1. Gradients and Directional Derivatives Chp 15.6

2. Putting it all together… Over the past several classes you have learned how to use and take partial derivatives Today we look at the essential difference between defining slope along a 2D curve and what it means on a 3D surface or curve

3. Example…What’s the slope of at (0,1/2)? What’s wrong with the way the question is posed? What’s the slope along the direction of the x- axis? What’s the slope along the direction of the y-axis?

4. Quick estimate from the contour plot: z Look at this in Excel:

5. The Directional Derivative We need to specify the direction in which the change occurs… Define, via a slightly modified Newton quotient: This specifies the change in the direction of the vector u =

6. The Gradient We can write the Directional derivative as: Gradient of f(x,y,z)

7. A Key Theorem Pg 982 – the Directional derivative is maximum when it is in the same direction as the gradient vector! Example: If your ski begins to slide down a ski slope, it will trace out the gradient for that surface!