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P3 Questions
1) y = x(sinx)^0.5, 0 < x < pi. The maximum point on the curve is A.
a) Show that the x-coordinate of the point A satisfies the equation 2tanx+x=0.
The finite region enclosed by the curve and the x-axis is shaded as shown in Fig. 1 (It's in the book, so I can't show it. Sorry.).
A solid body S is generated by rotating this region through 2pi radians about the x-axis.
b) Find the exact volume of S.
2) Relative to a fixed origin O, the point A has position vector 3i+2j-k, the point B has position vector 5i+j+k, and the point C has position vector 7i-j.
a) Find the cosine of angle ABC.
b) Find the exact value of the area of the triangle ABC.
The point D has position vector 7i+3k.
c) Show that AC is perpendicular to CD.
d) Find the ratio AD
B.
3) Fluid flows out of a cylindrical tank with constant cross section. At time t minutes, t is greater than or equal to 0, the volume of fluid remaining in the tank is Vm^3. The rate at which the fluid flows, in m^3 min^-1, is proportional to the square root of V.
a) Show that the depth h metres of fluid in the tank satisfies the differential equation
dh/dt = -k√h, where k is a positive constant.
b) Show that the general solution of the differential equation may be written as
h = (A-Bt)^2, where A and B are constants.
Given that at time t=0, the depth of fluid in the tank is 1m, and that 5 minutes later, the depth of fluid has reduced to 0.5m.
c) Find the time, T minutes, which it takes for the tank to empty.
d) Find the depth of water in the tank at time 0.5T minutes.
4) f(x) = 25/(1-x)(3+2x)^2, |x|=1.
a) Express f(x) as a sum of partial fractions.
b) Hence find the integral of f(x) dx.
c) Find the series expansion of f(x) in ascending powers of x up to and including the term in x^2. Give each coefficient as a simplified fraction.
Answers:
1b) pi[pi^2-4].
2a) -4/9.
2b) 0.5√65.
2d) 2:-1.
3c) 17.1 min.
3d) 0.25m
8b) -ln|1-x| + ln|2x+3| - (2x-8)(2x+3)^-1 + C.
8c) 25/9 - 25x/27 + 25x^2/9...
Cheers
a) Show that the x-coordinate of the point A satisfies the equation 2tanx+x=0.
The finite region enclosed by the curve and the x-axis is shaded as shown in Fig. 1 (It's in the book, so I can't show it. Sorry.).
A solid body S is generated by rotating this region through 2pi radians about the x-axis.
b) Find the exact volume of S.
2) Relative to a fixed origin O, the point A has position vector 3i+2j-k, the point B has position vector 5i+j+k, and the point C has position vector 7i-j.
a) Find the cosine of angle ABC.
b) Find the exact value of the area of the triangle ABC.
The point D has position vector 7i+3k.
c) Show that AC is perpendicular to CD.
d) Find the ratio AD
B.3) Fluid flows out of a cylindrical tank with constant cross section. At time t minutes, t is greater than or equal to 0, the volume of fluid remaining in the tank is Vm^3. The rate at which the fluid flows, in m^3 min^-1, is proportional to the square root of V.
a) Show that the depth h metres of fluid in the tank satisfies the differential equation
dh/dt = -k√h, where k is a positive constant.
b) Show that the general solution of the differential equation may be written as
h = (A-Bt)^2, where A and B are constants.
Given that at time t=0, the depth of fluid in the tank is 1m, and that 5 minutes later, the depth of fluid has reduced to 0.5m.
c) Find the time, T minutes, which it takes for the tank to empty.
d) Find the depth of water in the tank at time 0.5T minutes.
4) f(x) = 25/(1-x)(3+2x)^2, |x|=1.
a) Express f(x) as a sum of partial fractions.
b) Hence find the integral of f(x) dx.
c) Find the series expansion of f(x) in ascending powers of x up to and including the term in x^2. Give each coefficient as a simplified fraction.
Answers:
1b) pi[pi^2-4].
2a) -4/9.
2b) 0.5√65.
2d) 2:-1.
3c) 17.1 min.
3d) 0.25m
8b) -ln|1-x| + ln|2x+3| - (2x-8)(2x+3)^-1 + C.
8c) 25/9 - 25x/27 + 25x^2/9...
Cheers
Aristotle
a) Show that the x-coordinate of the point A satisfies the equation 2tanx+x=0.
The finite region enclosed by the curve and the x-axis is shaded as shown in Fig. 1 (It's in the book, so I can't show it. Sorry.).
A solid body S is generated by rotating this region through 2pi radians about the x-axis.
b) Find the exact volume of S.
1
a)
y = x(sinx)^½
dy/dx = x(½ cosx)(sinx)^-½ + (sinx)^½
turning point at dy/dx = 0
0 = (x/2)cosx(sinx)^-½ + (sinx)^½
-(sinx)^½ = [x/2(cosx)]/(sinx)^½
-2sinx = xcosx
-2tanx = x
0 = x + 2tanx
1
b) (integration by parts, twice)
V = pi ∫ y² dx
V = pi ∫x²sinx dx
u =x² ; v' = sinx
u' = 2x ; v =-cosx
∫ uv' = uv - ∫u'v
∫ x²sinx dx = x².(-cosx) - ∫ -2xcosx dx
∫ x²sinx dx = x².(-cosx) + 2 ∫ xcosx dx
∫ xcosx dx
let u=x ; v'=cosx
u' =1 ; v=sinx
∫ xcosx dx = xsinx -∫ sinx dx
∫ xcosx dx = xsinx + cosx
V = pi ∫x²sinx dx
V = pi {-x²cosx + 2[xsinx + cosx] }
limits are pi to 0.
V = pi (pi^2 -4)
b) (integration by parts, twice)
V = pi ∫ y² dx
V = pi ∫x²sinx dx
u =x² ; v' = sinx
u' = 2x ; v =-cosx
∫ uv' = uv - ∫u'v
∫ x²sinx dx = x².(-cosx) - ∫ -2xcosx dx
∫ x²sinx dx = x².(-cosx) + 2 ∫ xcosx dx
∫ xcosx dx
let u=x ; v'=cosx
u' =1 ; v=sinx
∫ xcosx dx = xsinx -∫ sinx dx
∫ xcosx dx = xsinx + cosx
V = pi ∫x²sinx dx
V = pi {-x²cosx + 2[xsinx + cosx] }
limits are pi to 0.
V = pi (pi^2 -4)
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