**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 *!