Consider the direction field of the differential equation dy/dx = x(y − 6)2 − 4, but do not use technology to obtain it. describe the slopes of the lineal elements on the lines x = 0, y = 5, y = 6, and y = 7.

The slopes shows that the direction of the field is from -2 to +2, with three point charges, q₁, q₂ and q₃ at -2, 0 and +2 respectively.

Explanation:

Given;

The slope, dy/dx = 2x(y-6) - 4

2x(y-6) - 4 = 2xy - 12x - 4, divide through by 'x'

dy/dx = 2y -12 - 4/x

The slopes of the linear elements on the lines, x =0, y = 5, y = 6, y = 7.

At x = 0, and y = 5

dy/dx = 2y -12 - 4/x

dy/dx = 2(5) - 12 = -2

At x = 0, and y = 6

dy/dx = 2y -12 - 4/x

dy/dx = 2(6) - 12 = 0

At x = 0, and y = 7

= 2y -12 - 4/x

dy/dx = 2(7) - 12 = 2

Therefore, the slopes shows that the direction of the field is from -2 to +2, with three point charges, q₁, q₂ and q₃ at -2, 0 and +2 respectively.

not too sure thinking it will always be directed towards the centre of the earth. could be wrong not sure

diagram is missing, but i will try to explain your doubt.

answer:

amplitude is the maximum displacement or distance moved on a point of a vibrating body or wave measured from its equilibrium position.it is equal to half the length of vibration path.