AP CALCULUS CHAPTER 6 PRACTICE EXAM
Hailey Kettel, Tom Maxson, Jackson Fair
Find the solution to the exact differential equation
1. 𝑑𝑦/𝑑𝑥 = secxtanx-𝑒 𝑥
Solution: anti dirivitve of secxtanx = secx
𝑌 = 𝑠𝑒𝑐𝑥 − 𝑒𝑥 + 𝐶
2. Use seperation of Variables to solve intial value problem
𝑑𝑢
= 𝑒 𝑉−𝑈
𝑑𝑣
Solution:
𝑑𝑢
𝑑𝑣
=
𝑒𝑢
𝑒𝑣
∫ 𝑒 𝑢 𝑑𝑢 = ∫ 𝑒𝑢𝑑𝑢
3. Construct a slope field or table for the differential equation
𝑑𝑦
𝑑𝑥
= 2x-y
Solution
𝑑𝑦
Y=-1
Y=0
Y=1
Y=2
X=-1
=-1
=-2
=-3
=-4
X=0
= -1
=0
=1
=2
𝑑𝑥
= 2x-y
X=1
=3
=2
=1
=0
X=2
=5
=4
=3
=2
4. Use U-substitution to evaluate the integral
∫(3𝑥 2 + 2𝑥)𝑒 (𝑥
3 +𝑥 2 )
Solution
U= 𝑥 3 + 𝑥 2
du= 3𝑥 2 + 2𝑥 dx
Rewrite equation now with U and du
∫ 𝑒 (𝑢) 𝑑𝑢 = 𝑒 𝑢 + C then plug original U back in = 𝑒
𝑥 3 +𝑥 2 )
+C
5. ∫ 𝑆𝑖𝑛𝑥5𝑥 𝑑𝑥
Solution:
1
U= 5x
∫ 𝑠𝑖𝑛𝑢 5 𝑑𝑢
du= 5 dx
1
5
𝑑𝑢
5
- 5 cos5x + C
- cosu + C
1
= 𝑑𝑥
6. Integrate equation by parts
∫ 𝑥(𝑥 − 1)𝑑𝑥 when y=2 and x=1
Solution:
U= x
Equation: ∫ 𝑢𝑑𝑣 = 𝑢𝑣 − ∫ 𝑣𝑑𝑢
𝑥2
𝑥2
− 𝑥) – ∫( 2 − 𝑥) 𝑑𝑥
2
𝑥2
𝑥3
𝑥2
= 𝑋 ( 2 − 𝑥) – ( 6 − 2 )
𝑥2
𝑥3
𝑥2
= ( 2 − 𝑥2) - 6 + 2
Du= 1 dx
= X(
𝑑𝑉 = (𝑥 − 1) 𝑑𝑥
V=
𝑥2
2
−𝑥
=
𝑥3
3
−
𝑥2
2
+ 𝐶
7. If you invest 1500 into a bank account which pays 8.8% interest how long will it take for
the investment to double and how much will you have in 30 years
A: it will take 7.9 years to double and in 30 years will be worth $21,019.81
8. dydx= 3xLN3+3x2+1
A: 3x+3tan-1x+c