**Question 1**

Question 2

Question 3

Question 4

Answer to question 1

consider the equation

Step 2

d2xdt2+x-y=0d2ydt2+y-x=0x=0 , x'(0)=-5 , y(0)=0 , y'(0)=1Taking the Laplace Transform of the given system x fieldssx1=-0 Sx2=0 of equivalently Sx1+x-y=0 sx2+y-x=-5solving this algebric system using cramers rule x1s=1x-y1y-x s x-y 2 y-xTaking the inverse Laplace transform of x1s and x2sX1=e2t-4te2tX2=e2t+4te2t

Answer to question 2

We will use a graphing calculator to graph the region

This graph properly shows the area between the given curves.

Then we find where do y=1/x and y=1/x^{2} meet

We set them equal and solve for x

For both functions, x can not be 0. So x=1

So we have to integrate from 1 to 8

Answer to question 3

Answer to question 4

Step 1: Concept

f(x+a)…….left by a units

f(x)-a…… down by a units

-f(x)……reflection across x axis

Step 2: Solution

Because of minus sign with the function, reflection across x axis.

Because of +6 inside the function, left 6 units.

Because of -5 with the function, down by 5 units

Option d