- Using the contour diagram provided:
- Locate and mark at least 2 peaks (local maximum points) and 2 bowls (local minimum points). How can these points be reliably identified in a contour diagram?
- Add gradient vectors at several points along the level curve nearest to one of the extreme points identified above. What do you notice about the gradient vectors? What is the magnitude of the gradient at the actual extreme point?
- Can you locate any other points on the contour diagram that have the same magnitude of the gradient as what you found at the extreme points? How do these differ from the gradient at the extreme points?
- Let $f(x,y) = x^2+xy+y^2-3x$.
- Find all $(x,y)$ locations where $\vec \nabla f = \vec 0$.
- At each location you found, does the function have a maximum, minimum, or saddle at that point. Use the eigenvalues of the second derivative to make your decision.
- You can find lots more practice with this in section 14.7 in the text.
- Let $f(x,y) = x^2+y^2$. Your goal is to find the smallest value of $f$ at points on the curve $g(x,y)=xy^2=16$.
- Draw the curve $xy^2=16$. Then add to your curve several level curves of $f$.
- Find $\vec \nabla f$ and $\vec \nabla g$.
- Solve the system $\nabla f=\lambda \nabla g$ together with $xy^2=16$. (Show the three equations you need to solve are $2x=\lambda y^2$, $2y=2\lambda xy$, and $xy^2=16$.)
- Repeat 1. with $f(x,y) = x^3-6x-y^2+4x$.
|
Sun |
Mon |
Tue |
Wed |
Thu |
Fri |
Sat |