Section 13.4: The Gradient and Directional Derivatives

Here's a general outline of what I plan to cover in class:

• The definition of the gradient and what it is used for
• The gradient of a linear function and how it acts as we expect
• Graphing gradient vectors on top of level curves
• Using the gradient for linear approximations to surfaces rather than just to functions (like example 2)
• Directional derivatives - amazing stuff! Rather than just giving the derivative in the x- or y- direction, we now take the derivative in ANY direction!!!! Pretty darn cool! Just to make sure that you all understand: the directional derivative at a point in the direction of the unit vector u gives the rate at which the function changes at that point in the direction of u.
• Will skip the last little section from the middle of page 736-end for the moment. So read it if you want, otherwise skip it.

When reading section 13.4, consider and try to answer the following questions. Take notes in your journal, so that you can discuss your thoughts in class.

1. The gradient of a function  f : Rⁿ --> R at an input (x₁, x₂, ... , x_n) is a vector. Which vector?
   (a) What are its components?  (Try #6,7 first to do this for an input (x₁, x₂) or (x₁, x₂, x₃) and then generalize.)
   (b) What does the direction of this vector tell you?
   (c) What does the length of this vector tell you?
2. Try exercise 1,2,3.
3. Can you use the idea of Example 2 to find the tangent plane to the surface z = f(x,y) = 3+cos(x)sin(2y) at the point (0,0,3) discussed in the previous section? (Hint: "gradient as normal vector")
   If yes, do it, if no, explain why not.
4. In your own words: What do we mean by "D_u f (x₀,y₀)", where z = f (x,y) is a function of two variables?
   What is u?
5. What, to you, are the two most amazing conclusions of this section?
6. What, to you, are the two most unclear ideas/concepts/conclusions in this section?
   

Besides the Exercises mentioned above, I recommend looking at #31a,32