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Iterations - Mathematics and Computing - Exam, Exams of Mathematics for Computing

Alagappa UniversityMathematics for Computing

Main points of this past exam are: Iterations, Parametric Equations, Tangent, Equation, Curve, Newton-Raphson Method, Approximate Percentage Error, Minimum Points, Turning Points, Centroid

Typology: Exams

2012/2013

Uploaded on 03/28/2013

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bg1
Cork Institute of Technology
Bachelor of Engineering in Mechanical Engineering โ€“ Stage 2
(EMECH_7_Y2)
Summer 2008
Mathematics & Computing
(Time: 3 Hours)
Answer FIVE questions Examiners: Ms. J. English
Dr. D. Cremin
Mr. A. Bateman
Dr. P. Delassus
Q1. (a) A curve is described by the parametric equations 56
3
t
xt
โˆ’
=
+
, 74
1
t
yt
+
=+.
Find the equation of the tangent to the curve at t = 2.4 . (7 Marks)
(b) If 33
74cos()3 0xxy y+โˆ’= , find dy
dx at the point (3, 2). (6 Marks)
(c) Show that the equation 4x2- 6x โ€“7=0 has a root near x = -0.5.Use the
Newton-Raphson method with three iterations to find the root correct to two decimal
places. (7 Marks)
Q2. (a) Given m = -5x3 +6x3y+8y2 find
,
mm
x
y
โˆ‚โˆ‚
โˆ‚โˆ‚
and
2
2
m
x
โˆ‚
โˆ‚ (6 Marks)
(b) You are given that
3
2
5kd
sb
= where k is a constant and d and b are variables. Use a
calculus method to find the approximate percentage error in s due to errors of +3.5% in d
and โ€“2.2% in b. (8 Marks)
(c) Locate the turning points on the curve
32
12 8 24
64
nn
rn
=
โˆ’โˆ’+ and establish whether they
are maximum or minimum points. (6 Marks)
pf3
pf4
pf5

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Cork Institute of Technology

Bachelor of Engineering in Mechanical Engineering โ€“ Stage 2

(EMECH_7_Y2)

Summer 2008

Mathematics & Computing

(Time: 3 Hours)

Answer FIVE questions Examiners: Ms. J. English Dr. D. Cremin Mr. A. Bateman Dr. P. Delassus

Q1. (a) A curve is described by the parametric equations

t x t

t y t

Find the equation of the tangent to the curve at t = 2.4. (7 Marks)

(b) If 7 x^3 + 4 x cos( y ) โˆ’ 3 y^3 = 0 , find

dy dx

at the point (3, 2). (6 Marks)

(c) Show that the equation 4x^2 - 6x โ€“7=0 has a root near x = -0.5.Use the Newton-Raphson method with three iterations to find the root correct to two decimal places. (7 Marks)

Q2. (a) Given m = -5x^3 +6x^3 y+8y 2 find

m m x y

and

2 2

m x

(6 Marks)

(b) You are given that

3 2

k 5 d s b

= where k is a constant and d and b are variables. Use a

calculus method to find the approximate percentage error in s due to errors of +3.5% in d and โ€“2.2% in b. (8 Marks)

(c) Locate the turning points on the curve

n n r = โˆ’ โˆ’ n + and establish whether they

are maximum or minimum points. (6 Marks)

Q3. Determine each of the following integrals:

(i) โˆซ 3 x^2 cos( ) x dx (ii)

(^6 ) 3 4

x x dx x

(iii)

(^4 ) 3 1

x dx x x

(iv)

2 2 2 3 4 1

x (^) e x dx x x

โˆซ โˆ’^ +^ โˆ’

(20 Marks)

Q4. (a) Find the position of the centroid of the figure bounded by the curve y = 3 x^2 , the x-axis,

and the ordinate at x = 0 and x= 2.

b

a b

a

X

xydx

ydx

=

b

a b

a

Y

y dx

ydx

=

(8 Marks)

(b) Calculate the area between the curve y = x^3 -2x 2 -8x and the x-axis. (6 Marks)

(c) Find the root mean square of the function

y x x

= + over the interval 1 โ‰ค x โ‰ค 2

(6 Marks)

Q7. (a) Concrete blocks are tested and it is found that, on average, 7% fail to meet the required specification. For a batch of 9 blocks, determine the probabilities that

(i) 3 blocks will fail to meet the specification. (ii) less than 4 blocks will fail to meet the specification. (iii) 3 or more blocks will fail to meet the specification.. (7 Marks)

(b) The mean mass of active material in tablets produced by a manufacturer is 5g and the standard deviation of the masses is 0.036g. In a bottle containing 250 tablets, find how many tablets are likely to have masses of (i) less than 4.9g? (ii) between 4.92 and 5.04g? (iii)more than 5.05g? (7 Marks)

(c) A manufacturer estimates that 2.2% of his output of a component is defective. The components are manufactured in packets of 200. Determine the probability of a packet containing:

(i) less than 3 defective components. (ii) two or more defective components. (6 Marks)

Probability Distributions

Binomial Distribution: P r ( ) = n^ C p qr r^ n โˆ’ r

Poisson Distribution:

e m^ mr P r r

โˆ’

Normal Distribution: Standard units, x X Z