Math 241, Spring 2006 (Alex Freire)
LIST 2: Partial derivatives, directional derivatives, chain rule,
differential, mean value theorem, Taylor approximations
1. Let f(x, y) = −3·x2+ 2 ·xy + 2 ·y2. Find fxxfyy −fxyfy x. (Answer
: -28)
2. Let f(x, y, z) = 24 ·x+ 1 ·y2+ 1 ·z2+ 4 ·yz. Find the directional
derivative of fat P(4,1,−5) in the direction of [2,2,1]. (Answer : 2)
3. Let f(x, y) = 2 ·x2y+ 2 ·y2−1·x,x(t) = 1 + 1 ·t,y(t) = 2 + 2 ·t.
Find the limit of f(x(t),y(t))−f(1,2)
tas tgoes to 0. (Answer : 27)
4. Let f(x, y) = 4 ·x2y+ 4 ·y2−1·x,x(t) = 1 + 2 ·t,y(t) = 2 + 5 ·t.
Find the second derivative of g(t) = f(x(t),y (t)) at t= 0. (Answer : 424)
5. Let f(x, y)=1·x2y+ 5 ·y2−1·x,x(t) = 4 + r·t,y(t) = 4 + s·t,
where r, s are constants. Find the second derivative of g(t) = f(x(t), y(t))
at t= 0. (Answer : r2·8+2rs ·8 + s2·10)
6. Let f(x, y) = 2 ·x3+ 42 ·x2+ 6 ·x·y−216 ·y+ 3 ·y2.Find Du(Du(f))
at (−4,40) if u= [a, b]. (Answer : 36 ·a2+ 12 ·a·b+ 6 ·b2)
7. If z=p28 −4·x2−2·y2and (x, y) changes from (1,2) to (1.04,1.88),
find 100 ·|dz−∆z|
z+∆z. (Answer : 0.125187)
8. Suppose f(x, y, z ) = x1·y4·z−3, 100 ·|∆x|
|x|≤0.5, 100 ·|∆y|
|y|≤0.5, and
100 ·|∆z|
|z|≤0.5. Estimate 100 ·|∆f|
|f|(Answer : 4)
9. Let f(x, y, z) be the distance from P(x, y, z) to Q(2,3,1). Find the
gradient of fat R(2,−2,3). (Answer : [0,−.928477,0.371391])
10. Let f(x,y,z) be the distance from P(x, y, z) to the line x= 2 ·
t, y = 3 ·t, z = 3 ·t. Find the gradient of fat R(3,5,−3). (Answer :
[0.316192,0.5571,−.767895])
11. Let f(x, y, z) be the distance from P(x, y, z ) to the plane 1·x+5·y+5·
z= 5. Find the gradient of fat R(−3,5,2). (Answer : [0.140028,0.70014,0.70014])
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