consider the linear non-homogeneous recurrence relation of order three x(n)+x(n-1)-4x(n-2)-4x(n-3)=16

find the eigenvalues of the recurrence relation, the closed form solution, and based on the dominant Eigen value determine if x(n) has longterm exponential growth

1. In its first 10 years the Janus Flexible Income Fund produced an average annual return of 9%. Assume that money invested in this fund continues to earn 9% compounded annually. How long (to the nearest year) will it take money invested in the fund to double?

Suppose that ?: ℝ → (0, ∞) satisfies ?(? + ?) = ?(?)?(?). Show that if ? is continuous at 0, then there is an ? ∈ (0, ∞) such that ?(?) = ?^x for all ? ∈ ℝ.

1. The population of a small country is growing 4.5% per year. If in 1997 the population was 140 thousand, determine the exponential expression for population as a function of the year. Letting t=0 correspond to 1997and find the population in 2008.

2. How much should be deposited today if it’s to accumulate to $44,900 in 30 years, if the account bears interest at 7.5% compounded monthly?

3. How long will it take an investment at 11.25% interest, compounded continuously to quadruple?

4. When advertising is discontinued, sales decrease at a rate proportional to sales at that instant. A magazine has an initial circulation of 1.7 million and 2 years later decreases 14%. When will circulation reach 850,000?

Use the elimination method to find a general solution for the given linear​ system, where differentiation is with respect to t.

x'=5x-6y+sin(t)

y'=3x-y-cos(t)

Consider a transformation ?:ℝ3→ℝ3T:R3→R3 defined by

?(?,?,?)=(?+?+?, 2?+?, ?−2?+?).T(x,y,z)=(x+y+z, 2x+y, x−2y+z).

(a) Find the standard matrix of ?T.

(b) Is ?T a linear transformation? Explain.

(c) Is ?T invertible?

(d) Find the image of (2,−1,1)(2,−1,1) under ?T.

Multiplication by Powers of t

1. f(t) = 4t sin 2t

2. f(t) = 4t^3 sint t

3. f(t) = t^3 cos t

For each n ∈ N, let f_n : [0, 1] → [0, 1] be defined by f_n(x) = 0, x > 1/n and f_n(x) = 1−nx if 0 ≤ x ≤1/n.

The collection {f_n(x) : n ∈ N} converges to a point, call it f(x) for each x ∈ [0, 1]. Show whether {f_n(x) : n ∈ N}
converges to f uniformly or not.

Write a script that uses the function below to find root brackets for ?(?) = cos(e^x) + sin(?), between ? = 0 ??? ? = ? with ns=100. Plot the output by first plotting the function and then plotting ‘*’ at each bracket point (on the x-axis). You may either give the plot() function two sets of inputs, or you can use hold on ... hold off to add plots to your figure.

function xb = incsearchv(func,xmin,xmax,ns)

Please solve both parts of the following question. Please show all work and all steps clearly and type out the solutions. Please do not copy and paste solutions from the other question posted on chegg that answers these questions. instead, please write your own individual answers solving them differently than the answers posted before on chegg. solve these problems differently and provide new solutions.

1a.) Estimate the first eigenvalue of x" + (lambda)(e^t)(x)=0, x(0)=x(2)=0

for part 1a, the solution should be ((pi^2)/4e)) <= lambda * e^t <= ((pi^2)/4)).

1b.) Find the eigenvalues of x" + (lambda)(x) = 0, x(0)=x'(pi)=0.

Q1. What is the z-score of the following values?

1. x1=2.7 when M(x)=3, SD(x)=0.5
2. x2=2.8 when M(x)=3, SD(x)=0.5
3. x3=3.3 when M(x)=3, SD(x)=0.5
4. x4=3.8 when M(x)=3, SD(x)=0.5

Q2. What are the percentiles for each z-score above?

1. x1
2. x2
3. x3
4. x4

Q3. You own a big apple tree farm and wondered how many apples each apple tree have on average.You counted the number of apples per apple tree for 25 trees to estimate the number per apple tree at your farm. You found that the mean number of apples per tree is 50 when the standard deviation is 10. What is 95 % confidence interval of the estimated number of apples per tree of your farm?

Q4. In the same situation with Q3, what is 99% confidence interval?

Q5. What do the answer from Q3 and Q4 mean? What is wider and why?

Consider a forced spring-mass equation of the form x′′ + x = cos(ωt) with initial conditions x(0) = 1 and x′(0) = 0.
a) Suppose ω doesnt = 1, find the solution to the IVP.

b)If ω = 1, find the solution to the IVP.

c)In which of the two cases does the phenomenon of pure resonance occur? Ex- plain your answer.

d)Verify that with ω = 9/10, x(t) = 100/19 (cos( 9t/10 ) − (81/100) cos t) solves the IVP.

e) Recall the trig identity cos(α) − cos(β) = −2 sin( (α+β)/2 ) sin( (α−β)/2 ). Observe that x(t) ≈ 100/19 (cos( 9t/10 ) − cos t). Use this last form and the trig identity above to approximate x(t) by a product of two sin functions. Make a sketch of the graph, carefully accounting for the periods of the two functions. What would you call the long-term behavior of this solution??

Set A = {1, 2, 3, 4}, write a binary relation R on A that is reflexive, symmetric and transitive, with (1, 2),(3, 2) ∈ R.

Use numerical integration to calculate the solution of Airy’s differential equation x''=tx from initial conditions x(-14.5)=0 and x'(-14.5)=1 until t =3.

Then plot the curve. Be sure to sample the curve frequently enough that it appears smooth.