1. (10pt) Let V and W be a vector space over R. Show that V × W together with (v0,w0)+(v1,w1)=(v0 +v1,w0 +w1) for v0,v1 ∈V, w0,w1 ∈W

   and

   λ·(v,w)=(λ·v,λ·w) for λ∈R, v∈V, w∈W is a vector space over R.

2. (5pt)LetV beavectorspaceoverR,λ,μ∈R,andu,v∈V. Provethat (λ+μ)(u+v) = ((λu+λv)+μu)+μv.

   (In your proof, carefully refer which axioms of a vector space you use for every equality. Use brackets and refer to Axiom 2 if and when you change them.)

The SIT Drug company produced two types of liquid pain killer, N(normal) and S (super). Each bottle of N requires 2 units of drug A, 1 unit of drug B and 1 unit of drug C

Each bottle of S requires 1 unit of A, 1 unit of B  and 3 units of C.The company is able to produce, each week, only 1400 units of A 800 units of B and 1800 units of C

The profit per bottle of N and S is $11 and $15 respectively.

Solve the LP model using MATLAB linprog ( ANS is S=500 and N=300 bottles) ( I just need the MATLAB codes)

prove that if you multiply 4 successive integers and take the square root of that + a number, the only number for which root(product + x) is a perfect square is 1.

Let {a_n} and {b_n} be bounded sequences. Prove that limit superior {a_n+b_n} ≦ limit superior {a_n} + limit superior{b_n}

Plot the Euler’s Method approximate solution on [0,1] for the differential equation
y* = 1 + y^2 and initial condition (a) y0 = 0 (b) y0 = 1, along with the exact solution (see
Exercise 7). Use step sizes h = 0.1 and 0.05. The exact solution is y = tan(t + c)

On the interval [-1,1], consider interpolating Runge’s function f(x) = 1/ (1 + 25x^2) By Pn(x), use computer to graph:

(c) Take 11 equally spaced nodes in [-1,1], starting at –1, ending at 1, and obtain the interpolating polynomial P10(x). Also, use 11 Chebyshev nodes in [-1,1] and obtain Pc(x), the corresponding interpolating polynomial. In the same graph, plot the three functions f(x), P10(x) and Pc(x) over the interval [-1,1] . Use different line-styles, so that f(x), P10(x) and Pc(x) look distinct.

b)Comment on the error formula for the nth degree interpolating polynomial by only computing the first three derivatives of f(x). Explain how accuracy will be lost (or gained) by interpolating f(x) at more points (by increasing n)

Let X = N^N endowed with the product topology. For x ∈ X denote x by (x₁, x₂, x₃, . . .).

(a) Decide if the function given by d : X × X → R is a metric on X where, d(x, x) = 0 and if x is not equal to y then d(x, y) = 1/n where n is the least value for which x_n is not equal to y_n. Prove your answer.

(b) Show that no compact set in X can contain a non-empty open set.

5) CHART - Create a 3-D pie chart showing the breakdown of total Hours Worked into total Overtime Hours and the NonOvertime hours located at the bottom of the database listing, row 37 columns E, F, G. The pie chart should show the values, the legend should indicate Overtime Hours and NonOvertime, and the chart title should be Total Daily Hours. Place this chart so that it will fill cells K1:R16. (Hint: When selecting the data for this chart remember that pie charts show parts of a whole so there’s no reason to include a total if you are displaying the parts making up that total.) [In spreadsheets, does it make sense to put totals at the bottom of the database?] 6) DATA TABLE - Create a Data Table that shows the average Daily Pay by gender for each department number. Start the criteria in cell E40 and the data table in cell E44. 7) SUBTOTALS - Create a Subtotal showing the total hours worked by department and the total overtime hours worked by gender within each department. Change to Outline View 3. [Which department works the most hours and do you see something odd about the overtime hours?] Copy the subtotal information to Sheet 2 using Copy/Paste Special Values (previously undisplayed/hidden data will appear when copied to Sheet 2). Remove the Subtotaling on Sheet 1 using the Remove All button before continuing. 8) GOAL SEEK - In cell E7, place a formula that will multiply the Overtime Rate (B3) by the Total Number of Overtime Hours. Perform a Goal Seek to determine the overtime rate necessary to achieve $400 in total overtime pay. Leave the goal seek in place so that it can be graded. 9) EXTRACT (Advanced Filter) - Sort the database by the employees' names in ascending order. Extract just the name and the department number (Dept. No.) of those who worked more than 8 hours. Start the criteria in cell A40 and the extract output in cell A44. 10) DATA TABLE – on SHEET 3 create a One-Variable Data Table that will take the ODD numbers from 1 to 55 and will show each of the following independent calculations: multiply by $6.64; divide by 3.245; add $8.23 to each number; subtract 4 from each number.

A certain bookstore chain has two stores, one in San Francisco and one in Los Angeles. It stocks three kinds of book: hardcover, softcover, and plastic (for infants). At the beginning of January, the central computer showed the following books in stock.

Suppose its sales in January were as follows: 200 softcover, 400 hardcover, and 400 plastic books sold in Los Angeles, and 1,300 softcover books, 1,900 plastic books, and 600 hardcover books sold in San Francisco.

(a) Write the given inventory table as a 2 ✕ 3 matrix.

(b) Write the given sales figures as a similar matrix.

(c) Compute the inventory remaining in each store at the end of January.

1. (Covering part 1 of the lecture slides)

Consider the subset A of natural numbers, defined by A={ all natural numbers n : n is congruent to 1 (mod 3) }

First consider G: N -------> {0,1} to be the characteristic function of A.

a) Determine G(23) and G(25)

b) determine which sequence of 0,1 is this set associated with. (Write about 9 terms of it, or describe it with a formula.)

c) Which real number in the interval (0,1) in base 2 is associated with this set?  

Hi, can you please verify with the basic code of a beginner MATLAB user, and try explaining how to approach if possible. Thanks a lot! :)

1. Answer the following questions regarding complex numbers. You must provide all handwritten working and MATLAB code/outputs for the problems below.

(a) Express the complex number w = √ 3+i in complex exponential form. (Provide a handwritten solution)

(b) (1 mark) Use MATLAB to check your answer for part (a) by calculating the length and argument of w. (Provide your MATLAB code/outputs)

(c) Find all solutions to z^5 = √ 3 + i. (Provide a handwritten solution)

(d) Use MATLAB to plot the solutions found above. The solutions must be plotted on the same figure. (Provide you MATLAB code/outputs) (e) What do you notice about the solutions in the plot? (Provide a handwritten solution)

2. Let A = {1,2,3,4}. Let F be the set of all functions from A to A. Recall that IA ∈ F is the identity function on A given by IA(x) = x for all x ∈ A. Consider the function E : F → A given by E(f) = f(1) for all f ∈ F.
(a) Is the function E one-to-one? Prove your answer.

(b) Is the function E onto? Prove your answer.

(c) How many functions f ∈ F are there so that E(f) = E(IA)? Explain.

(d) How many onto functions f ∈ F are there so that E(f) = E(IA)? Explain.

A mass that weight 5lb stretches a spring 3in. The system is acted on by an external force 6sin⁡(8sqrt2 t)lb. If the mass is pulled down 4in and then released, determine the position of the mass at any time t. Use 32ft/s2 as the acceleration due to gravity. Pay close attention to the units.

u(t)=

Let U and V be vector spaces, and let L(V,U) be the set of all linear transformations from V to U. Let T_1 and T_2 be in L(V,U),v be in V, and x a real number. Define vector addition in L(V,U) by (T_1+T_2)(v)=T_1(v)+T_2(v) , and define scalar multiplication of linear maps as (xT)(v)=xT(v). Show that under these operations, L(V,U) is a vector space.

Let f(x) = sin(πx).

• x0 = 1,x1 = 1.25, and x2 = 1.6 are given. Construct Newton’s DividedDifference polynomial of degree at most two.

• x0 = 1,x1 = 1.25,x2 = 1.6 and x3 = 2 are given. Construct Newton’s Divided-Difference polynomial of degree at most three.