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
‌ :
‌=‌(x3−3x2+2x−1)=(3x2−6x+2)‌Using Given Information:
We know
‌=6. Substitute this value into the differentiated equation:
(3x2−6x+2)‌=6Solving for
‌ :
Rearrange the equation to solve for
‌ :
‌=‌Substitute the Point
(2,−1) :
Substitute
x=2 into the equation to find
‌ at this specific point:
‌=‌=‌Simplify the Expression:
‌=‌=‌=3Therefore, the rate of change of the
x-coordinate of the point when at
(2,−1) is 3 units per second.