Problem 1. Verify that the Chain Rule (as stated in class) is true for the functions
Do this by first writing out their composition as a function and then computing the derivative matrix .
Next, compute the derivative matrices and and multiply them. Your final answer should equal .
Problem 2. Consider the function . Use the Chain Rule to compute where is our standard cylindrical coordinate variable.
Problem 3. (a) Suppose is a differentiable function and that at the point we know that and we know that
Compute the linear approximation, , for at the point .
(b) Suppose is a differentiable function and that at the point we know that
Compute the derivative matrix at the point where is the function from part (a).
Problem 4. Find the equation of the plane that passes through the point and is parallel to the plane .
Problem 5. Draw a picture of the set .
Problem 6. Draw a picture of the line parameterized by
Problem 7. Consider the function .
(a) Compute the derivative matrix .
(b) Compute the gradient .
(c) Form the new function defined by
where is the function that parameterized the line in problem 6. Note that this function is the kind of function we might see in a Calculus I class; for old time’s sake, compute the derivative .
(d) Use the Chain Rule to compute the derivative matrix
where is the same composed function from part (c).
Note: Your computation for part (d) could have been re-phrased in terms of the gradient of instead of its derivative matrix, . Had we done this, we would have found
This is an important note! It is our definition of something called a directional derivative! That is, we think of this computation as telling us how changes at the point when we move inputs in the direction of !