School of Mathematics & Applied Statistics

Family Name . . . . . . . . . . . . . . . First Name . . . . . . . . . . . . . . . . Student Number . . . . . . . . . . . . .

MATH141 Foundations of Engineering Mathematics

Tutorial Group . . . . . . . . . . . . . . Tutorial Day & Time . . . . . . . . . . .

Tutor . . . . . . . . . . . . . . . . . .

AUTUMN SESSION 20XX IN-SESSION EXAMINATION 2: Sample Total Time Allowed: Total Number of Questions: 100 minutes 4

DIRECTIONS TO STUDENTS 1. Questions 1 and 2 in the blue section of this examination paper are for Strand 1. Questions 3 and 4 in the yellow section of this examination paper are for Strand 2. 2. Write your name and tutorial details on both sections of this examination paper in the spaces provided. Do not detach any pages from this paper. 3. All four questions are to be attempted and are of equal value (individual parts within a question may not be of equal value). 4. The examination paper is printed on both sides allowing space for working. 5. All answers and required working for Questions 1 and 2 must be shown on the blue section of this examination paper. 6. All answers and required working for Questions 3 and 4 must be shown on the yellow section of this examination paper. 7. If you require more working paper, please raise your hand to request more. Write your name and student number on all extra sheets and indicate in the appropriate box the number of extra sheets used for each section. 8. All notation is as used in lectures. 9. A Table of Integrals is attached.

EXAMINATION MATERIALS/AIDS ALLOWED • Non-alphanumeric, non-programmable calculators. • A one-page, single-sided, A4 size summary sheet. EXAMINATION MATERIALS/AIDS TO BE SUPPLIED None THIS EXAMINATION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM Number of extra sheets used for STRAND 1

Blank for working

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Strand 1
Question 1: Answer Only For each part in this question, write down your answer in the space provided. You will receive marks for the answer only (the parts are of equal value). Use page 2 for working. (a) Find dy sin x for y = 2 . dx x

Answer: d2 y . dx2

(b) A curve is defined by the parametric equations x(t) = t2 and y(t) = sin t. Calculate

Answer:

(c) Differentiate f (x) = xx

2

−1

.

Answer:

(d) Evaluate the integral

2x

x2 − 1 dx.

Answer: sin x

(e) Differentiate the expression x √

te

sec2 t

dt .

Answer:
1

(f) Evaluate the integral
0

1 dx. x−2

Answer:

3

Strand 1
Question 2: Full Working For each part in this question, do all your working in the space provided. If you need more space, please first write on the reverse side of the blue pages of this examination paper. If you still require more space, please request extra sheets of paper by raising your hand. Make sure you clearly label all solutions. (a) Differentiate the following functions with respect to x, showing all your working. (i) e3x (ii) x2 sinh x (iii) sin−1 (2x)

4

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5

(b) If y is given implicitly by the equation xy 2 + y = cosh x find

d2 y in terms of x and y. dx2

6

(c) Consider the function f (x) = cosh x. (i) Sketch the curve y = f (x) = cosh x, stating the domain and range of f . (ii) Restrict the domain of f and find an expression for f −1 (x) = cosh−1 x in terms of a logarithm.

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(d) Evaluate the following integrals. (i) (ii) x cos x dx − sin − x dx 3

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TABLE OF INTEGRALS 1 xn+1 + c, n+1

[1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14]

xn dx =

n = −1

dx = ln |x| + c x ex dx = ex + c sin x dx = − cos x + c cos x dx = sin x + c tan x dx = ln | sec x| + c sec2 x dx = tan x + c cosec2 x dx = − cot x + c sinh x dx = cosh x + c cosh x dx = sinh x + c tanh x dx = ln(cosh x) + c x(ax + b)n dx = ax + b b 1 (ax + b)n+1 − + c, a2 n+2 n+1 n = −1, −2

x2 1 1 dx = 3 (ax + b)2 − 2b(ax + b) + b2 ln |ax + b| + c ax + b a 2 x2 1 b2 dx = 3 ax + b − − 2b ln |ax + b| + c (ax + b)2 a ax + b √ 2 (ax + b)5/2 b(ax + b)3/2 x ax + b dx = 2 − +c a 5 3 2ax − 4b √ x dx = ax + b + c 3a2 ax + b √ √ 1 ax + b − b 1 √ √ + c, dx = √ ln √ x ax + b b ax + b + b √ √ x dx = sin−1 +c 2 a −x

[15] [16]

[17]

b>0

[18] [19] [20]

a2

x dx 1 = tan−1 +c a2 + x2 a a a2 1 x+a 1 dx = ln +c 2 −x 2a x−a

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[21]

1 x 1 x+a dx = + ln +c (a2 − x2 )2 2a2 (a2 − x2 ) 4a3 x−a 1 a+ √ dx = − ln 2 − x2 a x a 1 √ a2 − x2 +c x

[22]

[23]

1 1 x dx = 2 √ +c a (a2 − x2 )3/2 a2 − x2 √ a2 − x2 dx = x a2 − x2 − a ln a+ √ a2 − x2 +c x

[24]

[25]

√

x2

1 dx = ln x + ± a2 1

x2 ± a2 + c √ x2 + a2 +c x

[26]

a+ 1 dx = − ln a x x2 + a2 √

[27] [28]

1 1 x +c dx = ± 2 √ 2 ± a2 a (x2 ± a2 )3/2 x x2 ± a2 dx = √ x2 + a2 dx = x 1 1 x x2 ± a2 ± a2 ln x + 2 2 x2 + a2 − a ln a+ √ x2 ± a2 + c

[29]

x2 + a2 +c x ab = 0

[30] [31] [32] [33] [34] [35] [36] [37] [38] [39] [40]

1 1 dx = ax − ln(b + keax ) + c, b + keax ab eax sin bx dx = eax cos bx dx = sinn x dx = − cosn x dx = tann x dx = secn x dx = a2

1 eax (a sin bx − b cos bx) + c + b2

1 eax (a cos bx + b sin bx) + c a2 + b2 sinn−2 x dx cosn−2 x dx

1 n−1 cos x sinn−1 x + n n

1 n−1 sin x cosn−1 x + n n 1 tann−1 x − n−1

tann−2 x dx secn−2 x dx sinm x cosn−2 x dx

secn−2 x tan x n − 2 + n−1 n−1

sinm x cosn x dx =

sinm+1 x cosn−1 x n−1 + m+n m+n xn−1 ex dx xn−1 cos x dx xn−1 sin x dx

xn ex dx = xn ex − n

xn sin x dx = −xn cos x + n xn cos x dx = xn sin x − n

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School of Mathematics & Applied Statistics

Family Name . . . . . . . . . . . . . . . First Name . . . . . . . . . . . . . . . . Student Number . . . . . . . . . . . . .

MATH141 Foundations of Engineering Mathematics

Tutorial Group . . . . . . . . . . . . . . Tutorial Day & Time . . . . . . . . . . .

Tutor . . . . . . . . . . . . . . . . . .

AUTUMN SESSION 20XX IN-SESSION EXAMINATION 2: Sample Total Time Allowed: Total Number of Questions: 100 minutes 4

DIRECTIONS TO STUDENTS (Repeated) 1. Questions 1 and 2 in the blue section of this examination paper are for Strand 1. Questions 3 and 4 in the yellow section of this examination paper are for Strand 2. 2. Write your name and tutorial details on both sections of this examination paper in the spaces provided. Do not detach any pages from this paper. 3. All four questions are to be attempted and are of equal value (individual parts within a question may not be of equal value). 4. The examination paper is printed on both sides allowing space for working. 5. All answers and required working for Questions 1 and 2 must be shown on the blue section of this examination paper. 6. All answers and required working for Questions 3 and 4 must be shown on the yellow section of this examination paper. 7. If you require more working paper, please raise your hand to request more. Write your name and student number on all extra sheets and indicate in the appropriate box the number of extra sheets used for each section. 8. All notation is as used in lectures.

EXAMINATION MATERIALS/AIDS ALLOWED • Non-alphanumeric, non-programmable calculators. • A one-page, single-sided, A4 size summary sheet. EXAMINATION MATERIALS/AIDS TO BE SUPPLIED None THIS EXAMINATION PAPER MUST NOT BE REMOVED FROM THE EXAMINATION ROOM Number of extra sheets used for STRAND 2

Blank for working

14

Strand 2
Question 3: Answer Only For each part in this question, write down your answer in the space provided. You will receive marks for the answer only (the parts are of equal value). Use page 14 for working. (a) Let a = (3, −1, 2) and c = (4, 3, −1). Find the unit vector of 2c − 3a.

Answer:

(b) Let a = (3, −1, 2) and b = (5, 2, −1). Find the component and projection of a on −b.

Answer:

(c) Find the equation of the line, L, passing through the points (1, 1, 1) and (0, −1, 1). Decide whether the point (2, 1, 3) also lies on this line.

Answer:

(d) Determine whether the lines

      1 1 x L1 :  y  =  1  +  2  t 0 1 z and       1 −2 x L2 :  y  =  −1  +  1  s z 2 −1

intersect, are parallel or are skew.

Answer:

(e) Find the distance between the line L1 (given in part (d)) and the point (1, −1, 2).

Answer:

(f) Find the normal to the plane P1 passing through the points (2, −1, 0), (3, −2, −2) and (1, −1, 1).

Answer:

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Strand 2
Question 4: Full Working For each part in this question, do all your working in the space provided. If you need more space, please first write on the reverse side of the yellow pages of this examination paper. If you still require more space, please request extra sheets of paper by raising your hand. Make sure you clearly label all solutions. (a) Let a = (3, −1, 2), b = (5, 2, −1) and c = (4, 3, −1). (i) Determine if the vectors a and c are perpendicular. (ii) Find a · c × b and give a geometrical interpretation of your result.

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(b) Consider the plane P1 passing through the points (2, −1, 0), (3, −2, −2) and (1, −1, 1). (i) Show that the linear form of the equation representing P1 is given by x − y + z = 3. (ii) Let the linear form of the equation representing the plane P2 be given by x − 2y + z − 3 = 0. Determine whether the planes P1 and P2 intersect or are parallel. If the two planes intersect, find the equation of the intersection line. If the two planes are parallel, find the distance between them. (iii) Find the distance between the plane P2 , from part (b)(ii), and the point (1, 3, −1).

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(c) The solution to the differential equation − 0, is given by

d2 u +u(x) = x subject to boundary conditions u(0) = u(1) = dx2 u(x) = x − sinh x . sinh 1

Using the finite element method of solution and assuming five equally spaced nodes (creating four equal subintervals) over the interval 0 ≤ x ≤ 1, then the finite element method basis functions are given by  x − xi−1  x −x ,   i i−1  xi+1 − x φi (x) =  xi+1 − xi ,     0, for xi−1 ≤ x ≤ xi , for xi ≤ x ≤ xi+1 , elsewhere.

Find an expression for φ3 (x) and evaluate α3 = u(x3 ).

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