Applied Math. Sample-Final. 2022
1. Determine if the series ∑ 𝑛2 𝑒 −𝑛 converges or diverges
2. Find the direction of the most rapid change of 𝑓(𝑥, 𝑦) = 𝑥 2 𝑦 + 𝑒 𝑥𝑦 sin⁡(𝑦) at P(1,0)
3. Determine by double integral the moment of inertia with respect to the x-axis 𝐼𝑥 = ∫𝑅 𝑦 2 𝑑𝐴
of a triangular plane section with vertices (0,1), (1,0) and (2,1).
4. Integrate 𝑓 = 1/(𝑥𝑦) over the square region with vertices (1,1), (2,1), (1,2), (2,2).
5. A tank initially holds 100 lt of brine(salty water) (V0=100 lt) containing 20 kg of salt (at t=0, Q=20kg).
A brine with 0.01 kg per lt (b=0.01) is poured into the tank at the rate of 5lt/min (e=5). The mixture
leaves the tank at the same rate (f=e). The differential equation for the amount of salt Q in the tank at
any time t is
1 𝑑𝑄 1 𝑓
+
𝑄=1
𝑏𝑒 𝑑𝑡 𝑏𝑒 𝑉0
Find the amount of salt Q in the tank at any time t.
6. Solve the differential equation
𝑑𝑦
𝑑𝑥
=
10𝑦
5+10𝑥
7. A body is thrown vertically into the air with an initial velocity of 𝑣(0) = 40⁡m/sec. The differential
equation of the velocity of this body is 𝑣 ′ + 10𝑣 = −10⁡,⁡⁡⁡⁡(⁡)′ = 𝑑(⁡)/𝑑𝑡. a) Find an expression
for the velocity 𝑣(𝑡), b) Find the time at which the body reaches its maximum height, i.e., the time
for 𝑣 = 0.
8. Solve the differential equation
−4.
𝑑2 𝑦
𝑑𝑥 2
𝑑𝑦
+ 𝑑𝑥 − 2𝑦 = 0 subject to initial conditions 𝑦(0) = 1, 𝑦 ′ (0) =