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Ermina Topintzi

ENGINEERING MATHEMATICS – Tutorial Questions (First Term)

2 Ermina Topintzi

Tutorial 1 (Functions I) 1. Show that:

a) 2

sincos 11 π=+ −− xx (2 Marks)

b)

−+=+ −−−abbaba

1tantantan 111 (3 Marks)

2. Determine the periods, where applicable, of the functions: a) xy 2sin3= b) ( )4tan2 += xy c) ( )23cos2 += − xey x d) xxy 2sin3sin +=

e)

+=3

2sin3cos2 xxy

3. Find dxdy if (5 Marks)

a) ( )32ln 2 ++= xxy b)

22

21ln

−+=

xxy

c) 2

1

2

3

11ln

−+=

xxy

d) 232 += xey

e) xey 3sin2

= f) xy 2tan 1−= g) ( )3sec 31 += − xy

4. Use the definitions in your notes to show that: a) 1sinhcosh 22 =− xx b) 1sectanh 22 =+ xhx c) xxech 22 cothcos1 =+

5. Sketch the curves of the functions xy coth= , xy tanh= , echxy cos= and their inverses. (5 Marks)

6. Show that:

a) ( ) xechxdxd 2coscoth −=

b) ( ) xechxechxdxd cothcoscos −=

c) ( ) 1,1

1coth 21 >

−=− x

xx

dxd


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d) ( ) 10,1

1sec2

1


4 Ermina Topintzi

Tutorial 2 (Limits & Differentiation) 1. Evaluate the following limits:

a) ( )xxx

4lim 22

−→

b) xxxx

x −

−

→ +−

3333lim

0

c) 4

2lim22 −

−→ x

xx

d) 5432lim

−+

∞→ xx

x

e) 5

lim 2 +∞→ xx

x

f) xxxx

x −

−

∞→ +−

3333lim

g) ( )x

xx −

−→ π

ππ

sinlim

h) xxx

cotlim0→

i) ( ) ( )h

fhfh

44lim0

−+→

where ( )( )21

1x

xf+

=

2. Find y ′ in the following: a) 6105 345 +−+= xxxy

b) xx

y 42

12 +=

c) xxy 22 += d) ( )651 xy −= e) ( ) 21243 xxy −+= f)

3223

++=

xxy

g) 5

1

+=

xxy

h) ( )( )11

+−=

xxy

i) ( ) 221 2 +−−= xxxy j) ( ) ( )3342 523 −+= xxy k) 2

2

32

xxy

−+=


5 Ermina Topintzi

Tutorial 3 (Functions II) 1. Find to 4dp:

a) 8.0sinh 1− b) 2cosh 1− c) ( )5.0tanh 1 −−

2. The speed V of waves in shallow water is given by L

dLV 3.6tanh8.12 = , where d is the

depth of the water and L the wavelength. If 30=d and 270=L , calculate the value of V .

3. The formula

−+=

atatatatat

coscoshsinsinh

2λ gives the increase in resistance of strip conductors

due to eddy currents at power frequencies. Calculate λ when 075.1=a and 1=t .

4. Sketch the graph of xx eey 2−− −= . Prove that the maximum of y is 41 and find the

corresponding value of x . Find the two values of x corresponding to 401=y .

Answers: 2. (a) 0.7327, (b) 1.3170, (c) –0.5493; 3. 17.1383; 4. 1.0074; 5. 3.6629, 0.02599


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Tutorial 4 (Differentiation) 1. Differentiate with respect to x :

a) x2cot

b) x31

sin c) xx cos3 d) xx 32 cossin

e)

x1cot

f)

+ 211tanx

g) xx

sin34sin43

++

h) xe x 3tan2

i)

− x2

cos π

2. If ( )xy lnsin= show that 022

2 =++ ydxdyx

dxydx (5 Marks)

3. If ( )21sin xy −= show that ( ) ydxdyx 41

22 =

− and ( ) 021 22

2 =−−−dxdyx

dxydx

4. If

++=2

tan22

tan xxBxAy show ( ) ydx

ydx =+ 22

cos1 (5 Marks)

5. Find y′ when (i) xyyx 333 =+ , (ii) ( )xyxey 1cosh−= , (iii) 1=nm yx

6. If 2

12

sin1sin2

−+=

xxy show that ( )

( )( )[ ] 21322

sin1sin22

sinsin22cos

xx

xxxy−+

−+=′ (5 Marks)

7. If

++= −

xbabxay

coscoscos 1 show

+−= −

xbaxbay

cossinsin

221 where 0>> ba and

( )xbabay cos22 +−=′ . 8. If tex = and ty 2sinh= show that te

ty 2cosh2=′ and 22 222 xyyxyx =−′+′′ (5 Marks)


7 Ermina Topintzi

Answers:

1. (a) xec 2cos2 2− , (b) xxcossin31 32− , (c) [ ]xxxx sincos32 − , (d)

[ ]xxxx 222 sin3cos2cossin − , (e)

xec

x1cos1 22 , (f) ( )2222 11

1sec2 xx

x +

+− , (g)

( )2sin34cos7

xx

+, (h) [ ]xxex 3sec33tan22 2+ , (i) xcos ;

5. (i) xy

xy−

−2

2

, (ii) ( )111

22

22

−−−−

yxexyyxe

y

y

, (iii) nxmy−

Coursework: Questions 2, 4, 6, 8 for week 7 of this term.


8 Ermina Topintzi

Tutorial 5 (Differentiation Applications) 1. Use Maclaurin’s theorem to show that:

a) ( ) �+−+−=+432

1ln432 xxxxx

b) �−+−=−53

tan53

1 xxxx

c) �+++=15

23

tan53 xxxx

2. Investigate the stationary values and points of inflection of the following: a) ( )122 −= xxy b) ( )( )212 +−= xxy c) x

xy ln2 +=

3. Find the value of x which gives a stationary value of xxy1

= . 4. A rectangular box with an open top is made in the following way. A piece of tin 10cm by

16cm has a small square cut from each corner and the edges are folded vertically. What should be the size of the square cut out if the box is to have as large a volume as possible?

5. A silo is to be built in the form of a tight circular cylinder surmounted by a hemisphere. If the cost of the material for the walls, floor and top is the same, find the most economical proportions for a given volume V .

6. Use Newton’s method to calculate 21 accurate to 4 decimal places by looking for the root of the function ( ) 221 xxfy −== . Take 4=ox .

7. Find the positive root of 021sin =− xx correct to 3 decimal places.

Answers: 3. (a) (2,-16) min. (-2,16) max. (0,0) inf. (b) (1,-4) min. (-1,0) max. (0,-2) inf.

(c) (2,1+ln2) min. (4, 2ln21 + ) inf.

3. x=e; 4. 2 cm by 2 cm; 5. Radius of base = height =3

1

53

πV ; 6. 4.5826; 7. 1.896


9 Ermina Topintzi

Tutorial 6 (Complex Numbers) 1. Find real numbers x and y such that

a) 3x +2iy – ix + 5y =7 +5i (2 Marks)

b) iiyxi

i +=+

+

−+ 11

11 2 (2 Marks)

c) iiyxi

i 31511 −=

++

+− (2 Marks)

2. Express in polar form: a) 2222 i+ b) 31 i+− c) 22 i− d) 232 i−−

3. Express in cartesian (real-imaginary) form: a) ( )( )ii 243 −+ b) ( ) ( ) ( )[ ]13122 −−+− iii c)

ii

−−

432

4. If iz 341 −= and iz 212 +−= obtain 21 zz + and 21 zz − .

5. Evaluate 43

32

928

ππ ciscis where θθθ sincos icis +≡ . (6 Marks)

6. Find the indicated roots and locate them on an Argand diagram:

a) ( ) 41232 i−− b) ( ) 32322 i+ c) ( ) 32−− i

7. Solve the equations: (i) 0273 =−z , (ii) 164 =z . 8. Find the cube roots of i−2 . (6 Marks)

9. If p is real and the complex number ii

piiz

+++

++=

332

21 show that z is represented in the

Argand diagram by a point on the line y=x provided 215 ±−=p .

10. Find the locus of (x, y) when iziz

++

1 is (i) imaginary, (ii) real.

11. The point representing z in an Argand diagram moves in such a way that ( ) ( ) 2112 =++ izz . Show that it describes a straight line. (7 Marks)


10 Ermina Topintzi

Answers:

1. (a) (x,y) = (-1,2); (b) (x,y) =

−51,

52 ; (c) (x,y) = (1,2);

2. (a) 44π∠ ; (b) 2

32π∠ ; (c)

4722 π∠ ; (d) 4

67π∠

3. (a) 14 - 2i, (b) 7i – 9, (c) 17

1011 i−

4. 10 & 25 ; 5. ( )i3116 +− ; 6. (a)

2472 π∠ ,

24192 π∠ ,

24312 π∠ ,

24432 π∠ ; (b)

924 3

2 π∠ , 9

84 32 π∠ ,

9144 3

2 π∠ ; (c)

3π∠ , π∠ ,

35π∠ ;

7. (a) 3, ( )i3123 ±− ; (b) i2± , 2± ;

8. 15.05 61

−∠ , 94.15 61

∠ , 03.45 61

∠ ; 10. (a) The y axis; (b) circle


11 Ermina Topintzi

Tutorial 7 (Partial Differentiation) 1. Find the first partial derivatives of the following functions:

a) xyxf 32 2 −= b) 6242 −+= xyyxf

c) 32 1 yxf ++=

d) 22 yxf +=

e) 22 yxxyf+

=

f)

+= −

yxf

1sin 1

g) ( )yxef cos= 2. If

=xyfyz 3 where f is an arbitrary function show that:

zyzy

yxzxy

xzx 62 2

22

2

2

22 =

∂∂+

∂∂∂+

∂∂

3. Given that ( )

−+= −xyyyxxz 122 tan2ln show that 02

2

2

2

=∂∂+

∂∂

yz

xz .

4. Find the first and second derivatives of ( )zyxzyxV

32

,, = at the point (1,2,1).

Answers:

1. (a) yxfx 34 −= , xf y 3−= ; (b) yxyfx 224 += , xyxf y 24

32 += ; (c) 12 +

=x

xfx ,

23yf y = ; (d) 22 yxxfx+

= , 22 yx

yf y+

= , (e) ( )( )22222

yxxyyfx

+−= , ( )( )222

22

yxyxxf y

+−= ; (f)

( ) 2211

xyfx

−+= ,

( ) ( ) 2211 xyyxf y

−++

−= ; (g) ( )yyx xeef sin−= , ( )yyy xexef sin−= ; 4. 16=xV , 12=yV , 8−=zV , 16=xxV , 12=yyV , 16=zzV , 24=xyV , 16−=xzV , 12−=yzV


12 Ermina Topintzi

Tutorial 8 (Differentials) 1. A container is in the form of a right circular cylinder of radius r and height h surmounted

by a hemisphere of radius r. If r and h are measured as 10m and 18m respectively and these measurements have an error of 1%, show that the error in the surface area of the container, calculated from the formula 232 rrhA ππ += is 2%. (10 Marks)

2. If ψ

φθsin

sinsin=z and z is calculated for the values �30=θ , �60=ψ , �45=φ , find

approximately the change in the value of z if each of the angles θ and ψ is increased by

the same small angle �a and φ is decreased by �a21 .

3. The rate of flow of gas in a pipe is given by 6

5

21

T

CWQ = , where C is a constant, W is the

pipe diameter and T is the absolute gas temperature. The measurement of W is subject to a maximum error of %6.1± and that of T to one of %2.1± . Find approximately the maximum percentage error in the value of Q. (10 Marks)

Answers:

2. ( )2108034 −aπ ; 3. %8.1±

Coursework: Hand in solutions to questions 1 and 3 in the second week of the next term.

ENGINEERING MATHEMATICS – Tutorial Questions (Second Term)

Ermina Topintzi

Tutorial 9 (Vectors I) 1. In the triangle OAB, C is the midpoint of AB and D is the midpoint of OB. Show using

vectors that OADC21= .

2. Given C and C ′ are the midpoints of any two non-intersecting and non-parallel straight lines AB and BA ′′ respectively, show that: CCBBAA ′=′+′ 2 .

3. Use vectors to show that the medians of a triangle are concurrent and that each median is divided in the ratio 2:1 by the point of intersection.

4. ABCD is a parallelogram. E is a point on diagonal BD such that BDBE32= and F is the

midpoint of side DC. If bAB = and cAC = show that ( )bcEF −= 261 .

5. If P and Q have co-ordinates (1,3,2) and (-4,-1,3) respectively referred to axes x, y, z, find the direction cosines of the vector PQ . If ( )0,1,4RR ≡ find the angle between the lines PQ and PR.

6. For what value of α are vectors kjiu +−= 2α and kjiv 42 −+= αα perpendicular? 7. Find the work done in moving an object from A (2,1,3) to B (4,-1,5) if the force applied is

kjiF 235 +−= .

Answers: 5. 425− ,

424− ,

421 ; angle =

×−−

17429cos 1 ;

6. a = -1, a = 2; 7. 20 units of work.


14 Ermina Topintzi

Tutorial 10 (Vectors II) 1. The two sides of a triangle are formed by vectors kjiA 263 −+= and kjiB 34 +−= .

Determine the angles of the triangle. 2. Given kjiu ++= 2 , kjiv +−= and vbuar += find: (5Marks)

a) the angle between u and v b) the relation between a and b if r is orthogonal to the vector kjiw ++−= 3 c) the values of a and b if r is a unit vector and b) holds.

3. Find the perpendicular distance of the point S (3, 2, 6) from the plane passing through points P (1, 3, 2), Q (-4, -1, 3) and R (2, 3, 4).

4. Given kjiA 432 ++= , kjiB 323 +−= find the angle between vectors A and B using (i) the dot product and (ii) the cross product. (5 Marks)

5. Find the area of the triangle ABC if A ≡A (-2, 3, 1), B ≡B (4, 2, -2) and C ≡C (2, 0, 1). 6. If kjiA 32 −+= and kjiB +−= 2 find a vector of magnitude 5 which is perpendicular

to the plane of A and B . (5 Marks) 7. Use vectors to show that ( ) αββαβα cossincossinsin ±=± . 8. A body rotates with constant angular velocity w radians per second about a line L which

passes through the origin of x, y, z space. If L is parallel to vector kji ++ 2 and the speed of a point on the body at the point (3, 2, 0) is 3 units per second, find w. (5 Marks)

Answers: 1. 757cos 1− ,

7526cos 1− ,

2π ; 2. (a)

2π , (b) b = 2a, (c)

231=a ,

32=b ; 3.

20111 ; 4. �61.635 ; 5. 421

21 ; 6. ( )kji ++±

35 ; 8.

2963

Coursework: Hand in solutions to questions 2, 4, 6, and 8 for marking in the FIFTH week of term.


15 Ermina Topintzi

Tutorial 11 (Vectors III) 1. If kjiA 32 −+= , kjiB +−= 2 and kjiC 4−+−= , find ( ) CBA ⋅⋅ , CBA ×⋅ and

( ) CBA ×× . Verify that ( ) ( ) ( ) ACBBCACBA ⋅⋅−⋅⋅=×× . 2. Show that ( ) ( ) ( ) 0=××+××+×× BACACBCBA 3. A particle moves along the space curve [ ]kjtiter t ++= − sincos . Find r� , r�� , r� and r�� at

t = 0. 4. If wtbwtar sincos += where a and b are constant non-parallel vectors and w is a scalar

constant show that: ( )bawdt

rdr ×=× .

5. If ( ) ( ) ktjtitr 31sin1cos α+−+−= is the position vector of a particle at time t, find the condition imposed on α by requiring that at t = 1, the acceleration is normal to the position vector.

6. Find the distance between the points P (-2, 4, 3) and Q (0, 1, -3). If PQ is extended to R so that QRPQ = find the co-ordinates of R.

7. Find the co-ordinates of the point P in which the line joining the points A (1, -2, 6) and B (2, -4, 3) meet the plane z = 0. What are the co-ordinates of the point where the line meets the plane 2x + y – z = 3?

8. A ≡A (1, 3, 2) and B ≡B (-1, 1, 1). Write down the equation of the line through A and B in both cartesian and vector form.

9. Find the equation of the plane through P (2, -1, 1), Q (3, 2, -1) and R (-1, 3, 2). Answers: 1. kji 1233 +− ; 20; kji 101525 −− ; 2. kjir −+−=� , 3=r� , kjr +−= 2�� ,

5=r�� ; 5. 6

1±=α ; 6. 7, (2, -2, -9); 7. (3, -6, 0), (4, -8, -3);

8. ( )kjikjir −−−+++= 2223 λ , 1

22

32

1 −=−=− zyx ; 9. 3013511 =++ zyx


16 Ermina Topintzi

Tutorial 12 (Differential Equations) 1. Solve

a) 12 =+ ydxdyx

b) xyydxdyx =−

c) ( ) 01)1( 22 =−++ yxdxdyxy

2. The acceleration of a particle moving in a straight line is xk 2 away from a point O when x

is its distance from O. By expressing the acceleration as dxdυυ , where υ is the velocity,

form a differential equation for the motion. Show that the solution consistent with the

velocity being zero when x = a is ( ) 2122 axk −=υ and if x = a when t = 0, the displacement is ktax cosh= at time t.

3. A particle is let fall from a great height h above the earth. If the acceleration of the particle

2xAf = , where x is the distance of the particle from the centre of the earth, provided

Rx ≥ , R: the radius of the earth, and if f = -g when x = R, show that it reaches the earth

with velocity ( )2

12

+ hRgRh . You may neglect air resistance.

4. Stokes law states that the resistance to motion of a sphere of radius a moving with speed υ in a fluid of viscosity µ is υπµa6 . Show that when a sphere of mass m falls under

gravity in a fluid through a distance x from rest we have ( )υυλυυ −= 0dxd , where 0υ is the

terminal velocity and m

aπµλ 6= . Show that the sphere reaches a velocity 021υ after falling

a distance λυ

−212ln0 .

5. Solve:

a) ( ) xydxdyx =++ 21

b) ( ) xxydxdyx =++12 given y(0) = 2

c) ( ) ( )22 11 xxxydxdyx +=−+

6. Solve:

a) ( ) yxdxdyyx −=+ 22


17 Ermina Topintzi

b) ( ) 3323 3 yxdxdyxyy +=− given that y(0) = 1.

7. A particle of unit mass is attached to a fixed point O by a force xµ when its distance is x from O in a medium which offers a resistance 2kv to its motion where v is its velocity.

Show that the equation of motion is 02 =+− xkvdxdvv µ . If the particle is initially at rest at

a distance a from O show that it reaches O with velocity [ ] 21222 212kaka kaee

k−− −−µ .

8. Radium has a half-life of 1690 years. How long will it take for 1 gram to reduce to 0.1

grams. (Assume that the amount M(t) present at time t satisfies the equation kMdt

dM −=

for decay constant k).

Answers: 1. a) 11

2

2

+−=

AxAxy , b) xAxey = , c) ( ) Aexy x =− 222 1 ;

5. a) ( ) cxxxy ′++=+ 232 3216 , b) ( ) ( ) 111 212 =−+ yx , c) ( ) 2122 11 xAxy +++= ; 6. a) Ayxyx =−− 22 , b) 014 434 =+−+ yxyx ; 8. 5,614 years


18 Ermina Topintzi

Tutorial 13 (Second Order Differential Equations) 1. Solve:

a) 045 =+′+′′ yyy b) 025204 =+′+′′ yyy c) 0134 =+′+′′ yyy

2. Solve: a) xeyyy 37086 =+′+′′ b) xeyyy 2886 −=+′+′′ c) xyy 2cosh124 =−′′ d) xyyy 3cos80134 =+′+′′ e) 284 xyy =+′′ f) xexyyy 2327134 −=+′+′′

3. Solve the differential equation tydtdy

dtyd 161682

2

=++ given y = 0, 0=dtdy when t = 0.

Answers: 1. a) xx BeAey 4−− += , b) ( ) 25xeBAxy −+= , c) ( )DxCey x += − 3cos2 ; 2. a) xxx eBeAey 342 2++= −− , b) xxx xeBeAey 242 4 −−− ++= , c) xBxxAy 2cosh2sinh)3( ++= ,

d) [ ] xxxBxAey x 3sin63cos23sin3cos2 +++= − , e) 122sin2cos 2 −++= xxBxAy , f) [ ]xxxBxAey x 233sin3cos 32 −++= − 3.

21

21 4 −+

+= − tety t


19 Ermina Topintzi

Tutorial 14 (Matrices I) 1. Evaluate:

a)

54

13

21

+

−− 2

330

12

, b)

0213

5 +2

−−

3514

, c)

−−

331

542

42

21

63

d)

−−

42

21

63

331

542

2. If matrices I, A and B are given as

=100010001

I ,

=415121312

A and

−=101

212

130

21

1B show that:

a) IA = AI =A b) IB =B c) BI is undefined

3. If

−=

1030

A find a relationship between a and b so that OAbAaA =++ 32 , where

=

0000

O .

4. If

=142

313

201

A ,

−−−

=21

1

42

2

131

B ,

−=

141

322

102

C verify that A(B+C) = AB+AC

and that A(BC) = (AB)C.

5. If

−

−=

132

201

A ,

−=

13

21

10

B verify that ( ) TTT ABAB =

6. Display the following equations in matrix form AX =B.

a) 14

923

42−

===

−++

+−+

zz

z

yyy

xxx

, b)

201

4

4323

5

2

42

==−=

=

+−+

−

−−

+

+

zzz

y

yy

xx

x

ww

w

, c) zyx

zz

z

yy

y

xxx

λλλ

===

++−

−++

24

323

2


20 Ermina Topintzi

Tutorial 14 (Determinants)

1. Evaluate the following determinants: a) 67656463

, b) θθθθ

cossinsincos

−, c)

987654321

,

c) 52319

11151372114

, e) 930627

315−− , f)

32142143

14324321

−−−−

−−−−−

, g)

107641523104

2793

−−

−

2. For what values of x are the following determinants zero?

a) x

xx

−−−−−

−

142141223

, b) 541

42123

−+−−+

−+−

xxxxxx

xxx

3. Show that

333

222

111

11

1

aaaaaaaaa

++

+= 3211 aaa +++

Answers: 1. a) 61, b)1, c) 0, d) 630, e) 0, f) 136, g) –326;

2. a) x = 1, 2, 3 b) x = 32


21 Ermina Topintzi

Tutorial 15 (Matrices II)

1. Find the inverse of

−−−−

=102114123

A and verify that IAAAA == −− 11 and ( ) AA =−− 11 .

2. Given

=230241121

A ,

−

301420221

B verify that ( ) 111 −−− = ABAB . (7 Marks)

3. If

−=7503146117

A ,

−−

−=

517751175

181C and 1−= BCA . Find the matrix B and

verifying that B is non-singular, find 1−B .

4. The displacement of a particle at the point 0r is given by 0rA , where

−−=

114410623

A .

If a particle is initially at

124

where does it move to? Another particle moves to

−82

3.

Where does it come from? (7 Marks) 5. Solve the following sets of equations by matrix inversion:

a) 283

3

2

342

5

===

−++

++−

zzz

yy

y

xxx

, b) 021

554

32

7

3

===

+−+

+−+

zzz

yyy

xxx

(6 Marks)

Answers: 1.

−−−=−

542752321

1A ; 3.

−−=013120

301B ,

−−−

−−−−=−

216193631

1911B ; 4.

13622

,

−10633

171 ; 5.

21=x ,

23=y , 1=z

Coursework: Hand in solutions to questions 2, 4, and 5 (i) in week 9 of term.


22 Ermina Topintzi

Tutorial 16 (Sequences and Series) 1. Find the eighth term of the arithmetic progression 65, 62, 59,… 2. Find the twenty-first term of the arithmetic progression 24.5, 23, 21.5,… 3. If the seventh term of an arithmetic progression is 15 and the twelfth term is 17.5 find the

first term. 4. Find the sum of the first 24 terms of the arithmetic progression 1, 7,13,… 5. Find the sum of the first 18 terms of 17, 16.8, 16.6,… 6. Find the seventh term of the geometric progression 1, 5, 25,…

7. Find the ninth term of the geometric progression 8, 2, 21 ,…

8. Find the sum of the first 8 terms of the geometric progression 2.4, 0.6, 0.15,…

9. Find the sum to infinity of the geometric progression 1, 251,

51 ,…

10. An oil company drills through rock and in the first hour drills 20m. The rate of drilling decreases by 15% per hour after this. How far have they drilled in 4 hours? How long will it take to reach a depth of 70m?

Answers: 1. 44; 2. –5.5; 3.12; 4. 1680; 5. 275.4; 6. 15,625; 7. 1.22× 410− ; 8. 3.19995; 9. 1.25; 10. 63.73m, 4.58 hours.


23 Ermina Topintzi

Tutorial 17 (Integration) 1. Integrate the following:

(1) 21

x , (2) xx

2sincos , (3) 23 6x2x − , (4) ( )1021 x− , (5)

1

3

−xx , (6) ( )323 −x

x ,

(7) 1323

−+

xx , (8) ( ) ( )24 4 −+ xx , (9) ( )33

2

2+xx , (10) ( )4

32 −xx , (11)

1−xx ,

(12) x

x2

1

3

cos

sin , (13) x3sinh , (14) )sin( 2xx , (15) xx 23 sectan , (16) 2)3( xx

ee+

,

(17) 2

2

4 xx−

, (18) )3)(1(

1xx −+

, (19) 24

1xx −

, (20) 102

12 ++ xx

,

(21) 256

12 ++ xx

, (22) x4sin , (23) x3cos , (24) xx 33 cossin , (25) 2xxe− ,

(26) xx ln2 , (27) xx 4tan 1−

Answers: (1) cx +23

32 , (2) cecx +− cos , (3) cxx +− 34 2

21 , (4) cx +−−

22)21( 11 ,

(5) cxxxx +−+++ )1ln(23

23

, (6) cx

x +−

−2)23(9

31 , (7) cxx +−+ )13ln( ,

(8) cxx ++− 5)4)(165(301 , (9) c

x+

+−

23 )2(61 , (10) cx +− 43 2 ,

(11) cxx +−+ 1)2(32 , (12) cxx +− 2

12

5cos2cos

52 , (13)

33cosh x , (14) cx +− )cos(

21 2 ,

(15) cx +4tan41 , (16) c

ex+

+−

31 , (17) cxxx +−−− 21 4

21

2sin2 , (18) cx +

−−2

1sin 1 ,

(19) cx +−1sec , (20) cx +

+−3

1tan31 1 , (21) cx +

+−4

3sinh 1 ,

(22) cxxx ++− )4sin2sin812(321 , (23) cxx +− 3sin

31sin ,

(24) cxx +− )2cos3

2cos(161 3 , (25) ce x +− −

2

21 , (26) cxx +− )

31(ln

3

3

,

(27) ( )[ ]xxx 44tan161321 12 −+ −


24 Ermina Topintzi

Tutorial 17 (Numerical Integration)

1. Evaluate ∫+

2

0 213)4( x

dx numerically using (a) the trapezoidal rule, (b) Simpson’s rule. Take

10 strips in both cases and give your answer correct to 4 decimal places. 2. Values of y are given at x = 1.0, 1.2, …, 3.0 in the following table

x 1.0 1.2 1.4 1.6 1.8 2.0 2.2 2.4 2.6 2.8 3.0 y 0.84 1.12 1.40 1.60 1.75 1.82 1.79 1.62 1.34 0.94 0.42

Evaluate ∫3

1

ydx using (a) the trapezoidal rule and (b) Simpson’s rule.

3. Evaluate ∫ +3

2

3 )1ln( dxx using Simpson’s rule and 10 strips. Work to 5 decimal places for

an answer correct to 4 decimal places. Answers: 1. (a) 0.8536, (b) 0.8541 2. (a) 2.802 (b) 2.815; 3. 2.7955


25 Ermina Topintzi

Tutorial 18 (Gradient of a Scalar Function) 1. If yxxz 242 −=φ find φ∇ and φ∇ at the point (2, -2, -1). 2. If kyzxjzxixyz 22323 32 ++=∇ φ find ),,( zyxφ given that 4)2,2,1( =−φ . 3. Find a unit vector which is perpendicular to the surface of the paraboloid 22 yxz += at

the point (1, 2, 5). 4. Find an outward drawn unit normal vector to the sphere 9)2()1( 222 =+++− zyx at the

point (3, 1, -4). 5. Find equations for the tangent plane and normal line to the surface 22 yxz += at the

point (2, -1, 5). 6. Find the equation of the tangent plane to the surface 122 −=+ zyxxz at the point (1, -3,

2). Answers: 1. kji 16410 −−=∇ φ , 932=∇ φ ; 2. 2032 += yzxφ ; 3. 21)42( kji −+± ;

4. 3)22( kji −+ ; 5. 4x + 2y – z = 5; 15

21

42

−−=+=− zyx ; 6. 2x – y – 3z + 1 = 0

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