Math 115H (Assignments)

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1. Consider the function $f\colon\R^2\to\R$ by $f(x,y)=2x^3y-3xy^2+y$, and the point $P=(2,-1)$.
   1. In what direction is the function increasing the fastest at $P$?
   2. How fast is $f$ increasing in that direction at $P$?
   3. In what direction(s) from $P$ does $f$ initially stay the same?
2. Consider the function $f(x,y)=x^2+y^2$ and the level set at height $k=2$. Verify that $\grad f$ is perpendicular to the level set at the points $(2,0)$, $(0,2)$, $(1,\sqrt 3)$, and $(-\sqrt 3,1)$. What can you say about the gradient at any point $(a,b)$ on the level set?
3. Consider the function $f(x,y)=x^2-y^2$ and the level set at height $k=0$. Verify that $\grad f$ is perpendicular to the level set at the points $(2,2)$, $(-1,1)$, $(3,-3)$ and $(-1,-1)$. Show that $\grad f$ is perpendicular to the level set for any point $(a,a)$ or $(a,-a)$ on the level set at height 0.

Math 115H (Fall 2013) web pages Created: 30 Oct 2013 Last modified: Oct 30, 2013 4:18:51 PM Comments to: dpvc@union.edu

Assignments