1.) a.) f(x) = (1 + 14x)/[(1 - x)(1 + 2x)] = [A(1 + 2x) + B(1 - x)]/[1 - x)(1 + 2x)]

-> A(1 + 2x) + B(1 - x) = 1 + 14x

x = 1: 3A = 15 -> A = 5

x = -1/2: 3B/2 = -6 -> B = -4

->

**f(x) = 5/(1 - x) - 4/(1 + 2x)**
b.) Int. (1/3 to 1/6) f(x) dx. = Int. (1/3 to 1/6) 5/(1 - x) - 4(1 + 2x) dx. = [5ln|1 - x| - 2ln|1 + 2x|] (Limits 1/3 & 1/6) = [5ln|2/3| - 2ln|5/3|] - [5ln|5/6| - 2ln|4/3|] = 5[ln|2/3| - ln|5/6|] + 2[ln|4/3| - ln|5/3|] = 7ln(4/5) =

**ln(16384/78125)**
c.) f(x) = (1 + 14x)(1 - x)^(-1).(1 + 2x)(-1) = (1 + 14x)[1 + x + x^2 + x^3][1 - 2x + 4x^2 - 8x^3] = (1 + 15x + 15x^2 + 15x^3...)(1 - 2x + 4x^2 - 8x^3...) = 1 - 2x + 4x^2 - 8x^3 + 15x - 30x^2 + 60x^3 + 15x^2 - 30x^3 + 15x^3... =

**1 + 13x - 11x^2 + 37x^3...**
2.) L1: r = [11, 5 , 6] + Lamba[4, 2, 4]

L2: [24, 4, 13] + mu[7, 1, 5]

a.) At POI: [11 + 4Lambda, 5 + 2Lambda, 6 + 4Lambda] = [24 + 7mu, 4 + mu, 13 + 5mu]

-> 11 + 4Lambda = 24 + 7mu -> Lambda = (13 + 7mu)/4 * (1)

-> 5 + 2Lambda = 4 + mu -> Lambda = (mu - 1)/2 * (2)

-> 6 + 4Lambda = 13 + 5mu -> 4Lambda = 7 + 5mu -> Lambda = (7 + 5mu)/4

Equate (1) and (2): (13 + 7mu)/4 = (mu - 1)/2 -> 26 + 14mu = 4mu - 4 -> 10mu = -30 -> mu = -3

Lamda = (7 - 15)/4 = -8/4 = -2

Check solutions with (3): -2 = (7 - 15)/4 = -2 -> True.

-> Equations are consistent -> POI exists.

b.) At POI, Lambda = -2:

-> POI = [11, 5, 6] + (-2)[4, 2, 4] = [11, 5, 6] + [-8, -4, -8] = [3, 1, -2]

-> POI: (3, 1 , -2).

c.) Let direction vector of L1 be: d1 = [4, 2, 4]

Let direction vector of L2 be: d2 = [7, 1, 5]

d1.d2 = 28 + 2 + 20 = 50

50 = |d1|*|d2|*cos(theta)

-> 50 = Sqrt[(4^2 + 2^2 + 4^2)(7^2 + 1^2 + 5^2)]*cos(theta)

-> 50 = Sqrt(2700)*cos(theta) = 30Sqrt(3)*cos(theta)

-> cos(theta) = 50/[30Sqrt(3)]

-> cos(theta) = 50Sqrt(3)/90

-> cos(theta) = 5Sqrt(3)/9

->

**k = 5/9**
3.) a.) x = cost, y = sin2t.

dx/dt = -sint, dy/dt = 2cos2t

dy/dx = 2cos2t * (-1)/sint = (-2cos2t)/sint

b.) dy/dx = 0

-> -2cos2t = 0

-> cos2t = 0

-> 2t = Pi/2, 3Pi/2, 5Pi/2, 7Pi/2

->

**t = Pi/4, 3Pi/4, 5Pi/4, 7Pi/4**
c.) When tangent is parallel to x-axis -> dy/dx = 0.

-> Hence at such points: t = Pi/4, 3Pi/4, 5Pi/4, 7Pi/4.

t = Pi/4: x = Sqrt(2)/2, y = 1 -> Point: [Sqrt(2)/2 , 1]

t = 3Pi/4: x = -Sqrt(2)/2, y = -1 -> Point: [-Sqrt(2)/2, -1]

t = 5Pi/4: x = -Sqrt(2)/2, y = 1 -> Point: [-Sqrt(2)/2, 1]

t = 7Pi/4: x = Sqrt(2)/2, y = -1 -> Point: [Sqrt(2)/2, -1]

d.) x = cost, y = sin(2t) = 2sintcost

-> y = 2sint.cost = 2xsint

-> y = 2x[√(1 - cos^2t)]

y = 2x.Sqrt(1 - x^2)

e.) y = -2x.Sqrt(1 - x^2)

4.) a.) dM/dt = -kM

b.) Don't understand notation here, sorry.

c.) 10/(10M - 1) dM = -k dt

-> ln|10M - 1| = c - kt

t = 0, M = 10: c = ln99

t = 10, M = 8.85: ln87.5 = ln99 - 10k -> 10k = ln[99/(175/2)] = ln(198/175) -> k = (1/10)ln(198/175)

Therefore: ln|10M - 1| = ln99 - (t/10)ln(198/175) = ln99 - ln[(198/175)^(t/10)] = ln{99/[(198/175)^(t/10)]}

-> 10M - 1 = 99/[(198/175)^(t/10)]

When t = 15: 10M - 1 = 99/[(198/175)^(1.5)]

-> 10M - 1 = 82.261

-> 10M = 83.261

->

**M = 8.33 g (3.S.F)**
5.) a.) R = Int. (2Pi to Pi) x^2.sin(x/2) dx.

Let u = x^2 -> du/dx = 2x

Let dv/dx = sin(x/2) -> v = -2cos(x/2)

R = [-2x^2.cos(x/2)] (Limits 2Pi and Pi) + 2 Int (2Pi to Pi) 2x.cos(x/2) dx.

To find Int. 2x.cos(x/2) dx.

Let u = 2x -> du/dx = 2

Let dv/dx = cos(x/2) -> v = 2sin(x/2)

-> Int. 2x.cos(x/2) dx. = 4xsin(x/2) - 4Int. sin(x/2) dx. = 4xsin(x/2) + 8cos(x/2)

Hence: R = [-2x^2.cos(x/2)] (Limits 2Pi and Pi) + [8xsin(x/2) + 16cos(x/2)] (Limits 2 Pi and Pi) = 2(4Pi^2) + [-16Pi - 8Pi] = 8Pi^2 - 24Pi =

**8Pi(Pi - 3)**
I think I've had enough.