Lee Cantrell, Ben Arrants, Jack Blahouse, Marcus Lynch
Chapter 6 practice test
6.1
1. Find the general solution to the exact differential equation:
𝑑𝑦
1
= 5𝑥 ln 5 + 2
𝑑𝑥
𝑥 +1
Solution:
𝑦 = 5𝑥 + tan−1 𝑥 + 𝑐
Find the anti-derivative
2. Solve the initial value problem explicitly:
𝑑𝑣
= 4 sec 𝑡 tan 𝑡 + 𝑒 𝑡 + 6𝑡 𝑎𝑛𝑑 𝑣 = 5 𝑤ℎ𝑒𝑛 𝑡 = 0
𝑑𝑥
Solution:
𝑣 = 4 sec 𝑡 + 𝑒 𝑡 + 3𝑡 2 + 𝑐
Find the anti-derivative
5 = 4 sec(0) + 𝑒 (0) + 3(0)2 + 𝑐
Plug in the points given
5= 4+1+0+𝑐
Simplify
0=𝑐
Solve for c
𝜋
𝜋
Answer: [𝑦 = 4 sec 𝑡 + 𝑒 𝑡 + 3𝑡 2 (− 2 < 0 < 2 )]
3. Solve the initial value problem using the Fundamental Theorem. (Your answer will contain definite
integral.)
𝐹 ′ (𝑥) = 𝑒 cos 𝑥 𝑎𝑛𝑑 𝐹(2) = 9
Solution:
𝑥
𝐹(𝑥) = ∫2 𝑒 cos 𝑡 𝑑𝑡 + 9
4. Use Euler’s method with increments of ∆𝑥 = 0.1 to approximate the value of y when 𝑥 = 1.3.
𝑑𝑦
= 𝑦 − 𝑥 𝑎𝑛𝑑 𝑦 = 2 𝑤ℎ𝑒𝑛 𝑥 = 1
𝑑𝑥
Solution:
(x, y)
(1, 2)
(1.1, 2.1)
(1.2, 2.2)
𝑑𝑦
=𝑦−𝑥
𝑑𝑥
1
1
1
∆𝑥
𝑑𝑦
∆𝑥
𝑑𝑥
0.1
0.1
0.1
∆𝑦 =
0.1
0.1
0.1
(𝑥 + ∆𝑥, 𝑦 + ∆𝑦)
(1.1, 2.1)
(1.2, 2.2)
(1.3, 2.3)
Answer: [𝐹(1.3) = 2.3]
5. Use Euler’s method with increments of ∆𝑥 = −0.1 to approximate the value of y when 𝑥 = 1.7
𝑑𝑦
= 𝑥 − 𝑦 𝑎𝑛𝑑 𝑦 = 2 𝑤ℎ𝑒𝑛 𝑥 = 2
𝑑𝑥
Solution:
(x, y)
(2, 2)
(1.9, 2)
(1.8, 2.01)
𝑑𝑦
=𝑥−𝑦
𝑑𝑥
0
-0.1
-0.21
∆𝑥
-0.1
-0.1
-0.1
𝑑𝑦
∆𝑥
𝑑𝑥
0
0.01
0.021
∆𝑦 =
(𝑥 + ∆𝑥, 𝑦 + ∆𝑦)
(1.9, 2)
(1.8, 2.01)
(1.7, 2031)
Answer: [𝐹(1.7) = 2.031]
Use the indicated substitution to evaluate the integral. Confirm your answer by differentiation
1.
sec(2x)tan(2x)dx, u=2x
Solution:
2.
sin(3x)dx, u=3x
Solution:
Use substitution to evaluate the integral
3.
Solution:
4.
Solution: