Solutions to Past AMS 161 Final Exams

 AMS 161 Final Exam Solutions for Spring, 2008

1. (6 pt each) a) -3(pi)^2, b) (-1/9)cos(e^(3x^2)), c) 2x + 6ln|x-3|, d) 
(3x+1)2sqrt(x+3) - 4(x+3)^(3/2).
2. (8 pt each) a) xsin(3x)/3 + cos(3x)/9, b) -ln(2x)/3x^3 - 1/(9x^3), c)
  -x^6*e^(-3x^6))/18 - e^(-3x^6)/54.
3. (6 pt each) a) 5/7, b) okay at x =2, ans = 2sqrt(3) - 2sqrt(-2), c) 1/50.
4. LH < MP < true integral < TP < RH.
5. a) int from 0 to 2 of pi*((y/2)^(2/3) - 3)^2, b) pi[(16-x)^2 - (x^2 + 5)^2].
6. a) int from 0 to 20 of 30+4(40-x), b) int from 0 to 200 of (200-y)(62.4)((y+1)^2 -1), 
c) int from 0 to 5 of 62.4*15(8h/5)*h.
7. a) 1 - 3x + 6x^2 - 10x^3, b) 1 -9x^2 + 54x^4 - 270x^6, c) 1 - (19/2)x + (1/24+45/2)x^2.
8. 1/3.
9. a) y = ln(x^2 + 1),  b) y = e^2x + e^4x.
10. a) dT/dt = k(T - To), T(t) = 60 -10e^kt, where k = ln(1/2)/10, 
b) dP/dt = 400 - P/20 = -(1/20)(P - 8000), P(t)= 8000 - 8000e^(-t/20).

AMS 161 Final Exam Solutions for Fall, 2007

1. (6 pt each) a) e^4/2 + 19/2, b) (-1/6)[ln(cos(x^3)]^2, c) 3x - 6ln(x+2), d) (4/3)(x+2)^3/2 - 6(x+2)^1/2.
2. (8 pt each) a) (1/2)sin(2x) + (1/4)cos(2x), b) (-1/(2x^2))[ln(2x) + 1/2], c) (-e^(-2x^5)/10)[x^5+1/2]. 
3. (6 pt each) a) 1/2, b) undefined at x=2, c) 2sqrt(2) - 2i (all right at x=1).
4. (8 pt) LH < TR < integral Solutions to AMS 161 Final Exam, Spring 2007
1. a) 3/2(pi^2), b) -(1/6)e^(cos(2x^3)),  c) 2x + 8ln(x-4); d) (solved by int. by parts: (2/3)(x+3)(x-2)^(3/2) - (4/15)(x-2)^(5/2).

2. a) (x/2)e^2x - (1/4)e^2x,  b) (1/9)(3x^3*ln(2x) - x^3), c)-x^4*cos(3x^4)/12 + sin(3x^4)/36.
3. a) 1/4, b) 1/6, c) diverges at x = 2.
4. LH < MP < True Integral < TP < RH.
5. a) int from y = 0 to y = 2 of pi(y/2), b) int from x = 0 to x = 2 of pi[(10-x^2)^2 - (9-2x)^2].
6. a') Int from 0 to 8 of 2pi*x*25.3(x+2)^(-5/2), a") Int from 15 to 50 of 
110 + 3(50-x), b) Int from 0 to 40 of 62.5pi(20+x)(25-25x/40)^2, c) Int from 
0 to 100 of 62.5(100-y)2(y/3)^(3/5).
7. a) 1/2 - x/4 + x^2/8 - x^3/16, b) 1/2(1 - x^2 + x^4 - x^6), c) 1/2 - (3/4)x^2 + (37/48)x^4.
8. |x| < 1/2.
9. a) y = 6e^((3/2)x^2), b) (11/5)e^t - (1/5)e^6t.
10. a) DE y' = k(y-75), y = 75 - 40e^((a/12)t), where a = ln(13/40), b) y = 7500(1 - e^(-t/15).