Question

At what point on the curve x = 3t² + 6, y = t³ − 1 does the tangent line have slope 1/2?

(x, y) = ?

Answer 1

To find the point on the curve where the tangent line has a slope of 1/2, take the derivatives of x and y with respect to t, find the slope function dy/dx, set it equal to 1/2 solving for t, and then calculate x and y. The point is (9, 0).

Step-by-step explanation:

To determine the point on the curve x = 3t² + 6, y = t³ − 1 where the tangent line has a slope of 1/2, follow these steps:

1. Compute the derivatives of x and y with respect to t to determine the slope of the tangent line, which is dy/dx.
2. Since dy/dx = (dy/dt) / (dx/dt), calculate dy/dt and dx/dt separately.
   
   • dy/dt = 3t²
   • dx/dt = 6t
3. Divide the derivatives to find the slope function: dy/dx = (3t²)/(6t) = t/2.
4. Set the slope function equal to 1/2 to solve for the value of t: t/2 = 1/2 → t = 1.
5. Plug this value of t into the original equations to find the x and y values at the point of tangency: x = 3(1)² + 6 = 9 and y = (1)³ − 1 = 0.

Therefore, the point on the curve at which the tangent has a slope of 1/2 is (9, 0).