Supplementary notes

Main Points:
The gradient of a function is a vector that consists of the partial derivative of the function in respect to x and the partial derivative in respect to y. The partial derivative in the x direction makes up the x component of the vector and the partial derivative in the y direction makes up the y component of the vector. Directional derivatives are a representation of the change in x and y in any direction given by a unit vector. To find the directional derivative one must find the gradient of the function and then multiply the x and y components of the gradient at a given point by the x and y components of the unit vector being used. If you look at a graph of gradient vectors of a function you'll notice that they are perpendicular to the function and they point in the direction of greatest increase and the direction opposite would represent the greatest decrease. As the slope of a function increases, so does the gradient vector's magnitude.

Challenges:
The statements made about the graphical representation of gradient vectors were a little confusing. The direction that they point is stated as the direction of greatest increase for the function, but when I tried to visualize this rule with the given graph, I couldn't figure out how this was true.

Reflections:
It seems to me that there are real world applications for math that can solve for the rate of change of a function in any direction. I hope we can see some of the applications of this math, so it will be more than just a bunch of numbers and coordinates.