Math 115H (Assignments)

# Homework on 23 October 2017:

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1. Consider the function $f\colon\R^2\to\R$ by $f(x,y)=x^2y^3$. Suppose you want to know the directional derivative for $f$ at the point $(2,-1)$ in the direction of $\langle 1,2\rangle$.

In class, we described a method of determining the directional derivative at $(x_0,y_0)$ in the direction of a unit vector, $\u$: first parameterize the line through $(x_0,y_0)$ in the direction of $\u$ as a function $L(t)$, then plug this line into $f$ to get a new function $F\colon\R\to\R$ by $F(t)=f(L(t))$. Then since $L(0)=(x_0,y_0)$, $F'(0)$ will give the directional derivative at $(x_0,y_0)$ in the direction of $\u$. Use this approach to compute the directional derivative described above.

(Hint: You need to make sure $\u$ is a unit vector. You may find it easier to use the product and chain rules rather than trying to multiply everything out before differentiating. Since you are going to be substituting $t=0$, there is not much point in simplifying the answer before evaluating it.)

2. Finish the WeBWorK assignment.

 Math 115H (Fall 2017) web pages Created: 23 Oct 2017 Last modified: Oct 23, 2017 5:41:28 PM Comments to: dpvc@union.edu