1. 110 ice creams of different flavors are purchased weekly in a student residence: vanilla, passion fruit, and coconut. The budget for this purchase is 540 euros and the price of each ice cream is 4 euros for vanilla, 5 euros for passion fruit and 6 euros for coconut. Once the students' tastes are known, it is known that between 20% passion fruit and coconut ice cream, vanilla must be purchased. to. A. Solve using a system of linear equations to calculate how many ice creams of each flavor are bought per week. b. Solve, using the Gauss Jordan Method

Exercise

Minimize            Z = X₁ - 2X₂

Subject to            X₁ - 2X₂ ≥ 4

                            X₁ + X₂ ≤ 8

                           X1, X2 ≥ 0

Solve the following linear programs graphically.

Minimize            Z = 4X₁ - X₂

Subject to            X₁ + X₂ ≤ 6

                            X₁ - X₂ ≥ 3

                           -X₁ + 2X₂ ≥ 2

                           X₁, X₂ ≥ 0

Solve the initial value problem. y'=(y^2)+(2xy)+(x^2)-(1), y(0)=1

Show that drastic sum and drastic product satisfy the law of excluded middle and the law of contradiction. [Hint: substitute the s-norm and t-norm operation for intersection and union in the two laws, respectively]

What are the square roots of 3+4i?

Find the inverse of the following 4x4 matrix:

1-j j 1+j 2

-j    4    2-j    3

1-j 2+j j 3-j

2 3 3+j 1

1. If f(x) = ln(x/4)
-(a) Compute Taylor series for f at c = 4
-(b) Use Taylor series truncated after n-th term to compute f(8/3) for n = 1,.....5
-(c) Compare the values from above with the values of f(8/3) and plot the errors as a function of n
-(d) Show that Taylor series for f(x) = ln(x/4) at c = 4 represents the function f for x element [4,5]

Exercise

Solve the following linear programs graphically.

Maximize            Z = X1 + 2X2

Subject to            2X1 + X2 ≥ 12

                            X1 + X2 ≥ 5

                           -X1 + 3X2 ≤ 3

                           6X1 – X2 ≥ 12

                           X1, X2 ≥ 0

1a. Proof by induction: For every positive integer n,
1•3•5...(2n-1)=(2n)!/(2n•n!). Please explain what the exclamation mark means. Thank you for your help!

1b. Proof by induction: For each integer n>=8, there are nonnegative integers a and b such that n=3a+5b

A tree is a circuit-free connected graph. A leaf is a vertex of degree 1 in a tree. Show that every tree T = (V, E) has the following properties: (a) There is a unique path between every pair of vertices. (b) Adding any edge creates a cycle. (c) Removing any edge disconnects the graph. (d) Every tree with at least two vertices has at least two leaves. (e) | V |=| E | +1.

R.A.T.-Create Your Own Water Park Apply your knowledge of polynomial functions to create a water park, with 6 waterslides - one for under 6 years old (highest point at least 5m above ground) two for ages 6 to 12 (highest point at least 10m above ground) three for over age 12 (highest point at least 20 m above ground)

A Create a polynomial equation for each waterslide. Show all of your work. The waterslide must begin at the y axis and the x axis must represent the ground. For each function, write the original function in factored form, then explain the transformations that were performed, in order to obtain the model function.

B. Graph (and print) each function using desmos. State the domain and range of each function.

C. Choose one of your waterslides and determine the interval(s) in which the height of the ride was above 3m. Explain your method.

D. Choose one of the waterslides for ages 12 and up and state the interval (from peak to trough) where the waterslide is steepest. Then determine the average rate of change for that interval (by using the equation). Next, determine the instantaneous rate of change at the point in the interval when the person is moving the quickest. Interpret the meaning of these numbers. Note: the maximum steepness of a ride should not exceed 4:1, rise to run. The waterslide should be decelerating as it comes to a stop.

Theta"(t)–Theta'(t)= tsint

That's it, no more information for this question.

Consider the differential equation y '' − 2y ' + 10y = 0;    e^x cos(3x), e^x sin(3x), (−∞, ∞).

Verify that the given functions form a fundamental set of solutions of the differential equation on the indicated interval.

The functions satisfy the differential equation and are linearly independent since W(e^x cos(3x), e^x sin(3x)) = _____ANSWER HERE______ ≠ 0 for −∞ < x < ∞.

Form the general solution.

y = ____ANSWER HERE_____