You have just started a new job and are thrilled to learn that your new employer offers a 401(k) retirement plan to its employees. Your annual salary is $40,000. Assume the IRS allows you to contribute up to $24,000 to your 401(k). You’ve decided to contribute 7% of your annual salary to the plan.

Questions:

1. How much more money would you need to contribute to meet the maximum allowable contribution set forth by the IRS?

2. The company offers you a $.50 match for each dollar that you contribute between 2 and 5 percent of your annual salary. How much is the company match based on your 7% contribution?

3. Is this a defined benefit plan or defined contribution plan? Why?

Find a general solution of the inhomogeneous equation y′′ + 2y′ + 5y = f(x) for
the following cases: (i) f(x) = 1 (ii) f(x) = x2 (iii) f(x) = e−x sin2x (iv) f(x) = e−x (v)
sin2x

COMPUTING LESLIE MATRIX

Example After one year, we have only 250 fishes left. And then 125 have reached their reproduction rate. If we set f₃ = 8, then we are back to n = (1000, 0, 0): We see that n1 = (0, 250, 0), n2 = (0, 0, 125), n3 = (1000, 0, 0)

Exercise Write down the Leslie matrix for the previous example and calculate for various choices of n the population vectors ni. What do you observe?

Exercise Show that you can find some n such that n⁺ = Ln = n. If n = (a, b, c) then

n⁺ = (8c, 0. 25a, 0. 5b). Then (a, b, c) = (8c, 0. 25a, 0. 5b) determines a unique stable distribution n amongst the age groups. n itself is unique up to a factor.

Exercise Now change f₃ = 8 to numbers smaller as well as larger than 8, say 6 and 10. Then calculate again for various choices of n the population vectors ni. Can you still find some n such that n⁺ = n?

Let f be a continuous function on the closed interval [0,1] with a range also contained in [0,1]. Prove that f that there exists an x in [0,1] such that f(x)=x. Is the same explanation still valid if f is not continuous?

8. What is a maximization linear program? Provide your own example of a linear program

that seeks to maximize profit. Be sure to provide some controllable and uncontrollable

inputs as well as some constraints and necessary figures (numbers, $, etc.)

1) Elipse. Calculate 5 pixels, if Rx= 4, Ry= 7, Xc= 0, Yc= 0

2) Circle line. Calculate 6 pixels, if R= 14, Xc= 1, Yc= -9

I just need matematical solunations

1. Find the real part, the imaginary part, and the modulus of the complex number 1 + 8i 2 + 3i , showing your work. 2. Find all three solutions of the equation 2z 3 + 4z 2 −z −5 = 0. (Hint: First try a few “simple” values of z.) You must show all working.

Use laplace transform in solving the ff.:

After cooking for 45 minutes, when a cake is removed from an oven its temperature is measured at 300°F. 3 minutes later its temperature is 200°F. The oven is not preheated, so at t=0, when the cake mixture is placed into the oven, the temperature inside the oven is also 70°F. The temperature of the oven increases linearly until t=4 minutes, when the desired temperature of 300°F is attained; thereafter the oven temperature is constant 300°F for t is greater than pr equal to 4 minutes.

a.) devised a mathematical model for the temperature of a cake while it is inside the oven and after it is taken out of the oven.

b.) how long will it take the cake to cool off to a room temperature of 70°F

Let (X,d) be a metric space. The graph of f : X → R is the set {(x, y) E X X Rly = f(x)}. If X is connected and f is continuous, prove that the graph of f is also connected.

An RC circuit with a 1-Ω resistor and a 0.000001-F capacitor is driven by a voltage E(t)=sin 100 t V. If the initial capacitor voltage is zero, determine the subsequent resistor and capacitor voltages and the current.

Find the general solution to the DE: y'''+8y''+16y=0 (Hint: Find the auxilary equation first)

(a) Prove that if v_1,v_2,v_3 is a basis for R^3, then so is u_1, u_2, u_3 where u_1=v_1, u_2=v_1+v_2, and u_3=v_1+v_2+v_3.

(b) State a generalization of the result in part (a).

10.3.6 Exercise: Product of Pairwise Comaximal Ideals. Let R be a commutative ring, and let {A1,...,An} be a pairwise comaximal set ofn ideals. Prove that A1 ···An = A1 ∩ ··· ∩ An. (Hint: recall that A1 ···An ⊆ A1 ∩···∩An from 8.3.8).

9.2.6 Exercise. Let R = Z and let I be the ideal 12Z of R.

1. (i) List explicitly all the ideals A of R with I ⊆ A.
2. (ii) Write out all the elements of R/I (these are cosets).
3. (iii) List explicitly the set of all ideals B of R/I (these are sets of cosets).
4. (iv) Let π: R → R/I be the natural projection. For each ideal A of R such that I ⊆ A, write out π(A) explicitly (this is a set of cosets). Confirm by direct calculation what the Third Isomorphism Theorem says: that the function A ?→ π(A) is a bijection from the set of such A that you found in step (i) and the set of ideals of R/I that you found in step (iii).
5. (v) For each ideal A of R with I ⊆ A, write out all the elements of the following three quotient groups (under addition): R/A, A/I, and (R/I)/(A/I) (the last consists of cosets of cosets!). Then confirm by direct calculation what the Third Isomorphism Theorem says: that the rule f : R/A → (R/I)/(A/I) with f(gA) = (gI)(A/I) makes a well defined function that is an isomorphism of rings.

For the given function f(x) = cos(x), let x0 = 0, x1 = 0.25, and x2 = 0.5. Construct interpolation polynomials of degree at most one and at most two to approximate f(0.15)