A point is moving on the curre y=x3−3x2+2x−1 and the y-coordinate of the point is increasing at the rate of 6 units per second. When the point is at (2,−1), the rate of change of x-coordinate of the point is
We are given that a point is moving along the curve described by the equation y=x3−3x2+2x−1 and that the y-coordinate of the point is increasing at a rate of 6 units per second. We need to determine the rate of chang of the x-coordinate of the point when the point is at (2,−1). Steps to Find the Rate of Change of the x-coordinate: Differentiating the Curve Equation: Given y=x3−3x2+2x−1, differentiate y with respect to t to express ‌
dy
dt
: ‌
dy
dt
=‌
d
dt
(x3−3x2+2x−1)=(3x2−6x+2)‌
dx
dt
Using Given Information: We know ‌
dy
dt
=6. Substitute this value into the differentiated equation: (3x2−6x+2)‌
dx
dt
=6 Solving for ‌
dx
dt
: Rearrange the equation to solve for ‌
dx
dt
: ‌
dx
dt
=‌
6
3x2−6x+2
Substitute the Point (2,−1) : Substitute x=2 into the equation to find ‌
dx
dt
at this specific point: ‌
dx
dt
=‌
6
3(2)2−6(2)+2
=‌
6
3×4−6×2+2
Simplify the Expression: ‌
dx
dt
=‌
6
12−12+2
=‌
6
2
=3 Therefore, the rate of change of the x-coordinate of the point when at (2,−1) is 3 units per second.