Problems

01

Chain rule over a curve and directional derivative

Let $f(x,y)=x^2-3xy$. Let $\mathbf{r}(t)=(\cos (t)-1,\sin (t))$ parametrize a particle traversing a circle which passes through the origin heading north. Notice that it is a unit-speed parametrization.

• (a) Find the derivative $\frac{d}{dt}f(\mathbf{r}(t))$ for $t=0$ when the particle is at the origin.
• (b) Find the tangent vector to $\mathbf{r}$ at $t=0$ and call it $\mathbf{v}$. Write a parametric equation of the line $L(t)$ through $(0,0)$ at $t=0$ with velocity vector $\mathbf{v}$. Evaluate the derivative $\frac{d}{dt}f(L(t))$ at $t=0$.
• (c) Compute $D_{\mathbf{v}}(f)$ at the origin.