Rapid recall
- If a rover moves along a straight line from point $(0,0)$ to point $(3,4)$, and drops 1 meters along this path, give the average slope along this path.
Group Problems
From measurements
A rover is located at the point $P$. Currently we've estimated that the slope moving east is $\frac{\Delta z}{\Delta x}\approx\frac{-1}{20}$ and the slope moving north is approximately $\frac{\Delta z}{\Delta y}\approx\frac{3}{20}$.
- If the rover moves west, what is the slope?
- Suppose the rover moves north 20 meters and east 20 meters? What is the actual displacement of the rover in the $xy$ plane (the run).
- Assuming the slopes stayed constant throughout this move, what happened to the height of the rover (the rise)?
- If the rover moves north east, then state the slope encountered in this direction.
- If the rover's displacement vector is $\vec u=(2,-3)$ meters, then how much did the height $dz$ change (assuming the slopes on the hill have stayed constant)?
- If the rover moves parallel to the direction $\vec u=(2,-3)$, then what is the slope in that direction?
- In which directions can the rover move so that the slope is zero?
- If the rover's displacement vector is $\vec u=(dx,dy)$, then give the change in height in the form $dz = ?dx+?dy$.
- If the rover moves parallel to the direction $\vec u=(dx,dy)$, then what is the slope in that direction?
- Give the direction $\vec u=(dx,dy)$ in which the rover will encounter the greatest slope.
From a function
A rover is located on a hill. The altitude at points on the hill is given by $z=f(x,y)= x^2+3xy$.
- Compute the differential $dz$ in terms of $x$, $y$, $dx$, and $dy$.
- Write $dz$ in the form $dz =?dx+?dy$, and then as a vector product $dz = (?,?)\cdot(dx,dy)$.
- If the rover is located at $P=(1,2)$, then what is the current height?
- What is the slope of the hill at $P=(1,2)$ if the rover moves east?
- What is the slope of the hill at $P=(1,2)$ if the rover moves north?
- What is the slope of the hill at $P=(1,2)$ if the rover moves north east?
- In which direction should the rover move so that the height does not change?
- What is the slope of the hill at $P=(1,2)$ if the rover moves in the direction $\vec u = (dx,dy)$?
- Give the direction $\vec u=(dx,dy)$ in which the rover will encounter the greatest slope.
Mathematica
Let's go through as many of these notebooks today as we have time.
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