Math 115H (Assignments)

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1. Consider the function $f(x,y,z)=3xy^2+x^2z-2yz^3$.
   1. In what direction is the function's value increasing the fastest at $(2,-1,5)$?
   2. What is the rate of change in $f$ in the direction of $\<{1,1,2}>$?
   3. Find a direction in which the function's value should not change.
2. Consider the ellipse given implicitly by $3x^2+y^2=12$.
   1. Show that $(1,3)$ is a point on the ellipse.
   2. Find a vector that is perpendicular to the ellipse at $(1,3)$.
   3. Find a vector that is tangent to the ellipse at $(1,3)$.
   4. Find an implicit equation for the tangent line to the ellipse at $(1,3)$.
   5. Find a parametric equation for the tangent line to the ellipse at $(1,3)$.
3. Consider the ellipsoid $x^2+2y^2+3z^2 = 20$.
   1. Show that $(3,2,1)$ is a point on the ellipsoid.
   2. Find a vector that is perpendicular to the ellipsoid at $(3,2,1)$.
   3. Find two non-paralel vectors that are tangent to the ellipsoid at $(3,2,1)$.
   4. Find an implicit equation for the tangent plane to the ellipsoid at $(3,2,1)$.
   5. Find a parametric equation for the tangent plane to the ellipsoid at $(3,2,1)$.

Math 115H (Fall 2013) web pages Created: 6 Nov 2013 Last modified: Nov 6, 2013 7:38:26 PM Comments to: dpvc@union.edu

Assignments