1. Consider the 2^nd order NHDE below

d2xdt2+k2x=F0eωt

            With the following initial conditions x0=0 and x'0=0. (Assume k≠ω)

1. Find the complimentary solution yc(t)

2. Find the particular solution yp(t)

3. Find the general solution for the IVP.

1 = Derivative of a Constant; 2 = Power Rule; 3 = Product Rule; 4 = Quotient Rule; 5 = Derivative of Exponential Function; 6 = Derivative of Logarithmic Function; 7 = Chain Rule
1. Circle the number(s) indicating the rule(s) used to find the derivative of each function. Then differentiate the function.
(a.) f(x) = ln7 1 2 3 4 5 6 7
(b.) p(y) = y3.7 1 2 3 4 5 6 7
(c.) g(x) = √x2ex 1 2 3 4 5 6 7
(d.) j(z) = 1 z2+1 1 2 3 4 5 6 7
(e.) h(x) = x lnx 1 2 3 4 5 6 7
2. Simplify each function, if possible. All exponents should be positive and factor out common factors. Do not find the derivative

. (a.) f(x) = x−4(x + 6)5

(b.) g(x) = e9x(x−2)2 + 9e9x(x−2)

(c.) h(x) = x x+2

(1 point) The count in a bacteria culture was 600 after 10 minutes and 11613 after 20 minutes. Assume the growth can be modelled exponentially by a function of the form Q(t)=AertQ(t)=Aert, where tt is in minutes.

(a) Find the relative growth rate, with at least the first 5 digits after the decimal point entered correctly:
r=r=

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(b) What was the initial size of the culture? Round your answer to the closest integer.

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(c) Find the doubling period (in minutes).

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(d) Find the population after 65 minutes. Use your answer to part (b) as the initial amount.

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(e) When will the population reach 13000? After

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Q-1)1 to 5 in bag A; 1 to 11 in bag B there are numbered cards. A random from the randomly selected bag card is selected. Since there is an odd number on the selected card, A
What is the probability of being chosen from the bag? Note: Make a tree diagram and
express your results with Bayes Theorem and
Confirm.

Q-2)ABCD is a rectangle whose long edge is twice the short edge. Long midpoint X of edge AB; The midpoint of the short edge AD is Y. This choice with the XAY triangle. A randomly selected point in a rectangle Find the probability of being selected in the XAY triangle.

Q-3)Ali and Ahmet are playing matches. Ali’s probability of winning the match
3 times the probability of winning. Ali and Ahmet’s chances of winning
Find and using the Binomial distribution:
a) In the event of 3 matches, the probability of Ali winning twice
You calculate.?
b) At least 1 win of Ali in case of 3 matches
Calculate the probability.?

Thanks

Find the point on the plane curve xy = 1, x > 0 where the curvature takes its maximal value.

Consider the helix r(t)=(cos(2t),sin(2t),−3t)r(t)=(cos(2t),sin(2t),−3t). Compute, at t=π/6

A. The unit tangent vector T=T= ( , , )

B. The unit normal vector N=N= ( , , )

C. The unit binormal vector B=B= ( , , )

D. The curvature κ=κ=

Aluminum, iron and magnesium are used to manufacture certain metal parts.The quantity of parts produced from x tons of aluminum y tons of iron and z tons of magnesium is Q (x, y, z) = xyz. Aluminum is $ 800 per ton, iron is $ 400 per ton, and magnesium is $ 600 per ton.

use the Lagrange multiplier method to determine the number of tons of each material that must be used to manufacture 5000 metal parts at the lowest possible cost.

1. FERTILIZER NUTRIENTS (Please Use Excel and show screenshots :) )

A produce farmer is purchasing fertilizer containing three nutrients: nitrogen, phosphorus, and potassium. The farmer’s minimum weekly requirements are 240 units of nitrogen, 120 units of phosphorus, and 80 units of potassium. There are two popular blends of fertilizer on the market. Blend A costs $8.00 a bag, and contains 4 units of nitrogen, 6 units of phosphorus, and 4 units of potassium. Blend B costs $10.00 a bag, and contains 12 units of nitrogen, 2 units of phosphorus, and 2 units of potassium.

(a) How many bags of each blend should the farmer purchase each week to minimize the cost of meeting the nutrient requirements?

(b) What is the minimum weekly cost?

Prove that if a diagonal of a parallelogram bisects the angles whose vertices it joins, the parallelogram is a rhombus

How do we interpret a confidence interval?

Let f (x, y) = -x³ - y³ + 9xy - 26. Check that (0,0) and (3,3) are stationary points of f and classify these points as maximum, minimum or saddle point. Obtain the maximum or minimum value of f.

Find the distance between the skew lines given by the following parametric equations:

L1: x=2t y=4t z=6t
L2: x=1-s y=4+s z=-1+s

The production costs, in $, per week of producing x widgets is given by C(x)=65000+4x+〖0.2x〗^2-〖0.00002x〗^3 and the demand function for the widgets is given by p=500-0.5x . Find the Marginal Revenue equation. Find the Marginal Cost equation. Find the Marginal Revenue and Marginal Cost for the firm when it is producing 300 widgets. Based on your numbers, would you advise the company to increase, decrease, or make no change to the level of production? Explain why.

Find the equation (in terms of x and y) of the tangent line to the curve r=2sin5θ at θ=π/3.

A farmer wants to fence in a rectangular plot of land adjacent to the north wall of his barn. No fencing is needed along the barn, and the fencing along the west side of the plot is shared with a neighbor who will split the cost of that portion of the fence. If the fencing costs $16 per linear foot to install and the farmer is not willing to spend more than $4000, find the dimensions for the plot that would enclose the most area.

(width, length) =