Solutions to Past AMS 161 Final Exams

 AMS 151 Final Exam Solutions for Spring 2009 
1. Test A: a) (3/2)pi^2, b) -(1/6)[1 + e^(cos(x^2))]^3, c) 2x +8ln|x-4|, 
d) (2/3)(4x-1)(x-1)^3/2 - (16/15)(x-1)^(5/2); 
Test B: a) pi^2, b) (1/16)[1 + e^(sin(x^4))]^4, c) 3x -15ln|x+5|, d) (2/3)(3x+2)(x-3)^3/2 - 4/5(x-3)^5/2.
2. Test A:a)  -e^(-4x)/4[x + 1/4], b) ln(6x)x^3/3 - x^3/9, c) -x^4cos(3x^4)/12+ sin(3x^4)/36;
Test B: a) [e^3x/3](x - 1/3), b) ln(4x)x^6/6 -x^6/36, c) -x^2cos(4x^2)/8 + sin(4x^2)/32.
3. Test A: a) 2sqrt(3), b) 1/6, c) undefined at x = 2; 
Test B: a) (3/2)(4)^2/3,  b) 3/5, c) undefined at 5.
4. Test A: RH < MP < integral < TP < LH;
Test B: RH < TP < Integral < MP < LH.
5. Test A: a) int 0 to 2 of pi[(y/2)^(1/3) - 1]^2, b) int from 0 to 2 of pi[10-x^2]^2 - pi[8-2x}^2;
Test B: a) int 0 to 1 of pi[(y/4)^(1/4) - 1]^2, b) in from 0 to 1 of pi[5-x^3]^2 - pi[3-3x]^2.
6. Test A: a) int from 15 to 50 of 150+3(50-x), b) int from 0 to 40 of 62.4pi*h(25-5h/8)^2,
 c) int from 0 to 100 of 62.4(100-y)2(y/3)^3/4;
Test B: a) int from 10 to 80 of 180+6(80-x), b) int from 0 to 50 of 62.4pi*h(15-3h/10)^2, 
c) int from 0 to 150 of 62.4(100-y)2(y/4)^5/4.
7. Test A: a) 1/3 - x/9 + x^2/27 - x^3/81, b) 1/3 - 2x^3/9 + 4x^6/27, 
c) x/3 -x^3/18 -2x^4/9 + x^5/360 + 2x^6/54;
Test B: a) 1/4 - x/16 + x^2/64 - x^3/256, b) 1/4 - 3x^3/16 + 9x^6/64,
c) x/4 - x^3/24 - 3x^4/16 + x^5/480 + 3x^6/96.
8. Test A: radius of conv = 1/2; Test B: radius of conv = 1/3.
9. Test A: a) y=3*x^(1/3), b) (-1/3)e^(4t) + (10/3)e^t;
Test B: a) y = 5*x^(1/2), b) e^(3t) + e^t.
10. Test A: a) y = 75-40e^(kt), where k=ln(3/8)/12, b) y = 7500(1-e^(-t/15));
Test B: a) y = 70 - 30e^(kt), where k=ln(1/3)/8, b) 6000(1-e^(-t/30)). 

AMS  161 Final Exam Solutions for Fall, 2008
1.  a) -2(pi)^5 , b) -(3/8)e^(2cos(x^4)), c) 4x - 4ln|x+1|, d) 2(2x-1)(x+
1)^(1/2) - (8/3)(x+1)^(3/2) (done by parts);
2. a) -(1/2)xe^(-2x) - (1/4)e^(-2x), b) (1/2)[ln(3x)]^2 (trick: is in the
 form of u*du where u = ln(3x)) , c) (1/15)x^5sin(3x^5) + (1/45)cos(3x^5)
3. a) 3/8, b) 3(2^(1/3)-(-3)^(1/3)), c) undefined  at x= 6;
4. RH < TP < Integral < MP < LH; 
5. a) int 0 to 2 of pi[(y/3)^2 - 5]^2, b) int from 1 to 2 of pi{[11-x^2]
^2 - [7-x]^2};
6. a) int 0 to 30 of 50 + 5(40-x), b) int 0 to 150 of 62.4(150-y)2sqrt((
ln(y+1))/2), c) int 0 to 12 of 62.4h*pi(8-8h/12)^2; 
7. a) 1 + 2x +3x^2 + 4x^3, b) 1 + 4x^2 + 12 x^4 + 32x^6, c) x +; (4-1/6)
x^3;
8. infinite radius of convergence.
9. a) y = 2e^(2-2e^x)), b) (13/2)e^3t - (7/2)e^5t;
10. Test A: a) dT/dt = k(T-70), T= 70 + 80e^(ln(1/2)t/10), b) dy/dt = 450 - y/20
 9000(1-e^(-t/20));




 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) 3/8, 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^2 + (1/24+9/2+54)x^4.
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).