The problems below show that first substituting and then finding differentials gives the same results as first finding differentials and then substituting. This fact is given the fancy name "Chain Rule."
- Consider the elevation function $f(x,y)=xy^2$ and the path $\vec r(t) = (2t-1,t^2)$.
- Compute $f(\vec r(t))$ and then compute $\frac{df}{dt}$.
- Find $df$ in terms of $dx$ and $dy$. Then find $dx$ and $dy$ in terms of $dt$.
- Use substitution to find $df$ in terms of $dt$. Then state $\frac{df}{dt}$.
- In you previous work, label each of $f_x$, $f_y$, $\frac{dx}{dt}$ and $\frac{dy}{dt}$.
- Repeat 1. with $f(x,y)=e^x\sin y$ and $\vec r(t) = (t^2,t^3)$.
- Repeat 1. with $f(x,y)=ax+by$ and $\vec r(t) = (ct+d, et+f)$.
- If you finish early, make up your own function $f(x,y)$ and path $\vec r(t)$ and repeat.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |