(2) More inclusion-exclusion counting: How many bit strings of length 15 have bits 1, 2, and 3 equal to 101, or have bits 12, 13, 14, and 15 equal to 1001 or have bits 3, 4, 5, and 6 equal to 1010? (Number bits from left to right. In other words, bit #1 is the left most bit and bit #15 is the right most bit.) Hint: The fact that the third bit appears in two of the required patterns means some special care will be needed to get the count correct.

For which values of λ does the system of equations

(λ − 2)x + y = 0

x + (λ − 2)y = 0

have nontrivial solutions? (That is, solutions other than x = y = 0.) For each such λ find a nontrivial solution.

Given f(x) = sin^2(x) and g(x) = sin^4(x) (give exact answers for all parts): a) Plot the functions on the x-interval [0, π]. Find the volume when the region enclosed by the curve and the x-axis is rotated about the line x = π. b) Find the area of the region. 1?b Aa and y = 1? b 1(f(x)2 −g(x)2)dx. Find the x-coordinate of the center of Aa2 mass of the region. In a print statement, explain why this answer makes sense based on the graph in part a). d) When the region rotates about the line x = π, how far does the center of mass travel? Multiply this value by the area. What do you notice when you compare your answer to part a)?

Python

Define error function and complementary error function, mathematically. Explain how these functions can be tabulated.

For each system described below say as much as possible about each system’s solution
set. Note which theorems you are using to reach your conclusions.
(a) A consistent system of 8 equations in 5 variables.
(b) A consistent system of 5 equations in 8 variables.
(c) A system of 4 equations in 9 variables.
(d) A system with 15 equations in 35 variables.
(e) A system with 8 equations and 5 variables. The reduced row-echelon form of the
augmented matrix of the system has 6 pivot columns.

Use LU decomposition to solve the following system of equations (show your work). Do not use a pivoting strategy, and check your results by using the matrix inverse to show that [A][A]-1= [I].

8x+ 2y−z=10

- 2x+4y+z=5

3x−y+ 6z=7

Using a power series methodology, obtain the general solution (form u = c1u1 + c2u2 + f(x)) to the equation u” + 4u = x.

2. Consider a homogeneous system of linear equations with m equations and n variables.

   1. (i) Prove that this system is consistent.

   2. (ii) Prove that if m < n then the system has infinitely many solutions. Hint: Use r (the number of pivot columns) of the augmented matrix.

A voltage of 10 sin(3t) volts is impressed on a series circuit containing a 20Ω resistor, 10^-3 H inductor and a 1μF capacitor. Obtain expressions for the charge q on the capacitor and current I in the circuit if q=0=I at t=0.

Scenario 7.2 - Gulab Greatness
Historical demand for gulab jamun from a sweet stall on Commercial Road is as displayed in the table.

What is the trend component of Holt's model for period 0?

Key information:

Box dimension: 12” x 10” x 6” O.D.

Pallet dimensions: 48” x 42” x 6”

Constraints: (1) No pallet overhang

                    (2) Pallet unit load <= 48” high, including the pallet, for storage in a rack system

Trailer dimensions: 53’ long x 8’ 6” wide (b/w the hinges) x 9’ high

Deliverables:

1. Draw a 3-D sketch of the pallet unit load on a wood pallet. Include dimensions for a box and the pallet unit load.
2. Show your calculations for the quantity of boxes in a unit load.
3. Show your calculations for the number of pallets per trailer load.
4. Show your calculations for the quantity of boxes per trailer load.
5. Explain why these results are important to a producer who ships goods to customers by freightliner.

Let V = R^2×2 be the vector space of 2-by-2 matrices with real entries over
the scalar field R. We can define a function L on V by
L : V is sent to V
L = A maps to A^T ,
so that L is the “transpose operator.” The inner product of two matrices B in R^n×n and C in R^n×n is usually defined to be
<B,C> := trace (BC^T) ,
and we will use this as our inner product on V . Thus when we talk about
elements B,C in V being orthogonal, it means that <B,C> := trace (BC^T) = 0.
Problem 1.
1. First show that L is linear, so that L in B (V ).
2. Now choose a basis for the vector space V = R^2×2, and find the matrix of
L with respect to your basis.

Consider the initial value problem

y′ = 18x − 3y, y(0) = 2

(a) Solve it as a linear 1st order ODE with the method of the integrating factor.

(b) Solve it using a substitution method.

(c) Solve it using the Laplace transform.

Use the Runge-Kutta method with step sizes h = 0.1, to find approximate values of the solution of

y' + (1/x)y = (7/x^2) + 3 , y(1) = 3/2 at x = 0.5 .

And compare it to thee approximate value of y = (7lnx)/x + 3x/2

Consider a general system of linear equations with m equations in n variables, called system I. Let system II be the system obtained from system I by multiplying equation i by a nonzero real number c. Prove that system I and system II are equivalent.