The prior probabilities for events are ?₁, ?₂,and?3?(?₁)=0.20, ?(?₂)=0.50,?(?₃)=0.30. (Note the events are mutually exclusive and collectively exhaustive). The conditional probabilities of event B given are ?₁, ?₂,and?₃?(?|?₁)=0.50, ?(?|?₂)=0.40,?(?|?₃)=0.30

1. Compute and .?(?⋂?₁), ?(?⋂?₂) and ?(?⋂?₃)

2. Compute ?(?)

3. Apply Bayes’ theorem to compute the posterior probability, and .?(?₁|?), ?(?₂|?) and ?(?₃|?)

general solution :

5?2?′′+23??′+16,2=0

Euler-Cauchy :

?2?′′+3??′+?=0 , ?(1)=3,6 , ?′(1)=0,4

If f(x) = (x+1)^3(2x-1)^2(x+2)

a. Identify the degree, the leading term, the constant, and the actual zeros with their multiplicities

b. Graph without using value table

1-Find the relative maximum and minimum for the function, 1/3*x^3-9x+2

2-Determine the intervals on which the function f(x)=1+2x+8/x

3- Identify the critical number of f(x)=x^2-5x+6 on interval [-2,2]

4- Identify the intervals where the function, f(x)=2x^3-24x+2

5- ln(sin^2x) find f'(x)

solve for matrix B

Let I be Identity matrix

(I-2B)^-1=

Problem 2. Construct a model of propane. Look down the axis of the C(1)-C(2) bond-C(1) should be the front carbon. Arrange the model in the staggered conformation A.

Now, keeping the front carbon, C(1), and its attached hydrogens stationary, make a rotation of 60° about the C(1)-C(2) bond. Sketch Newman projection B of the conformation which results. Rotate another 60° about the C(1)-C(2) bond and sketch Newman projection C of the resulting conformation. Continue rotating 60° until conformations D, E and F result and sketch Newman projections of each.

Are all of the staggered conformations of propane of equal potential energy?

Are all of the eclipsed conformations of propane of equal potential energy?

If S = 1-x/1! + x^2/2! - x^3/3! + .....  
where n! means factorial(n) and x is a variable that will be assigned.
Use matlab to compute S for x = 7 and n (number of terms) = 5.   Write the value below as the one displayed when you issue "format short" in matlab. Explain the process and result of the question.

1. Sketch a connected 2-dimensional complex with three 2-cells, 8 1- cells, and 5 0-cells.

2. Draw three examples of a connected 1-dimensional complex with 7 0-cells and 6 1-cells.

3. Sketch a connected 1-dimensional complex with 7 0-cells and 7 1-cells.

4. Sketch a connected 1-dimensional complex with 7 0-cells and 8 1-cells.

5. Is it possible to construct a connected 1-dimensional complex with 8 0-cells and 6 1-cells? Explain.