Suppose a change of coordinates t: r2→r2 from the uv-plane to the xy-plane is given by x=4v−2u−3, y=−1+5u+4v.

(a) find the absolute value of the determinant of the jacobian for this change of coordinates. ∣∣∂(x, y)/∂(u, v)∣∣= ∣det |=

(b) if a region d∗ in the uv-plane has area 7.04, find the area of the region t(d∗) in the xy-plane. area =

a) ∣∣∂(x,y)/∂(u,v)∣∣ = 28

b)Area = 197.12

Step-by-step explanation:

a) Find the absolute value of the determinant of the Jacobian

Change of coordinates

x = 4v -2u-3

y = -1+5u+4v

Then

∣∣∂(x,y)/∂(u,v)∣∣=∣det | =| |

∣∣∂(x,y)/∂(u,v)∣∣ = ∣(-2)(4)-(5)(4)∣ = 28  

b) The area of the region in the xy -plane is the area of the region in the uv-plane multiplies by the absolute value of the determinant of the Jacobian.

Thus

Area = (7.04)(28)

Area = 197.12    

step-by-step explanation:

let

we know that

the y-intercept is the y-coordinate of the point when the value of x is equal to zero

so

the y-coordinate of the point a is the y-intercept

the answer part a) is

the value of y-intercept is

part b) find the slope m

we know that

the value of the slope is equal to

substitute the values

therefore

the answer part b) is

the value of the slope is equal to

the general answer is

the slope is

the value of y-intercept is