MATH120 2023-2 Recitation Problems - Week 10
Note that some questions given below may NOT be solved during recitation.
1. Find โf(a, b) for a differentiable function fif Di+jf(a, b)=3โ2 and D3iโ4jf(a, b) = 5.
2. Suppose you are standing at the point (โ1,5,8) on a hill whose equation is z= 74 โx2โ7xy โ4y2.
The y-axis points north and the x-axis east, and distances are measured in meters.
(a) If you move to the south, are you ascending or descending? At what rate?
(b) If you move to the northwest, are you ascending or descending? At what rate?
(c) In what direction is the steepest downward path?
3. Find all points at which the direction of fastest change of the function f(x, y) = x2+y2โ2xโ4yis i+j.
4. Find โz
โx and โz
โy if yz = ln(x+z).
5. Find and classify all critical points of the function f(x, y) = Zy2
x
(t+ 1)etdt.
6. Find the maximum and minimum values of the function f(x,y ) = ln(xy2+ 1) on the triangular region
R={(x, y)โR2:xโฅ1, x +yโฅ2, x โyโค2}.
โโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโโ
OPTIONAL QUESTIONS
1. Find the directions in which the function f(x, y) = x4yโx2y3decreases fastest at the point (2,โ3).
2. Find the directions in which the directional derivative of f(x, y ) = x2+sin xy at the point (1,0) has the value
1.
3. Find โz
โx and โz
โy .
(a) x2+y2+z2= 3xyz
(b) xโz= arctan(yz)
4. Let f(x, y) = ax2+ (a+ 1)(y+ 1)2where ais a nonzero real number.
(a) Find all critical points of f.
(b) Find a value for asuch that fhas a saddle point.
(c) Find a value for asuch that fhas a local maximum.
5. Find absolute maximum and minimum points of the function f(x, y) = 2(x2+y2โ1)2+x2โy2on the region
R={(x, y) : R2:|x| โค 2,|y| โค 2}.
1
