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)

In each case, determine the number of ways

(a) 10 identical candies must be distributed among 4 children

(b) A 15-letter sequences must be made up of 5 A's, 5 B's and 5 C's

(c) 10 identical rings must be placed on your 10 fingers

(d) 3 red, 3 green and 3 blue flags are to be arranged along the street for the parade

The question is: Let G be a finite group, H, K be normal subgroups of G, and H∩K is also a normal subgroup of G. Using Homomorphism theorem ( or First Isomorphism theorem) prove that G/(H∩K) is isomorphism to a subgroup of (G/H)×(G/K). And give a example of group G with normal subgroups H and K such that G/(H∩K) ≆ (G/H)×(G/K), with explanation.

I was trying to find some solutions for the isomorphism proof part, but they all seems to have the condition with H∩K = {e} . I can ensure that there is no missing condition in my question. As there is another subquestion which I've already know the solution, is about given H∩K = {e} and show G/(H∩K) is isomorphism to (G/H)×(G/K).

Soundex produces two models of satellite radios. Model A requires 15 minutes of work on Assembly Line I and 10 minutes of work on Assembly Line II. Model B requires 10 minutes of work on Assembly Line I and 12 minutes of work on Assembly Line II. At most 25 hours of assembly time on Line I and 22 hours of assembly time on Line II are available each day. Soundex anticipates a profit of $12 on Model A and $10 on Model B. Because of previous overproduction, management decides to limit the production of Model A satellite radios to no more than 80 per day.

1. Find the range of values that the resource associated with the time constraint on Assembly Line I can

   assume.