In: Advanced Math

A vector y =
[R(t) F(t)]^{T}
describes the populations of some rabbits R(t)
and foxes F(t). The populations obey the system
of differential equations given by y′ =
Ay whereA =
The rabbit population begins at 84000. If we want the rabbit population to grow as a simple exponential of the form R(t) =
R_{0}e^{8t} with no
other terms, how many foxes are needed at time t =
0?(Note that the eigenvalues of A are λ = 8 and
2.) |

A vector y =
[R(t) F(t)]T
describes the populations of some rabbits R(t)
and foxes F(t). The populations obey the system
of differential equations given by y′ =
Ay where
A =
[−2
15]
[−2
9 ]
The rabbit population begins at 6000. If we want the rabbit
population to grow as a simple exponential of the form
R(t) =
R0e3t with no
other terms, how many foxes are needed at time t =
0?
(Note that the eigenvalues of A...

Given is a population of wolves (W) and rabbits (R). R[t+1] =
R[t]+ g*R[t] * (1 – R[t]/K) - sR[t]W[t] W[t+1] = (1-u)W[t] +
vR[t]W[t] Where the carrying capacity of rabbits is 1 million. The
growth rate of rabbits is 10% a year and s is equal to 0.00001, v
is 0.0000001, and u is equal to 0.01. How many wolves and how many
rabbits exist in the equilibrium?

Calculate the flux of the vector field F? (r? )=9r? , where r?
=?x,y,z?, through a sphere of radius 4 centered at the origin,
oriented outward.
Flux =

The position vector F(t) of a moving particle at time t[s] is
given by F(t)= e^t sin(t)i-j+e^t cos(t)k a) Calculate the
acceleration a(t). b) Find the distance traveled by the particle at
time t = 3π/2, if the particle starts its motion at time t = π/2.
c) Find the unit tangent vector of this particle at time t = 3π/2.
d) Find the curvature of the path of this particle at time t =
3π/2.

(1 point) For the given position vectors r(t)r(t) compute the
unit tangent vector T(t)T(t) for the given value of tt .
A) Let r(t)=〈cos5t,sin5t〉
Then T(π4)〈
B) Let r(t)=〈t^2,t^3〉
Then T(4)=〈
C) Let r(t)=e^(5t)i+e^(−4t)j+tk
Then T(−5)=

Let V be the vector space of all functions f : R → R. Consider
the subspace W spanned by {sin(x), cos(x), e^x , e^−x}. The
function T : W → W given by taking the derivative is a linear
transformation
a) B = {sin(x), cos(x), e^x , e^−x} is a basis for W. Find the
matrix for T relative to B.
b)Find all the eigenvalues of the matrix you found in the
previous part and describe their eigenvectors. (One...

Plot the solution of the equation6ẏ + y = f(t) if f (t) = 0 for t + 0 and f(t) = 15 for t ≥ 0. The initial condition is y(0) = 7.

Given the vector function r(t)=〈√t , 1/(t-1) ,e^2t 〉 a) Find: ∫
r(t)dt b) Calculate the definite integral of r(t) for 2 ≤ t ≤ 3
can you please provide a Matlab code?

Please Consider the function f : R -> R given by f(x, y) = (2
- y, 2 - x).
(a) Prove that f is an isometry.
(b) Draw the triangle with vertices A = (1, 2), B = (3, 1), C =
(3, 2), and the triangle with vertices f(A), f(B), f(C).
(c) Is f a rotation, a translation, or a glide reflection?
Explain your answer.

Let T : P3(R) → P4(R) be defined by T(f(x)) = 5f′(x)-∫ f(t)dt
(integral from 0 to x)
1. Show that T is a linear transformation.
2.Find dim (P3(R)) and dim (P4(R)).
3.Find rank(T). Find nullity(T)
4. Is T one-to-one? Is T onto? Justify your answers.

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