Day 13 in Mat329

• I've given you a new assignment for Friday.
• Homework: graded 7, 22, 36.
  • #7: equation of plane (linearization), and graph....
  • #22: Using the linearization to estimate $f$. Patrick.
  • #36: how do we maximize the differential? Colton or Susannah or Caleb -- you each made the right choice.
  Remember to reality-check your answers. Some of you gave impossible values. Anytime you're dealing with a physical problem, a story problem, you have the opportunity to ask "Does my answer make sense?"

• Quiz is returned.
  • A few of you still don't know the most important definition in calculus!
  • Some of you started talking about discontinuity.
  • Use partial derivatives as well as full....
  • Some of you arrived at a conclusion in defiance of your answer to part b!

• The First Major/Minor Lunch of the 2015/2016 Academic year!

  Friday 09/18 from 11:30a.m. to 1:30 p.m.

  MEP 4th Floor Atrium

  All Mathematics and Statistics Majors and Minors are invited (or even if you're thinking about it!).

  Friday's menu: Reuben Sandwiches, Brats, Mets, a little sauerkraut for the Brats and Mets, a few vegetarian hotdogs, Potato pancakes, hot slaw, baked beans, and very chunky applesauce. Cookies for dessert.

  This is a great chance to meet your faculty and fellow Mathematics and Statistics Majors and Minors

In this section we are interested in the slope of the surface along directions other than $x$ or $y$: hence the idea behind directional derivatives. At a given point at which a function is differentiable, one natural choice for two directions might be the direction in which the function is increasing fastest, and the direction perpendicular to this (a level curve).

One of the biggest pieces of news is that we're going to be working with vectors -- e.g. $u$ -- and with vector-valued functions; and that will place another demand upon your visualization skills.

But gradients are just another kind of multivariate function: one where the domain is points in the plane, like many of our other functions, but the range is the set of vectors.

At each point in the plane there is a gradient vector pointing -- indicating what?

Are you comfortable with

• the two properties of a vector?
• unit vectors?
• dot products?
• norms?

1. The directional derivative of $f$ at $(x_0,y_0)$ in the direction of unit vector ${\bf{u}} = \langle a,b \rangle$ is \[ D_{\bf{u}} f(x_0,y_0) = \lim_{h \to 0}\frac{f(x_0+ha,y_0+hb)-f(x_0,y_0)}{h} \] if this limit exists.

2. If $f$ is a function of two variables $x$ and $y$, then the gradient of $f$ is the vector function $\nabla f$ defined by \[ \nabla f(x,y) = \langle f_x(x,y), f_y(x,y) \rangle = \frac{\partial f}{\partial x} \hat{i} + \frac{\partial f}{\partial y} \hat{j} \]

3. Theorem: \[ D_{\bf{u}} f(x,y) = \nabla f(x,y) \cdot {\bf{u}} \]

4. More generally (in higher dimensions), we can write \[ D_{\bf{u}} f({\bf{x}}_0) = \lim_{h \to 0}\frac{f({\bf{x}}_0+h{\bf{u}} )-f({\bf{x}}_0)}{h} \] where $\bf{u}$ is a unit vector. (Isn't that a beautiful analogy with the univariate derivative?)

5. Theorem: Suppose $f$ is a differentiable multivariate function. The maximum value of the directional derivative $D_{\bf{u}}f({\bf{x}})$ is $|\nabla f({\bf{x}})|$ and it occurs when ${\bf{u}}$ has the same direction as the gradient vector $\nabla f({\bf{x}})$. A Mathematica example.

1. #1, p. 967
2. #5
3. #7