)

Then find the eigenvalues for every steady state and classify them

Formulate the Chinese Remainder Theorem for polynomials over Z/p and prove it. (Please don't just copy and paste another solution onto here)

Consider polynomial interpolation of the function f(x)=1/(1+25x^2) on the interval [-1,1] by (1) an interpolating polynomial determined by m equidistant interpolation points, (2) an interpolating polynomial determined by interpolation at the m zeros of the Chebyshev polynomial T_m(x), and (3) by interpolating by cubic splines instead of by a polynomial. Estimate the approximation error by evaluation max_i |f(z_i)-p(z_i)| for many points z_i on [-1,1]. For instance, you could use 10m points z_i. The cubic spline interpolant can be determined in MATLAB; see "help spline". Use m=10 and m=20. Compute splines that interpolate at equidistant nodes and at Chebyshev nodes. Provide tables of the errors and plots of the function f and the interpolating polynomials and splines.

Define Type I error and Type II error. Explain why both ‘‘false positive’’ and ‘‘false negative’’ should be avoided in the analysis and monitoring of environmental contaminants?

Describe in what cases the use of ‘‘median’’ can be preferred over that of ‘‘mean’’ in environmental data reporting?

a) The triangle △ ABC is right-angled with right angle at corner C and angle α at corner A. Calculate a = | BC |, given that c = | AB | = 8, and that tan α = 12.

Calculate a= ?

b) In the triangle △ ABC before the designations a = | BC |, b = | CA |, c = | AB |, and ∠A = α, ∠B = β and ∠C = γ.
Find c, given that the triangle has an obtuse angle at C and that a = 1, b = 9 and sin γ = 58.
calculate c =?

1.) Consider the following. (Assume that the coins are distinguishable and that what is observed are the faces or numbers that face up.) HINT [See Examples 1-3.]

Three coins are tossed; the result is at most one head.

Which of the following sets of elements are included in the sample space? (Select all that apply.)

HTT

TTH

THT

THH

HHT

HHH

TTT

HTH

List the elements of the given event. (Select all that apply.)

THT

TTT

THH

HTT

HHH

TTH

HHT

HTH

2.) Consider the following. HINT [See Examples 1-3.]

You are deciding whether to enroll for Psychology 1, Psychology 2, Economics 1, General Economics, or Math for Poets; you decide to avoid economics.

Which of the following sets of elements are included in the sample space? (Select all that apply.)

Economics 1

General Psychology

Psychology 2

General Economics

Psychology 1

Math for Poets

List the elements of the given event. (Select all that apply.)

Math for Poets

Psychology 1

General Psychology

Economics 1

Psychology 2

General Economics

Let S = {2 k : k ∈ Z}. Let R be a relation defined on Q− {0} by x R y if x y ∈ S. Prove that R is an equivalence relation. Determine the equivalence class

Let A be incidence matrix of graph G. prove if G has a cycle, then null space of transpose of A is not {0} (there exists non-trivial solution of (A^T)y=0)

How do you define a function that tests if a number is even using lambda calculus? The function should return true if the number is even, and false otherwise.