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CORK INSTITUTE OF TECHNOLOGY
INSTITIÚID TEICNEOLAÍOCHTA CHORCAÍ
Semester 1 Examinations 2008/
Module Title: MATH7004: Engineering Mathematics 201
Module Code: MATH
School: School of Mechanical & Process Engineering School of Manufacturing, Biomedical & Facilities Engineering
Programme Title: Bachelor of Engineering (Honours) in Mechanical Engineering-Stage 2 Bachelor of Engineering (Honours) in Biomedical Engineering-Stage 2
Programme Code: EMECH_8_Y EBIOM_8_Y
External Examiner(s): Dr.P.Robinson Internal Examiner(s): Mr. T. O Leary
Instructions: Select any four questions. The questions carry equal marks.
Duration: 2 Hours
Sitting: Winter 2009
Requirements for this examination:
Note to Candidates: Please check the Programme Title and the Module Title to ensure that you have received the correct examination paper. If in doubt please contact an Invigilator.
- (a) State the Mean Value Theorem for derivatives. Show that the function f(x)=x^4 +2x 2 satisfies the criteria of the theorem over the interval [0,2]. Find the critical value that satisfies the conclusion of the theorem. This value is close to x=1.1 (7 marks)
(b) The current i in an LR circuit at any instant is found by solving the differential equation
dtdi^ +3i=25sin4t i(0)=^0 Solve this differential equation. Express the steady state current as a single periodic function of the form Rsin(4t-α). Note: sin(A-B)=sinAcosB-cosAsinB (11 marks)
(c) Find the maximum/minimum value of V=x^2 -4xy+y^3 +4y+2 (7 marks)
- (a) Show that Taylor Series expansion of f(x,y)=xln(2x-y) about the values x=2,y=3 is given by f(x,y)=4(x-2)-2(y-3)-2(x-2) 2 +3(x-2)(y-3)-(y-3) 2 +… (10 marks)
(b) Variables r and θ are related to variables x and y by the formulae
θ= (^)
x arctan y r= x 2 +y^2
Find the partial derivatives of θ and r with respect to x and y. (i) If stress T is an arbitrary function in r and θ write the relationships between the partial derivatives of T with respect to x and y and those with respect to r and θ. (ii) Estimate the value of r where x=3±0.05 and y=4±0.10 (8 marks)
(c) In constructing an open rectangular tank with a square base it costs €12m-2^ to construct the base and €6m-2^ to construct all other sides. If the tank has a volume of 8m^3 find the dimensions of the tank that can be constructed so that this cost is to be minimised. Also find this minimum cost. You are required to use a Lagrangian Multiplier. No marks will be awarded if any other method is used. (7 marks)
- (a) Find the Fourier Series for the periodic function below
f(t)=
1 tif 0 t (^1) f(t+2)=f(t)
(10 marks)
2 2 2 2
Note: (1-t)cos(nπt)dt= (1-t)sin(nπt)^ - cos(nπt) nπ n π (1-t)sin(nπt)dt=- (1-t)cos(nπt)^ - sin(nπt) nπ n π
(b) Solve for x where dx =3x+y x(0)= dt dy =-x+y y(0)= dt
(10 marks)
(c) Write down the first two terms of a Taylor Series expansion of the function f(x) about the point x=a. Include a remainder. Establish the approximation f(a h) f(a) h f (a)^ O(h)
If f(1.0)=1.000 and f(1.1)=1.331 estimate the value of f (1)′. (5 marks)
DERIVATIVES
f(x) a=constant f ′ (x) x n nx n−^1 lnx x
e ax a eax sinx cosx cosx -sinx tanx (^) sec 2 x secx secxtanx
tan −^1 ( )x
x 1
tan −^1 x a x 2 a +a 2 sin-1^ (x) 1 x^2
uv dx vdu dx u dv+
v
u v^2 dx
udv dx v du−
INTEGRALS
f(x) a=constant ∫ f(x)dx
x n xn+ n+1 if n^ ≠-
x
1 lnx
e ax^1 a a^ eax sinx -cosx cosx sinx
A (^0) π 2 π sinA 0 0 0 cosA 1 -1 1
