Let C be the curve of intersection of the surfaces y=x^3 and z=x^2

Let \(C\) be the curve of intersection of the surfaces \(y=x^{3}\) and \(z=x^{2}\). (a) Parameterize \(C\) by \(\mathbf{r}(t)=\langle t,\ t^{3},\ t^{2}\rangle\). Verify that \(\mathbf{r}(t)\) lies on both surfaces for every real \(t\). (b) At the point \(P=(1,1,1)\), viewing \(y=x^{3}\) as a cylindrical surface with generators parallel to the \(z\)-axis, find an equation of the tangent plane to this surface at \(P\). Viewing \(z=x^{2}\) as a cylindrical surface with generators parallel to the \(y\)-axis, find an equation of the tangent plane to this surface at \(P\). (c) Compute \(\mathbf{r}'(1)\), and show that this tangent vector is orthogonal to both surface gradients at \(P\). Conclude that the tangent line to \(C\) at \(P\) is the line of intersection of the two tangent planes. (d) Write parametric equations of the tangent line to \(C\) at \(P\).

Each surface is a cylinder: \(F(x,y)=y-x^{3}\) is independent of \(z\), and \(G(x,z)=z-x^{2}\) is independent of \(y\). The tangent plane is the kernel of the gradient.