Task 17.1

We'll focus this task on making sure we understand how differentials can help us approximate changes in a function.

A forest ranger needs to estimate the height of a tree. The ranger stands 50 feet from the base of tree and measures the angle of elevation to the top of the tree to be about 60$^\circ$.

  1. If this angle of 60$^\circ$ is correct, then what is the height of the tree? Explain in general why the height of the tree is $h(\theta) = 50 \tan \theta$.
  2. Compute the differential $dh$ in terms of $\theta$ and $d\theta$.
  3. The ranger's angle measurement is mostly likely off by some amount. If the error in the ranger's measurement could be as much as $d\theta = 5^\circ$ (so $\frac{5\pi}{180}$ radians), then use differentials to estimate how large the error in the height could be (so compute $dh$). If your answer here is quite large (much larger than the height of the tree), then look back at your work and see if using radians instead of degrees makes a difference. Why does it? Feel free to ask in class.
  4. Compute the height if the angle were exactly 65 instead of 60. What's the actual difference between these two heights?

The US mint creates coins that are roughly a cylindrical shape, with volume $V = \pi r^2h$. Unfortunately, not every coin is exactly the same size, and small errors in $r$ (given by $dr$) and small errors in $h$ (given by $dh$), affect the amount of material needed to mint these coins.

  1. Compute $dV$ to give an approximate for the change in volume given by the errors $dr$ and $dh$.
  2. The radius of a coin is much larger than the height. Will an error in the radius, or an error in the height, cause a larger change in volume? Explain using your differentials.
  3. A soda can company has a cylindrical shape that instead has a large $h$ with small $r$. Will an error in the radius, or an error in the height, cause a larger change in volume in this situation.

Task 17.2

Suppose that our rover is located at point $P=(x,y)$ on a hill whose elevation is given by $z=f(x,y)$. The rover will be moving in the direction parallel to $\vec u$.

  1. Explain why the slope of the hill at $P$ in the direction $\vec u = (dx,dy)$ is given by $$\frac{dz}{\sqrt{(dx)^2+(dy)^2}}.$$
  2. Prove that this slope can be written, using gradients, as $$\vec \nabla f(P) \cdot \frac{\vec u}{|\vec u|}.$$
  3. Use the above fact to compute the slope of a hill given by $f(x,y) = x^2+3xy$ at $P=(2,-1)$ in the direction $\vec u = (3,4)$. (We call this the directional derivative of $f$ at $P$ in the direction $\vec u$, written $D_{\vec u}f(P)$.

The directional derivative of $f$ in the direction of the vector $\vec u$ at a point $P$ is defined to be $$D_{\vec u} f(P)=\vec \nabla f \cdot \frac{\vec u}{|\vec u|}.$$ We can simplify the above to just $f(P)=\vec \nabla f \cdot \hat u$ if $\hat u$ is a unit vector. We dot the gradient of $f$ with a unit vector in the direction of $\vec u$.

  1. Show that the partial derivative of $f$ with respect to $x$ is precisely the directional derivative of $f$ in the $(1,0)$ direction.
  2. Show that the partial derivative of $f$ with respect to $y$ is precisely the directional derivative of $f$ in the $(0,1)$ direction.

Please watch this short 2 part video that discusses the gradient a bit more, and how you can connect the gradient to the slope in various directions.

Task 17.3

Suppose our rover is located at a point $P$ on a hill whose elevation is given by $z=f(x,y)$. Recall that the directional derivative of $\vec f$ at $P$ in the direction $\vec u$ is the dot product $D_{\vec u} f(P)=\vec \nabla f(P)\cdot \frac{\vec u}{|\vec u|}.$ Also recall that we can compute dot products using the law of cosines $\vec \nabla f(P)\cdot \vec u= |\vec \nabla f(P)| |\vec u|\cos\theta,$ where $\theta$ is the angle between $\vec \nabla f(P)$ and $\vec u$.

  1. Give a formula for the angle $\theta$ between the two vectors $\vec \nabla f$ and $\vec u$?
  2. Given a direction $\vec u$, the directional derivative will give the slope of $f$ at $P$ in the direction $\vec u$. We want to know which direction we should be pick to obtain the largest slope (directional derivative). Explain why the angle between $\vec u$ and $\vec \nabla f(P)$ must be 0, in order to obtain the largest slope.
  3. State a vector $\vec u$ that yields the largest directional derivative.
  4. When $\vec u$ is parallel to $\vec \nabla f(P)$, show that $D_{\vec u}f(P) = |\vec \nabla f(P)|$. In other words, explain why the length of the gradient is precisely the slope of $f$ in the direction of greatest increase (the slope in the steepest direction).
  5. Which direction points in the direction of greatest decrease? What is the slope in that direction?
  6. In your own words, summarize what facts this task helped you learn about the gradient.

Task 17.4

The last problem for prep each day will point to relevant problems from OpenStax. Spend 30 minutes working on problems from the sections below.

  • Return to any of the previous day's OpenStax problems to locate extra practice.