Jordan Lindamood 6.1
Bret Stoneman 6.2
Nathaniel Kreiman 6.3
Valerie 6.4
6.1-6.4 Practice Test
6.1 Jordan Lindamood
Brett Stoneman
6.2 Section Practice Test
Evaluate the Integral
1. ∫dx/(1-x)2
2. ∫sec(ϴ+л/2)tan(ϴ+л/2)dϴ
3.∫√(cotx)csc2xdx
6.3 Nathaniel
1) Find the indefinite integral
∫ 2𝑡 cos(3𝑡) 𝑑𝑡
Solution:
2) Solve the initial value problem
𝑑𝑢
= 𝑥 sec 2 (𝑥) , 𝑤ℎ𝑒𝑛 𝑢 = 1 𝑎𝑛𝑑 𝑥 = 0
𝑑𝑥
Solution:
Integrate by parts to find the unknown integral
∫ 𝑒 𝑥 cos(2𝑥) 𝑑𝑥
Solution:
6.4 Valerie
𝑑𝑦
1. 𝑑𝑥 = cos 𝑦 2
𝑑𝑦
2. 𝑑𝑥 = (2𝑥 + 1)(𝑦 + 1)
and y=0 when x=0
Answer: y= 𝐭𝐚𝐧−𝟏 𝒙
and y=1 when x=-1
𝒙𝟐
Answer: y= 𝟐𝒆 𝒆𝒙 − 𝟏
1
∫
𝑑𝑦 = ∫(2𝑥 + 1) 𝑑𝑥
(𝑦 + 1)
ℓ𝓃 (𝑦 + 1) = 𝑥 2 + 𝑥 + 𝑐
2
𝑒 ℓ𝓃(𝑦+1) = 𝑒 𝑥 𝑒 𝑥 𝑒 𝑐
2
Y= 𝑒 𝑥 𝑒 𝑥 𝑒 𝑐 − 1
C= ℓ𝓃2 (Plug in C and get answer)
3. Find the amount of time required for a $2000 investment to double if the annual interest rate r is
compounded (𝑎)𝑎𝑛𝑛𝑢𝑎𝑙𝑙𝑦, (𝑏)𝑚𝑜𝑛𝑡ℎ𝑙𝑦, (𝑐)𝑞𝑢𝑎𝑟𝑡𝑒𝑟𝑙𝑦, 𝑎𝑛𝑑 (𝑑)𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠𝑙𝑦. r= 4.75%
Answer: a: t=14.94 years, b: t=14.62 years, c: 14.68 years, d: 14.59 years.
𝑟
Plug information into equation: A(t)=𝐴𝑜 (1 + 𝑘) .𝑘𝑡 Solve.
4. Suppose that a cup of soup cooled from 90℃ to 60℃ in 10 min in a room whose temperature was
20℃. Use Newton’s Law of Cooling to answer the following questions.
(𝑎) How much longer would it take the soup to cool to 35℃?
(𝑏) Instead of being left to stand in the room, the cup of 90℃ soup is put into the freezer whose
temperature is -15℃. How long will it take the soup to cool from 90℃ to 35℃?
Answer: a: 17.53 minutes, b: 13.26 minutes.
A:
Find K:
B:
5. Half-life. The radioactive decay of Sm-151 can be modeled by the differential equation
Dy/dx= -0.0077y, where t is measured in years. Find the half-life of Sm-151.
Answer: t= 90 years.
Use the equation 𝑦 = 𝐴𝑒 𝑘𝑡
1
2
Make y= 𝐴
1
= 𝑒 −0.0077𝑡
2
1
ℓ𝓃 2 = -0.0077t