Problems

01

For a given parametric curve $\gamma (t)=(x(t),y(t))$, there is a formula for the second derivative:

$$\frac{d^{2}y}{d{x}^2}=\frac{x^{\prime }(t)y^{\prime \prime }(t)-y^{\prime }(t)x^{\prime \prime }(t)}{x^{\prime }(t{)}^3}.$$

Show how to derive this formula. Verify the formula for the parametric curve $(x,y)=(t,t^3)$ using the fact that this curve is the graph of a function. (You know how to find $\frac{d^{2}y}{d{x}^2}$ for graphs of functions.) Now compute $\frac{d^{2}y}{d{x}^2}$ for the parametric curve $(t^2,t^6)$, which has the same image.

02

Do all 5 parts of the “Discovery Project” about Bézier Curves (p. 684 of 9e, 697-8 of 8e, 677-78 of 7e).