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Cork Institute of Technology
Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2
(NFQ – Level 8)
Autumn 2007
Mathematics
(Time: 3 Hours)
Answer FIVE questions. All questions carry equal marks.
Examiners: Prof.M.Gilchrist Mr.P.Clarke Mr. T O Leary
- (a) The current i in a circuit at an instant t is found by solving the differential equation
dtdi^ +4i=^10 0cos3t i(0)=^0 Solve this differential equation. Express the steady state current as a single function of the form Rcos( 3 t-α). Write down the maximum and minimum values of this function. Find the smallest positive values of t for which these extreme values hold. (8 marks)
(b) Write down three terms of a Taylor Series expansion of f(x) about x=a with a remainder and show that f(a h) f(a - h) 2h f (a)^ O(h )
Estimate the value of f ′(2)^ where
f(x) 4.56 4.84 5.00 5.17 5. x 1.90 1.95 2.00 2.05 2. (4 marks)
(c) Using a double integral find the area and the second moment of area about the y-axis of the sector of the elliptical region x 16
y 9 1
2 2
with vertices (0,0), (4,0)and ( 2 2 , 3 22 ) (8 marks)
- (a) Find a Taylor Series expansion of the function
f(x,y)= (^)
y arctan 2x
about the values x=1,y=2. The series is to contain terms deduced from second order partial derivatives. (7 marks)
(b) (i) Find the partial derivatives of u and v with respect to x and y where
u= ln ^ xy
v= x -y^2
(i) If T=f(u) is an arbitrary function in u where u is defined above show that
x T^ y T 0 x y
(ii) If P=g(u,v) is an arbitrary function in u and v show that
v v P y y P x x P ∂
(iii) Estimate the value of u if the values of x and y were estimated to be 5 and 3 with maximum errors of 0.04 and 0.02 respectively. (8 marks)
(c) Find the dimensions of the closed box of maximum volume that can be constructed from 24m^2 of material. (5 marks)
- (a) Find the general solution for x and for y where
dx (^) x y x(0) 6 dt dy (^) 2x+2y y(0) 0 dt
By using a second method solve for y. (9 marks) (b) Find the maximum/minimum value of x 3 -3xy+y 2 -3x+10 (6 marks) (c) By using the Three Term Taylor Method with a step of 0.1 estimate the value of y at x=0.1 where
dy =2xy y(0)= dx Solve the differential equation and calculate the error in the approximation above. (5 marks)
- (a) Evaluate the line integral
2 C
∫12xydx^ +24y dy
(i) where C is the line segment passing from (0,1) to (1,3), (ii) where C is the arc of the circle x^2 +y^2 =4 passing from (2,0) to (0,2) in an anticlockwise direction. (8 marks) (b) (i) By evaluating a double integral locate the centroid of the triangular region with vertices (-1,0), (1,0) and (0,2). (ii) By evaluating a double integral find the second moment of area of this triangular region about the x-axis. Sum horizontally. (iii) Evaluate the triple integral
V
∫∫∫4xzdV
where V is the volume with a triangular cross sectional area described above and
where 0 ≤z ≤ 3. (12 marks)
- (a) A variate x can only assume values between 0 and 2 and its probability density function of a variate x is given by p(x)=A(2x-x 2 ). Find the value of A. Show that the mean value, the median value and the modal value of the distribution are all equal. (7 marks)
(b) Lengths of screws produced by a certain machine are Normally distributed with a mean length of 90mm and with a standard deviation of 0.03mm. Calculate the percentage of screws with lengths greater than 90.04mm and the percentage with lengths between 89. and 90.02 mm. If 0.1% of these screws are rejected for being undersized, that is, the lengths are below some critical limit calculate this critical limit. By using the Binomial distribution calculate the probability of a batch of 80 of these screws contains three or more undersize screws. (9 marks)
(c) In a computer laboratory there is a printer and the number of files sent to that printer in a single hour were counted and are recorded below.
Number of files 1 2 3 4 5 > Number of students 5 12 16 12 5 0
Calculate the mean number of files per student sent to the printer in each hour. By using the Poisson Distribution calculate the probability that any student will send two, three or four files to the printer in any ten minute period. (4 marks)
LAPLACE TRANSFORMS
For a function f(t) the Laplace Transform of f(t) is a function in s defined by
F(s) e stf(t)dt 0
∞
∫ where s>0.
f(t) F(s) A=constant A s tn n 1 s
n!
e at^1 s −a sinhkt k s 2 −k^2 coshkt s s 2 −k^2
sin ωt^ ω
s 2 + ω^2
cos ωt s
s 2 + ω^2
e atf(t) F(s-a) f (t) ′ (^) sF(s)-f(0) f (t) ′′ s F(s)^2 − sf(0) − f (o)′ f(u)du 0
t
F(s) s
f(u)g(t u)du 0
t
∫ − F(s)G(s)
U(t-a) e s
-as
f(t-a)U(t-a) e^ −asF(s) δ ( t − a) e -as
Note: coshA e^2 e sinhA e^2 e
A A A A = +^ = −
− −
