SUNY Geneseo Department of Mathematics
Monday, March 27
Math 223
Spring 2023
Prof. Doug Baldwin
Reading the table in problem set 8?
Rows correspond to y values, columns to x. So, for example, f(3, 1) is 5.5, found in the third column and first row.
No responses yet, but I’ll keep checking.
Finish the last example from Friday.
The first thing left to the listener was to express the derivative we found solely in terms of s and t, the actual independent variables in the problem. To do this, just replace all occurrences of x with its definition, s2 + t2, and all of occurrences of y with st:
The next part was to find the derivative of z with respect to t. We had the expression for this from the tree diagram we built Friday, and we’d already calculated the derivatives of z with respect to x and y. We had to calculate the derivatives of x and y with respect to t, and then put everything together, and finally replace x and y with their definitions in terms of s and t:
Based on “Directional Derivatives” in section 3.6.
The definition of a directional derivative is a standard limit-based derivative definition, except in a form that no longer needs to be parallel to one of the coordinate axes. See the textbook for the precise definition.
The alternative formula, in terms of products of partial derivatives and sines or cosines of a direction angle, is easier than the limit definition to calculate. See the textbook or the examples below for this formula.
Directions can also be given by unit vectors.
Suppose f(x, y) = 3x2 - 5y2. What’s the directional derivative at (1, 2) in a direction 30° from the x axis?
Notice one thing this example illustrates about the direction angle vs direction vector: they’re interchangeable, in the sense that the direction you’re interested in can be given either as an angle relative to the x axis, or as a vector that makes that angle with the axis, or even (as in the above sketch) as both.
Start by working out the partial derivatives of f, then evaluate them at point (1, 2), and finally multiply by the cosine and sine of the angle:
Geometrically, interpret the number we got as how fast you would rise or descend if you were walking across the graph of this function from point (1, 2, f(1,2)), moving at a 30 degree angle to the positive x axis.
What’s the general function for the directional derivative in that direction?
Use the same approach as the directional derivative at a point, except use the partial derivative functions instead of values at a point:
If g(x, y) = (sin x)(cos y), what’s its directional derivative in direction ⟨3, 4⟩?
Start by thinking about how to get an angle from the vector. If you draw the vector in the xy plane, it forms the hypotenuse of a triangle, from which you can use the “SOHCAHTOA” definitions of cosine and sine to get those values. But as you do that, notice that what you’re doing is exactly the same calculation you’d use to find a unit vector in the direction of the original vector.
Moral: If you express a direction vector as a unit vector, its components are the cosine and sine you need, and you don’t need to work out the angle explicitly.
Finish the last directional derivative example.
Weather permitting, we’ll have a “calculus field trip.”
Then start talking about the vector ⟨∂f/∂x, ∂f/∂y⟩, which is called the “gradient” of f, and has lots of interesting properties.