Solutions to Past AMS 161 Final Exams

AMS 161 Final Exam Test A and Test B Solutions for Fall, 2010

Test A<
1. (6 pt each) a) 3pi^2, b) (-1/9)[cos(e^(3x^3))], c) 2x - 6ln(x-2), d) (2x-10)(x+3)^(1/2).
2. (8 pt each) a) (1/3)sin(3x) + (1/9)cos(3x), b) (-1/(3x^3))[ln(2x) + 1/3], c) (-e^(-3x^6)/18)[x^6+1/3]. 
3. (6 pt each) a) 3/8, b) 2{sqrt(3) -sqrt(-2)), all right at x = 2, c) 1/10.
4. (8 pt) RH < MP < integral Test B
1. (6 pt each) a) -cos(5x)/5 + 1/5 - 3pi^2, b) (1/8)sin(e^(2x^4)), c) 2x + 2ln(x-1), d) (4x-10)(x+3)^(1/2). 
2. (8 pt each) a) -(1/5)x*cos(5x) + (1/25)sin(2x), b) (-1/(3x^3))[ln(3x) + 1/3], c) (-e^(-3x^7)/21)[x^7+1/3].
3. (6 pt each) a) 5/7, b) c) 2sqrt(6) - 2sqrt(-2) (all right at x=2), c) 1/21.
4. (8 pt) RH < TP < integral  AMS 161 Final Exam Solution for Spring 2010 
1. a) (e^9/3 - 9^2/2)-(1/3), b) -cos(e^(3x^3))/9, c) 4x - 4ln(x+1), d) (5/3)sqrt(2x+2)(x-2) (done by parts and simplified)
2. a) e^4x[x/4 - 1/16], b) (x^8/8)[ln(x) - 1/8], c) -x^2*cos(2x^2)/4+sin(2x^2)/8
3. a) -1/[2(x-2)^2] undefined at x=2, b) 1/7, c) -1/(x-1) undefined at x=1
4. LH < MP < TI < TP < RH
5. a) int from 1 to 3 of pi[(2/y^2)^2], b) int from -1 to +1 of pi[(6-x^3)^2 - (2-x)^2]
6. a) int from 20 to 80 of 50 + 5(80-x), b) int from 0 to 12 of 62.5pi(10-10h/12)^2*h, 
  c) int from 0 to 250 of 62.5(250-y)4ln(y+1)
7. a) ln(2) + x/2 - x^2/8 + x^3/24, b) ln(2) + x^2 - x^4/2 + x^3/6,
  c) xln(2) + (1- ln(2)/6)x^3 + (-1/2 + 1/6 + ln(2)/120)^x^5 
8. an = 5^n/n!, limit of 5/(n+1) -> 0, rad of covergence is unbounded
9. a) y = Ae^[-(2/3)e^(-3x)], where A = 5e^(2/3),  b) 6e^t - e^(4t)
10. a) T = 70 + 60e^(kt) where k = (1/8)ln(1/2),  b) y = 4000(1-e^(-t/20)

 AMS 161 Final Exam Solutions for Fall 2009 
1.{6 pts each} a) e^8/4 - 25/4, b) (1/6)sin(e^(3x^2)), c) 2x + 6ln(x-3), d) (done by substitution) (1/9)((4/3)(3x-2)^(3/2) + 8(3x-2)^(1/2))
2.{8 pts each} a) (4/3)xe^(3x) - (4/9)e^(3x), b) (1/8)ln(x)*x^8 - (1/64)x^8, c) -x^5cos(2x^5)/10 + sin(2x^5)/20
3.{6 pts each} a) Undefined at x = 2, b) 1/3, c) Undefined at 1
4. {6 pts} LH < TP < True Integral < MP < RH
5. {10 pts} a) Integral from 1 to 4 of pi*(2/y)dy, b) Integral from -1 to 1 of pi*[(8-x^2)^2 - (4-x)^2]
6. a) (8 pt) integral from 10 to 40 of 25 + 2(40-h), b) (12 pts) integral from 0 to 30 of 62.5*pi*(20-2h/3)^2(h+40),
  c)(12 pts) integral from 0 to 125 of 62.5(125-y)6ln(y+1)
7. a) (9 pts) x - x^2/2 + x^3/3, b) (4 pts) 3x^2 - 9x^4/2 + 9x^6, c) (5 pts) 3x^3 + 5x^5 + (9 + 3/4 + 1/40)x^7
8. {5 pts} radius of conv is 3/4
9. a) (10 pts) y = 0e^(1.5e(-2x)) = 0, b) (12 pts) (7/2)e^t - (1/2)e^(3t)
10. a) (10 pts) T = 60 + 80e^(ln(.5)t/80, b) (15 pts) dy/t = 200 -t/15, y = 3000(1-e^(-t/15)