Question 1:
For f(x,y) = y + 𝑥 2
a.) Identify the Domain:
y + 𝑥 2 is defined for all real numbers.
b.)Sketch the level curves for k = -1, 0, and 1:
c.) Find the rate of change of f at the point (1,-2,-1) in the direction of
⃑ = <2,3>:
v
7
√13
d.) Sketch the vector in the direction of maximal rate of change at (1,-2)
on the level curve at k = -1.
Question 2:
Evaluate the following Limits, if they exist. If a limit doesn’t exist, show
why:
a.)
lim
𝑥 3𝑦
(𝑥,𝑦)→(0,0) 𝑥 6 +𝑦 2
DNE, Use y = x to get a limit of 0, and use y = 𝑥 3 to get a limit of
b.)
lim
𝑥 4 +𝑦 2
(𝑥,𝑦)→(0,0) √𝑥 4 +𝑦2 +9−3
Use conjugate. The limit is 6.
c.)
lim
𝑥 3𝑦
(𝑥,𝑦)→(0,0) 𝑥 5 𝑦 3 +𝑥 3 𝑦
Cancel out common factors. The limit is 1.
1
2
Question 3:
a.) Find the tangent plane to f(x,y) = 𝑥 ln | sin(𝑥𝑦) | − 𝑦 2 𝑒 𝑥
𝜋
at the point (1, ):
2
−𝑒𝜋 2
𝜋
𝑒𝜋 2
(𝑥 − 1) − 𝑒𝜋 (𝑦 − ) −
𝑧=
4
2
4
b.) Find the linear approximation to f at (2, 𝜋):
𝐿(2, 𝜋) = −𝑒𝜋 2
Question 4:
Given f(x,y) = 𝑥 4 𝑦 − ln |𝑥𝑦 3 |, find the following:
𝑓x : 4𝑥 3 𝑦 −
𝑓y : 𝑥4 −
1
𝑥
3
𝑦
𝑓xx : 12𝑥2 𝑦 +
𝑓yy :
3
𝑦2
𝑓xy : 4𝑥3
𝑓yx : 4𝑥3
𝑓yxy : 0
1
𝑥2
Question 5:
Determine whether the following functions satisfy this equation:
𝑓xx + 𝑓yy = 0
(State YES or NO):
a.) f(x,y) = 𝑥 sin(𝑦) + 𝑦 cos(𝑥)
NO
b.) f(x,y) = sin(3𝑥)𝑒 −3𝑦
YES
c.) f(x,y) = 𝑥𝑒 −𝑥𝑦
NO
Question 6:
Find the equation of the tangent plane to z = ln(1 + 𝑥𝑦) + 𝑥𝑦 2
At the point (1,2):
𝑧=
14
13
(𝑥 − 1) +
(𝑦 − 2) + 4 + ln 3
3
3
Question 7:
Given z = sin(𝑒 𝑥𝑦 ) where
𝑥 = 𝑟 2 + cos(𝑡𝑠) 𝐴𝑁𝐷 𝑦 = 𝑠 3 𝑟 + cos(𝑟 2 )
Draw a tree diagram, and then use the chain rule to find:
𝜕𝑧
a.) 𝜕𝑠
[𝑦𝑒 𝑥𝑦 cos 𝑒 𝑥𝑦 ][−𝑡 sin 𝑡𝑠] + [𝑥𝑒 𝑥𝑦 cos 𝑒 𝑥𝑦 ][3𝑠 2 𝑟]
𝜕𝑧
b.) 𝜕𝑟
[𝑦𝑒 𝑥𝑦 cos 𝑒 𝑥𝑦 ][2𝑟] + [𝑥𝑒 𝑥𝑦 cos 𝑒 𝑥𝑦 ][𝑠 3 − 2𝑟 sin 𝑟 2 ]
𝜕𝑧
c.) 𝜕𝑡
𝑦𝑒 𝑥𝑦 cos 𝑒 𝑥𝑦 [−𝑠 sin 𝑡𝑠]
Question 8:
Determine and categorize the critical points of
f(x,y) = 4 + 𝑥 3 + 𝑦 3 − 3𝑥𝑦:
(0,0) is a saddle point
(1,1) is a Relative Minimum
Question 9:
Evaluate these Integrals:
a.) ∬
1+𝑥 2
1+𝑦 2
𝑑𝐴 over 𝑅 = {(𝑥, 𝑦)|0 ≤ 𝑥 ≤ 1, 0 ≤ 𝑦 ≤ 1}:
𝜋
3
b.)
-6
∬ 4𝑥 3 − 9𝑥 2 𝑦 2 𝑑𝐴 over 𝑅 = {(𝑥, 𝑦)|0 ≤ 𝑥 ≤ 1, 1 ≤ 𝑦 ≤ 2}:
Question 10:
Evaluate the following Integral:
39
3
∬ 𝑥 3 𝑒 𝑦 𝑑𝑦 𝑑𝑥
0 𝑥2
1 729
(𝑒
− 1)
12