Calculus with Parametric curves

Question

At what point(s) on the curve  x = 6t² + 5, y = t³ − 7 does the tangent line have slope 1? (x, y) = ?

Answer 1

To find the point(s) on the parametric curve where the tangent line has a slope of 1/2, calculate dy/dx and set it equal to 1/2.

dx/dt = 12t

dy/dt = 3t²

dy/dx = 3t²/12t = t/4

Set dy/dx equal to 1/2:

t/4 = 1/2

t = 2

After substituting t = 2 back into the parametric equations x(t) and y(t), you get:

x = 29

y = 1

Therefore, the point on the curve where the tangent line has a slope of 1/2 is: (x,y) = (29,1)

Hope this was helpful.