Answer:
Explanation:
To show that the curve C parameterized by R(t) = (t, t^3, 2t + 3t^3) lies in a plane, we need to show that the points on this curve satisfy a linear equation in three variables, which is the defining property of a plane.
Let (x, y, z) be an arbitrary point on the curve C. Then, by definition of R(t), we have:
x = t
y = t^3
z = 2t + 3t^3
We can eliminate t between the first two equations to get:
t = x^(1/3)
t^3 = y
Substituting these expressions for t into the equation for z, we get:
z = 2t + 3t^3
= 2(x^(1/3)) + 3y
Therefore, the curve C lies in the plane with equation z = 2(x^(1/3)) + 3y. This is a linear equation in three variables, which confirms that C is a planar curve.