Question

A point moves on the curve \( y = x^3 - 3x^2 + 2x - 1 \) and its y-coordinate increases at a rate of 6 units per second. When the point is at (2,-1), the rate of change of its x-coordinate is:

• A. \( 3 \)
• B. \( \frac{1}{2} \)
• C. \( -\frac{1}{2} \)
• D. \( -3 \)

Hint:

For related rates problems, differentiate implicitly and substitute known values to solve for the required rate.

Answer 1

The Correct Option is A

Step 1: Differentiating implicitly Differentiating \( y = x^3 - 3x^2 + 2x - 1 \) with respect to \( t \): \[ \frac{dy}{dt} = 3x^2 \frac{dx}{dt} - 6x \frac{dx}{dt} + 2\frac{dx}{dt}. \] Given \( \frac{dy}{dt} = 6 \) and \( x = 2 \), substituting: \[ 6 = 3(2)^2 \frac{dx}{dt} - 6(2) \frac{dx}{dt} + 2\frac{dx}{dt}. \] \[ 6 = (12 - 12 + 2) \frac{dx}{dt}. \] \[ 6 = 2 \frac{dx}{dt}. \] \[ \frac{dx}{dt} = 3. \]