[Not to be turned in]
- Now that we understand derivatives for vector-valued functions, we can analyze the derivatives of vector operations. For example, the dot product satisfies a "product rule". That is, if $f\colon\R\to\R^n$ and $g\colon\R\to\R^n$ are two vector-valued functions, then $$ {d\over dt}\Bigl(f(t)\cdot g(t)\Bigr) = f(t)\cdot {d\over dt}\bigl(g(t)\bigr) + g(t)\cdot {d\over dt}\bigl(f(t)\bigr) $$
- Verify that this rule is true for the functions $f\colon \R\to\R^3$ by $f(t)=(\cos t,\sin t,t)$ and $g\colon\R\to\R^3$ by $g(t)=(t,t^2,t^3)$.
- Use this product rule to prove our observation from class that if the acceleration vector is perpendicular to the velocity vector for every $t$, then the speed is constant.
(Hint: You are trying to show that $|\v(t)|$ is constant. How is the norm related to dot products? And what is the derivative of a constant function? Be sure you use your if-then statements properly.)
- In class, we observed that a curve with a constant velocity vector is a line.
- Show that for a line $L(t)=p_0+t\v$, the velocity, $DL(t)$, is constant.
- Show that if $f\colon\R\to\R^3$ has $Df=\v$ for a constant vector $\v=\<{a,b,c}>$, then $f(t)=p_0+t\v$ for some point $p_0=(x_0,y_0,z_0)$.
(Hint: Integrate each coordinate of $\v$ separately. Don't forget your constant of integration.)
|
|