modern engineering mathematics, 4th ed. - monash...

Laborartory class exercises

Modern Engineering Mathematics, 4th ed.

Glyn James

Topic Exercises Questions

Vectors, Lines & Planes 4.2.8 17-20,23,254.2.10 31-344.3.3 52-55,59,60,62,63

Linear algebra 5.2.3 1,6,75.2.5 11,12,165.2.7 22

Matrices, Determinants & Matrix inverses 5.4.1 58,594.2.12 43-455.3.1 34,35,44

Eigenvalues & Eigenvectors 5.7.3 96,975.7.5 98-100,1025.7.8 105

Hyperbolic Functions 2.7.6 82,848.3.13 37,38

Integration by parts 8.8.4 105-107

Improper integrals 9.2.3 1

Topic Exercises Questions

Sequences & Series 7.2.3 1,2,4,5,12,137.3.4 19,21,22,247.6.4 41,449.4.4 8-17

Introduction to ODEs 10.3.6 1,210.4.5 3-5

1st Order ODEs 10.5.4 11,13,15,1710.5.6 18,2010.5.11 31-35

2nd Order homogenous ODEs 10.9.2 55-61

2nd Order inhomogenous ODEs 10.9.4 62-65

Multivariable Calculus 9.6.4 37-469.6.6 47,48,50-559.6.8 56-649.6.10 65-72

Maxima & Minima 9.7.3 76,78

Supplementary exercises

The questions on the following pages contain no new material not already coveredby the exercies in James. They are intended for students who want to practice theircraft as far as they can (i.e., to do as many questions as possible, well done). Thesequestions may also be helpful for students who do not have a copy of James to hand(but make no mistake: James is an essential book for this unit, you should obtain acopy or least know where to find copies in the library).

Please note that the exercises provided by James for Improper Integrals is ratherthin (just one question). So you are encouraged to complete the supplementaryexercises on Improper Integrals. This will be sufficient study for questions of thiskind should they appear on the final exam.

SCHOOL OF MATHEMATICAL SCIENCES

ENG1091

Mathematics for Engineering

Laboratory Class 1

Vectors, dot product, cross product

1. Find all the vectors whose tips and tails are among the three points with coordinates(2,−2, 3), (3, 2, 1) and (0,−1,−4).

2. Let v˜

= (3, 2,−2). How long is −2v˜

. Find a unit vector (a vector of length 1) inthe direction of v

˜.

3. For each pair of vectors given below, calculate the vector dot product and the angleθ between the vectors.

(a) v˜

= (3, 2,−2) and w˜

= (1,−2,−1)

(b) v˜

= (0,−1, 4) and w˜

= (4, 2,−2)

(c) v˜

= (2, 0, 2) and w˜

= (−3,−2, 0)

4. Given the two vectors v˜

= (cos(θ), sin(θ), 0) and w˜

= (cos(φ), sin(φ), 0), use the dotproduct to derive the trigonometric identity

cos(θ − φ) = cos(θ) cos(φ) + sin(θ) sin(φ).

5. Use the dot product to determine which of the following two vectors are perpendic-ular to one another: u

˜= (3, 2,−2), v

˜= (1, 2,−2), w

˜= (2,−1, 2).

6. For each pair of vectors given below, calculate the vector cross product. Assumingthat the vectors define a parallelogram, calculate the area of the parallelogram.

(a) v˜

= (3, 2,−2), w˜

= (1,−2,−1)

(b) v˜

= (0,−1, 4), w˜

= (4, 2,−2)

(c) v˜

= (2, 0, 2), w˜

= (−3,−2, 0)

7. Calculate the volume of the parallelepiped defined by the three vectors u˜

=(3, 2,−2), v

˜= (1, 2,−2), w

˜= (2,−1, 2).

8. Verify that v˜× w˜

= −w˜× v˜

.

Lines and planes

9. Consider the points (1, 2,−1) and (2, 0, 3).

(a) Find a vector equation of the line through these points in parametric form.

(b) Find the distance between this line and the point (1, 0, 1). (Hint: Use theparametric form of the equation and the dot product.)

10. Find an equation of the plane that passes through the points (1, 2,−1), (2, 0,−1)and (−1,−1, 0).

11. Consider a plane defined by the equation 3x+ 4y − z = 2 and a line defined by thefollowing vector equation (in parametric form)

x(t) = 2− 2t, y(t) = −1 + 3t, z(t) = −t.

(a) Find the point where the line intersects the plane. (Hint: Substitute the para-metric form into the equation of the plane.)

(b) Find a normal vector to the plane.

(c) Find the angle at which the line intersects the plane. (Hint: Use the dotproduct.)

12. Find the distance between the parallel planes defined by the equations 2x−y+3z =−4 and 2x− y + 3z = 24. (Hint: Use the cross product to construct a line normalto both planes, then use problem 11.)

13. Consider two planes defined by the equations 3x+ 4y− z = 2 and −2x+y+ 2z = 6.

(a) Find where the planes intersect the x, y and z axes.

(b) Find normal vectors for the planes.

(c) Find an equation of the line defined by the intersection of these planes. (Hint:Use the normal vectors to define the direction of the line.)

(d) Find the angle between these two planes.

14. Find the minimum distance between the two lines defined by

x(t) = 1 + t, y(t) = 1− 3t, z(t) = −2 + 2t

andx(s) = 3s, y(s) = 1− 2s, z(s) = 2− s

(Hint: Use scalar projection as demonstrated in the lecture notes. Alternatively,define the lines within parallel planes and then go back to problem 12.)


ENG1091


Laboratory Class 1 Solutions

Vectors, dot product, cross product

1. (0, 0, 0) ± (−1,−4, 2) ± (2,−1, 7) ± (3, 3, 5)

2. | − 2v˜| = 2

√17 , v

˜|v˜| = 1√

17(3, 2,−2)

3. (a) v˜· w˜

= 1, θ = arccos(

1√6·17

)≈ 1.4716 radians

(b) v˜· w˜

= −10, θ = arccos(−10√17·24

)≈ 2.0887 radians

(c) v˜· w˜

= −6, θ = arccos(−6√8·13

)≈ 2.1998 radians

4. v˜· w˜

= |v˜||w˜| cos(θ − φ) = 1 · 1 · cos(θ − φ) = cos(θ) cos(φ) + sin(θ) sin(φ)

5. u˜

and w˜

6. (a) v˜× w˜

= (−6, 1,−8) |v˜× w˜| =√

101

(b) v˜× w˜

= (−6, 16, 4) |v˜× w˜| = 2

√77

(c) v˜× w˜

= (4,−6,−4) |v˜× w˜| = 2

√17

7. (u˜× v˜

) · w˜

= 4

8. Yes, it is correct!

Lines and planes

9. (a) x(t) = 1 + t, y(t) = 2− 2t, z(t) = −1 + 4t

(b) 27

√14

10. 2x+ y + 7z = −3

11. (a) (2,−1, 0)

(b) (3, 4,−1)

(c) π2− arccos

(√9126

)≈ 0.37567 radians

12.√

56

13. (a) (2/3, 0, 0), (0, 1/2, 0), (0, 0,−2) and (−3, 0, 0), (0, 6, 0), (0, 0, 3)

(b) (3, 4,−1) and (−2, 1, 2)

(c) x(t) = −2 + 9t, y(t) = 2− 4t, z(t) = 11t

(d) arccos(−239

√26)≈ 1.835 radians

14.√

3


ENG1091


Laboratory class 2

Row operations and linear systems

Solve each of the following system of equations using Gaussian elimination withback-substitution. Be sure to record the details of each row-operation (for example,as a note on each row of the form (2)← 2(2)− 3(1).)

1.J + M = 75J − 4M = 0

2.x + y = 5

2x + 3y = 1

3.x + 2y − z = 6

2x + 5y − z = 13x + 3y − 3z = 4

4.x + 2y − z = 6x + 2y + 2z = 3

2x + 5y − z = 13

5.2x + 3y − z = 4x + y + 3z = 1x + 2y − z = 3

6. Repeat the last two questions, this time using Gaussian elimination (i.e. no back-substitution).

Under-determined systems

7. Using Gaussian elimination with back-substitution to find all possible solutions forthe following system of equations

x + 2y − z = 6x + 3y = 7

2x + 5y − z = 13

8. Find all possible solutions for the system (sic) of equations

x + 2y − z = 6

(Hint : You have one equation but three unknowns. You will need to introduce twofree parameters).

Matrices

9. Evaluate each of the following matrix operations

2

[1 11 −4

]−[

2 −13 1

],

[1 11 −4

] [2 −13 1

],

[1 1 31 −4 2

] 2 −13 11 2

10. Rewrite the equations for questions 1,2 and 3 in matrix form. Hence write down the

coefficient and augmented matrices for questions 1,2 and 3.

11. Repeat the row-operations part of questions 4 and 5 using matrix notation (shouldbe easy).

Matrix inverses

12. Compute the inverse A−1 of the following matrices

A =

[1 11 −4

]A =

2 3 −11 1 31 2 −1

Verify that A−1A = I and AA−1 = I.

13. Use the result of the previous question to solve the system of equations in questions1 and 5.

Matrix determinants

14. Compute the determinant for the coefficient matrices in questions 7 and 8. Whatdo you observe?

15. For the matrix

A =

2 3 −11 1 31 2 −1

compute the determinant twice, first by expanding about the top row and secondby expanding about the second column.

16. Given

A =

[1 11 −4

], B =

[2 −13 1

]compute det(A), det(B) and det(AB). Verify that det(AB) = det(A) det(B).


ENG1091



Row operations and linear systems

1. J = 60,M = 15 2. x = 14, y = −9 3. x = 7, y = 0, z = 14. x = 1, y = 2, z = −1 5. x = −1, y = 2, z = 0

Under-determined systems

7. Solution is x(t) = 4 + 3t, y(t) = 1− t, z(t) = t where t is a parameter, −∞ < t <∞.

8. Solution is x(u, v) = u − 2v + 6, y(u, v) = v, z(u, v) = u where u, v are parameters,−∞ < u, v <∞.

Matrices

9. Solutions are,[0 3−1 −9

] [5 0

−10 −5

] [8 6−8 −1

]

10. Coefficient and augmented matrices are

Q1.

[1 11 −4

],

[1 1 751 −4 0

]

Q2.

[1 12 3

],

[1 1 52 3 1

]

Q3.

1 2 −12 5 −11 3 −3

, 1 2 −1 6

2 5 −1 131 3 −3 4

Matrix inverses

12. Inverses are,

A−1 =1

5

[4 11 −1

]A−1 =

1

3

7 −1 −10−4 1 7−1 1 1

Matrix determinants

14. First add rows of zeroes to make the coefficient matrices square. Then compute thedeterminants, both are zero. This tells you that the system is under-determined andthat you will need to introduce parameters during the back-substitution.

15. Determinant = −3.

16. det(A) = −5, det(B) = 5 and det(AB) = −25.


ENG1091


Laboratory class 3

Matrices and Determinants Pt 2.

1. Compute the following determinants using expansions about any suitable row orcolumn.

(a)

∣∣∣∣∣∣1 2 33 2 20 9 8

∣∣∣∣∣∣ (b)

∣∣∣∣∣∣4 3 21 7 83 9 3

∣∣∣∣∣∣(c)

∣∣∣∣∣∣∣∣1 2 3 21 3 2 34 0 5 01 2 1 2

∣∣∣∣∣∣∣∣ (d)

∣∣∣∣∣∣∣∣1 5 1 32 1 7 51 2 1 03 1 0 1

∣∣∣∣∣∣∣∣2. Recompute the determinants in the previous question this time using row operations

(ie., Gaussian elimination).

3. Which of the following statements are true? Which are false?

(a) If A is a 3 × 3 matrix with a zero determinant, then one row of A must be amultiple of some other row.

(b) Even if any two rows of a square matrix are equal, the determinant of thatmatrix may be non-zero.

(c) If any two columns of a square matrix are equal then the determinant of thatmatrix is zero.

(d) For any pair of n × n matrices, A and B, we always have det(A + B) =det(A) + det(B)

(e) Let A be an 3× 3 matrix. Then det(7A) = 73 det(A).

(f) If A−1 exists, then det(A−1) = det(A).

4. Given

A =

[1 k0 1

]Compute A2, A3 and hence write down An for n > 1.

5. Assume that A is square matrix with an inverse A−1. Prove that det(A−1) =1/ det(A)

6. Let

A =

[5 22 1

]

Show thatA2 − 6A+ I = 0

where I is the 2× 2 identity matrix. Use this result to compute A−1.

7. Consider the following pair of matrices

A =

11 18 7a 6 3−3 −5 −2

, B =

3 1 12b −1 −5−2 1 −6

Compute the values of a and b so that A is the inverse of B while B is the inverseof A.

8. Here is a 2× 2 matrix equation[a bc d

]=

[e fg h

] [p qr s

]Show that this is equivalent to the following sets of equations[

ac

]= p

[eg

]+ r

[fh

]

and [bd

]= q

[eg

]+ s

[fh

]

9. Use the result of the previous question to show that if the original 2 × 2 matrixequation is written as

A = EP

then the columns of A are linear combinations of the columns of E.

10. Following on from the previous two questions, show that the rows of A can be writtenas linear combinations of the rows of P .


ENG1091


Laboratory class 3 Solutions

Matrices and Determinants Pt 2.

1. If possible, use a row or column that contains one or more zeros.

(a) 31 =

∣∣∣∣∣∣1 2 33 2 20 9 8

∣∣∣∣∣∣ (b) −165 =

∣∣∣∣∣∣4 3 21 7 83 9 3

∣∣∣∣∣∣(c) 0 =

∣∣∣∣∣∣∣∣1 2 3 21 3 2 34 0 5 01 2 1 2

∣∣∣∣∣∣∣∣ (d) 162 =

∣∣∣∣∣∣∣∣1 5 1 32 1 7 51 2 1 03 1 0 1

∣∣∣∣∣∣∣∣3. Which of the following statements are true? Which are false?

(a) False (b) False (c) True

(d) False (e) True (f) False

4. Compute A2 and A3 and note the pattern.

An =

[1 nk0 1

]

6.

A−1 = 6I − A =

[1 −2−2 5

]

7. Require that AB = I and BA = I. Then a = 4 and b = −1.


ENG1091


Laboratory class 4

Matrix operations

1. Suppose you are given a matrix of the form

R(θ) =

[cos θ − sin θsin θ cos θ

]Consider now the unit vector v

˜= [1, 0]T in a two dimensional plane. Compute

R(θ)v˜

. Repeat your computations this time using w˜

= [0, 1]T . What do you observe?Try thinking in terms of pictures, look at the pair of vectors before and after theaction of R(θ).

2. You may have recognised the two vectors in the previous question to be the familarbasis vectors for a two dimensional space, i.e., i

˜and j

˜. We can express any vector

as a linear combination of i˜

and j

˜, that is

u˜

= a i˜

+ bj

˜for some numbers a and b. Given what you learnt from the previous question, whatdo you think will be result of R(θ)u

˜? Your answer can be given in simple geometrical

terms (e.g., in pictures).

3. Give reasons why you expect R(θ + φ) = R(θ)R(φ). Hence deduce that

cos(θ + φ) = cos θ cosφ− sinφ sin θ

sin(θ + φ) = sin θ cosφ+ sinφ cos θ

4. Give reasons why you expect R(θ)R(φ) = R(φ)R(θ). Hence prove that the rotationmatrices R(θ) and R(φ) commute.

5. Show that detR(θ) = +1.

6. Given the above form for R(θ) write down, without doing any computations, theinverse of R(θ).

Eigenvectors and eigenvalues

A square matrix A has an eigenvector v with eigenvalue λ provided

Av = λv

The vector v would normally be written as a column vector. Its transpose vT is arow vector.

The eigenvalues are found by solving the polynomial equation

0 = det(A− λI)

7. Compute the eigenvalues and eigenvectors of the following matrices.

(a)

[4 −25 −3

](b)

[6 1−3 2

](c)

[5 3−3 −1

]

8. Given that one eigenvalue is λ = −4, compute the remaining eigenvalues of thefollowing matrices.

(a)

−1 3 −3√

2

3 −1 −3√

2

−3√

2 −3√

2 2

(b)

3 −1 −3√

2

−1 3 −3√

2

−3√

2 −3√

2 2

9. Compute the eigenvectors for each matrix of the previous question. Verify thatthe eigenvectors of part (b) are mutually orthogonal (i.e., 0 = vT1 v2, 0 = vT1 v3 and0 = vT2 v3).

10. Suppose the matrix A has eigenvectors v with corresponding eigenvalues λ. Showthat v is an eigenvector of An. What is its corresponding eigenvalue?

11. If λ, v are an eigenvalue-eigenvector pair for A then show that αv is also an eigen-vector of A.

12. Suppose the matrix A has eigenvectors v with corresponding eigenvalues λ. Deducethe eigenvectors and eigenvalues of R−1AR where R is a non-singular matrix.

13. Let A be any matrix of any shape. Show that ATA is a symmetric square matrix.


ENG1091



Matrix operations

1. Each of the vectors will have been rotated about the origin by the angle θ in acounterclockwise direction.

2. The rotation observed in the previous question also applies to the general vector u˜

.Thus R(θ) is often referred to as a rotation matrix. Matrices like this (and their 3dimensional counterparts) are used extensivly in computer graphics.

3. Any object rotated first by θ and then by φ could equally have been subject to asingle rotation by θ+φ. The resulting objects must be identical. Hence R(θ+φ) =R(θ)R(φ).

4. Regardless of the order in which the rotations have been applied the nett rotationwill be the same. Thus R(θ)R(φ) = R(φ)R(θ). Equally, you could have started bywriting θ + φ = φ+ θ, then R(θ + φ) = R(φ+ θ) and so R(θ)R(φ) = R(φ)R(θ).

5.

detR(θ) =

∣∣∣∣cos θ − sin θsin θ cos θ

∣∣∣∣ = 1

6. The inverse of R(θ) is R(−θ).

Eigenvectors and eigenvalues

7. (a) λ = −1 and 2 (b) λ = 3 and 5 (c) λ = 2 (a double root)

8. (a) λ = 8 and − 4 (a double root) (b) λ = 8, 4 and − 4

9. In part (a) there is a double root λ = −4. In this case there are two linearlyindependent eigenvectors. Your may answers may appear different from those givenhere, you will need to check that your eigenvectors are linear combinations of thosegiven here. Also, remember that any scaling is allowed for an eigenvector.

(a) λ = 8 v = (−1,−1,√

2)T

λ = −4 v = (2, 0,√

2)T

λ = −4 v = (−1, 1, 0)T

(b) λ = 8 v = (−1,−1,√

2)T

λ = 4 v = (−1, 1, 0)T

λ = −4 v = (1, 1,√

2)T

10. The eigenvalue of An will be λn.

11. This is trivial, just multiply the eigenvalue equation Av = λv by α.

12. The matrix R−1AR will have λ as an eigenvalue with eigenvector R−1v.

13. Use (PQ)T = QTP T and (AT )T = A to show that (ATA)T = ATA. Hence ATA issymmetric.


ENG1091


Laboratory class 5

Integration by parts

1. Evaluate each of the following using integration by parts. Recall that

∫fdg

dxdx = fg −

∫gdf

dxdx

(a)

∫x cos(x) dx (b)

∫xe−x dx

(c)

∫y√y + 1 dy (d)

∫x2 log(x) dx

(e)

∫sin2(θ) dθ (f)

∫cos2(θ) dθ

(g)

∫sin(θ) cos(θ) dθ (h)

∫θ sin2(θ) dθ

2. Use integration by parts twice to find∫ex sin(x) dx and

∫ex cos(x) dx.

3. Use a substitution and an integration by parts to evaluate each of the following

(a)

∫(3x− 7) sin(5x+ 2) dx (b)

∫cos(x) sin(x)ecos(x) dx

(c)

∫e2x cos (ex) dx (d)

∫e√x dx

4. Spot the error in the following calculation.

We wish to compute∫dx/x. For this we will use integration by parts with u = 1/x

and dv = dx. This gives us du = −dx/x2 and v = x. Thus using∫udv = uv−

∫vdu

we find ∫dx

x= 1 +

∫dx

x

and thus 0 = 1. (If this answer does not cause you serious grief then a career inaccountancy beckons).

Improper integrals

5. Decide which of the following improper integrals will converge and which will diverge.

(a)

∫ 1

0

1

xdx (b)

∫ 1

0

1

x1/4dx

(c)

∫ 1

0

1

y4dy (d)

∫ ∞0

e−2x dx

(e)

∫ ∞0

1

1 + θ2dθ

Comparison test for Improper integrals

6. Use a suitable comparison function to decide which of the following integrals willconverge and which will diverge.

(a)

∫ 1

0

ex

xdx (b)

∫ 1

0

1

1− x1/4dx

(c)

∫ 1

0

e−y

y4dy (d)

∫ ∞0

sin2(x)e−2x dx

(e)

∫ ∞0

e−θ

1 + θ2dθ (f)

∫ 1

0

1

x(1− x2)dx


ENG1091



Integration by parts

1. (a)

∫x cos(x) dx = cos(x) + x sin(x) + C

(b)

∫xe−x dx = −e−x − xe−x + C

(c)

∫y√y + 1 dy =

2

3y (y + 1)3/2 − 4

15(y + 1)5/2 + C

(d)

∫x2 log(x) dx =

x3

3log(x)− x3

9+ C

(e)

∫sin2(θ) dθ =

1

2(θ − cos(θ) sin(θ)) + C

(f)

∫cos2(θ) dθ =

1

2(θ + cos(θ) sin(θ)) + C

(g)

∫sin(θ) cos(θ) dθ =

1

2sin2(θ) + C

(h)

∫θ sin2(θ) dθ =

−θ2

cos(θ) sin(θ) +1

4sin2(θ) +

1

4θ2 + C

2. (a)

∫ex sin(x) dx =

ex

2(sin(x)− cos(x)) + C

(b)

∫ex cos(x) dx =

ex

2(sin(x) + cos(x)) + C

3. (a)

∫(3x− 7) sin(5x+ 2) dx =

3

25sin(5x+ 2) +

1

5(7− 3x) cos(5x+ 2) + C

(b)

∫cos(x) sin(x)ecos(x) dx = ecos(x) (1− cos(x)) + C

(c)

∫e2x cos (ex) dx = cos(ex) + ex sin(ex) + C

(d)

∫e√x dx = 2e

√x(√

x− 1)

+ C

4. Did we forget an integration constant? (And so with the natural order restored,fears of a career in accountancy fade from view.)

Improper integrals

5. Decide which of the following improper integrals will converge and which will diverge.

(a)

∫ 1

0

1

xdx diverges (b)

∫ 1

0

1

x1/4dx converges to 4/3

(c)

∫ 1

0

1

y4dy diverges (d)

∫ ∞0

e−2x dx converges to 1/2

(e)

∫ ∞0

1

1 + θ2dθ converges to π/2 (f)

∫ 2

0

1

1− x2dx diverges

(g)

∫ 2

0

1

x(x+ 2)dx diverges (h)

∫ 2

0

1

x(x− 2)dx diverges

Comparison test for Improper integrals

6. Use a suitable comparison function to decide which of the following integrals willconverge and which will diverge.

(a)

∫ 1

0

ex

xdx diverges, use

1

x<ex

xover 0 < x < 1

(b)

∫ 1

0

1

1− x1/4dx diverges, use x < x1/4 over 0 < x < 1

(c)

∫ 1

0

e−y

y4dy diverges, use

1

3y4<e−y

y4over 0 < y < 1

(d)

∫ ∞0

sin2(x)e−2x dx converges, use sin2(x)e−2x < e−2x over 0 < x <∞

(e)

∫ ∞0

e−θ

1 + θ2dθ converges, use

e−θ

1 + θ2<

1

1 + θ2over 0 < θ <∞

(f)

∫ 1

0

1

x(1− x2)dx diverges, use

1

x<

1

x(1− x2)over 0 < x < 1


ENG1091


Laboratory class 6

Sequences

1. Find the limit, if it exists, for each of the following sequences

(a) −1,+12,−1

3,+1

4, · · · , (−1)

n

n+1, · · ·

(b) 12, 23, 34, · · · , n+1

n+2, · · ·

(c) an = 1n+1

, n ≥ 0

(d) an = 1n+2− 1

n+1, n ≥ 0

(e) an =

1 + 1

n+1, n even

1− 1n+1

, n odd

(f) an =

e−n, n ≥ 100

en, 0 ≤ n < 100

(g) an = sin(nπ4

) (Hint : Write out the first few terms.)

2. Consider the sequence defined by

an+1 = an +

(1

2

)n+1

, n ≥ 0

with a0 = 1.

(a) Write out the first few terms a0, · · · , a4.

(b) Can you express a5 in terms of 12a4?

(c) Generalize this result to express an+1 in terms of 12an.

(d) Can you express an as a sum∑n

k=0 bk for some set of bk?

(e) Suppose the limit limn→∞ an exists. Use the result of (c) to deduce thelimit.

(f) Determine the values of λ for which the sequence an+1 = an +λn converges.

Series

3. Which of the following statements are true?

(a) The infinite series∑∞

n=0 an converges whenever limn→∞ |an| = 0.

(b) The harmonic series∑∞

n=0 1/(n+ 1) converges.

(c) If the series∑∞

n=0 |an| converges then∑∞

n=0 an also converges.

(d) If∑∞

n=0 an diverges then∑∞

n=0 (−1)nan converges.

(e) If limn→∞ |an+1

an| > 1 then

∑∞n=0 an converges.

The Integral Test

4. Establish the convergence (or divergence) of the following series using the integraltest.

(a)∑∞

n=01√n+1

(b)∑∞

n=01

(n+1)γ, γ > 1

(c)∑∞

n=01

n2+1

(d)∑∞

n=01

(n+1)(n+2)

(Hint : First establish a comparison with∑∞

n=0 (n+ 1)−2 then use

the integral test.)

The Comparison Test

5. Determine the convergence or otherwise of the following series using the suggestedseries for comparison.

(a)∑∞

n=0n+2n+1

compare with∑∞

n=0 1

(b)∑∞

n=01

(2+1/(n+1))n+1 compare with∑∞

n=01

2n+1

(c)∑∞

n=02+sinnn+1

compare with∑∞

n=01

n+1

(d)∑∞

n=03−n

n+1compare with

∑∞n=0

(13

)n

The Ratio Test

6. Use the ratio test to examine the convergence of the following series.

(a)∑∞

n=0 λ−n, |λ| > 1

(b)∑∞

n=0xn

n+1, |x| < 1

(c)∑∞

n=0 n1−n

(d)∑∞

n=0n3

en+2

7. What does the ratio test tell you about the convergence of

∞∑n=0

1

(n+ 1)2

Can you establish the convergence of this series by some other method?

8. The Starship USS Enterprise is being pursued by a Klingon warship. The dilithiumcrystals couldn’t handle the warp speed and so it would appear that Captain Kirkand his crew are about to become as one with the inter-galactic dust cloud.

Spock : Captain, the enemy are 10 light years away and are closing fast.

Kirk : But Spock, by the time they travel the 10 light years we will havetravelled a further 5 light years. And when they travel those 5 lightyears we will have moved ahead by a further 2.5 light years, and so onforever. Spock, they will never capture us!

Spock : I must inform the captain that he has made a serious error of logic.

What was Kirk’s mistake? How far will Kirk’s ship travel before being caught?


ENG1091



Sequences

1. (a) 0 (b) 1 (c) 0 (d) 0(e) 1 (f) 0 (g) Limit does not exist

2. This is the geometric series. It converges for |λ| < 1.

Series

3. (a) False (b) False (c) True (d) False(e) False

The Integral Test

4. (a) Diverges (b) Converges (c) Converges (d) Converges

The Comparison Test

5. (a) Diverges (b) Converges (c) Diverges (d) Converges

The Ratio Test

6. (a) Converges (b) Converges (c) Converges (d) Converges

7. The series converges and this could also be established using the integral test.

8. Clearly the fast ship must catch the slow ship in a finite time. Yet Kirk has put anargument which shows that his slow ship will still be ahead of the fast ship aftereach cycle (a cycle ends when the fast ship just passes the location occupied by theslow ship at the start of the cycle). Each cycle takes a finite amount of time. Thetotal elapsed time is the sum of the times for each cycle. Kirk’s error was to assumethat the time taken for an infinite number of cycles must be infinite. We know thatthis is wrong – an infinite series may well converge to a finite number.

Given the information in the question we can see that the fast ship is initially 10light years behind the slow ship and that it is traveling twice as fast as the slow ship.

Suppose the fast ship is traveling at v light years per year. The distance traveledby the fast ship decreases by a factor of 2 in each cycle. Hence the time interval foreach cycle also decreases by a factor of 2 in each cycle. The total time taken willthen be

Time =10 + 5 + 2.5 + 1.25 + ...

v

=10

v

(1 +

1

2+

1

4+

1

8· · ·)

=10

v

1

1− 12

=10

v/2

We expect that this must be time taken for the fast ship to catch the slow ship. Thefast ship is traveling at speed v while the slow ship is traveling at speed v/2. Thusthe fast ship is approaching the slow ship at a speed v/2 and it is initially 10 lightyears behind. Hence it will take the Klingon’s 10/(v/2) light years to catch Kirk’sstarship.


ENG1091


Laboratory class 7

Power series

1. Find the radius of convergence for each of the following power series

(a) f(x) =∑∞

k=0kxk

3k(b) g(x) =

∑∞k=0

xk

3kk!

(c) h(x) =∑∞

k=0 k2xk (d) p(x) =

∑∞k=0

x2k

log(1+k)

(e) q(x) =∑∞

k=0k!(x−1)k2kkk

(f) r(x) =∑∞

k=0 (1 + k)kxk

Maclaurin Series

2. Find the first 4 non-zero terms in Maclaurin series for each of the following functions

(a) f(x) = cos(x) (b) f(x) = sin(2x)

(c) f(x) = log(1 + x) (d) f(x) = 11+x2

(e) f(x) = arctan(x) (f) f(x) =√

1− x2

3. Use the previous results to obtain the first 2 non-zero terms in the Maclaurin seriesfor the following functions.

(a) f(x) = cos(x) sin(2x) (c) f(x) = log(1 + x2)

(d) f(x) = 11+cos2(x)

(e) f(x) = arctan(arctan(x))

As the algebra in some parts of this question is rather tedious, you might like to dothis question using Scientific Notebook.

Taylor Series

4. Compute the Taylor series, about the the given point, for each of the followingfunctions.

(a) f(x) = 1x, a = 1 (b) f(x) =

√x, a = 1

(c) f(x) = ex, a = −1 (d) f(x) = log x, a = 2

5. (a) Compute the Taylor series for ex

(b) Hence write down the Taylor series for e−x2

(c) Use the above to obtain an infinite series for the function

s(x) =

∫ x

0

e−u2

du

6. (a) Compute the Taylor series, around x = 0, for log(1 + x) and log(1− x).

(b) Hence obtain a Taylor series for f(x) = log(1+x1−x

)(c) Compute the radius of convergence for the Taylor series in part (b).

(d) Show that the function defined by y(x) = 1+x1−x has a unique inverse for

almost all values of y.

(e) Use the above results to obtain a power series for log(y) valid for 1 < |y| <∞.


ENG1091



Power series

1. (a) R = 3 (b) R =∞(c) R = 1 (d) R = 1(e) R = 2e, note limn→∞(1 + x/n)n = ex (f) R = 0

Maclaurin Series

2. (a) cos(x) = 1− 12x2 + 1

24x4 − 1

720x6 + · · ·

(b) sin(2x) = 2x− 43x3 + 4

15x5 − 8

315x7 + · · ·

(c) log(1 + x) = x− 12x2 + 1

3x3 − 1

4x4 + · · ·

(d) 11+x2

= 1− x2 + x4 − x6 + · · ·

(e) arctan(x) = x− 13x3 + 1

5x5 − 1

7x7

(f)√

1− x2 = 1− 12x2 − 1

8x4 − 1

16x6 + · · ·

3. (a) cos(x) sin(2x) = 2x− 73x3 + · · · (c) log(1 + x2) = x2 − 1

4x4 + · · ·

(d) 11+cos2(x)

= 12

+ 14x2 + · · · (e) arctan(arctan(x)) = x− 2

3x3 + · · ·

Taylor Series

4. (a) 1x

= 1− (x−) + (x− 1)2 − (x− 1)3 + (x− 1)4 + · · ·

(b)√x = 1 + 1

2(x− 1)− 1

8(x− 1)2 + 1

16(x− 1)3 + · · ·

(c) ex = e−1(1 + (x+ 1) + 12(x+ 1)2 + 1

6(x+ 1)3 + · · ·

(d) loge x = loge(2) + 12(x− 2)− 1

8(x− 2)2 + 1

24(x− 2)3 + · · ·

5. (a) ex = 1 + x+ 12x2 + 1

6x3 + 1

24x4 + · · ·

(b) e−x2

= 1− x2 + 12x4 − 1

6x6 + 1

24x8 + · · ·

(c) s(x) =∫ x0e−u

2= x− 1

3x3 + 1

10x5 − 1

42x7 + 1

216x9 + · · ·

6. (a) loge(1 + x) = x− 12x2 + 1

3x3 − 1

4x4 + · · · =

∑∞n=1

(−1)(n+1)

nxn

loge(1− x) = −x− 12x2 − 1

3x3 − 1

4x4 + · · · = −

∑∞n=1

1nxn

(b) loge(1+x1−x

)= 2x+ 21

3x3 + 21

5x5 + · · · = 2

∑∞n=1

12n−1x

2n−1, R = 1

(c) x = y−1y+1

, y 6= −1

(d) loge(y) = 2∑∞

n=11

2n−1x2n−1, x = (y − 1)/(y + 1)


ENG1091


Laboratory class 8

Separable first order ODEs

1. Find the general solution for each of the following seperable ODEs

(a)dy

dx= 2xy (b) y

dy

dx+ sin(x) = 0

(c) sin(x)dy

dx+ y cos(x) = 2 cos(x) (d)

1 + dy/dx

1− dy/dx=

1− y/x1 + y/x

Non-separable first order ODEs

2. For each of the following ODEs find any particular solution.

(a)dy

dx+ y = 1 (b)

dy

dx+ 2y = 2 + 3x

(c)dy

dx− y = e2x (d)

dy

dx− y = ex

(e)dy

dx+ 2y = cos(2x) (f)

dy

dx− 2y = 1 + 2x− sin(x)

3. Find the general solution of the homogenous equation for each of the ODEs in theprevious question. Hence obtain the general solution of the ODE.

Integrating factor

4. Use an integrating factor to find the general solution for each of the following ODEs

(a)dy

dx+ 2y = 2x (b)

dy

dx+

2

xy = 1

(c)dy

dx+ cos(x)y = 3 cos(x) (d) sin(x)

dy

dx+ cos(x)y = tan(x)

Second order homogenous ODEs

5. Find the general solution for each of the following ODEs.

(a)d2y

dx2+dy

dx− 2y = 0 (b)

d2y

dx2− 9y = 0

(c)d2y

dx2+ 2

dy

dx+ 2y = 0 (d)

d2y

dx2+ 6

dy

dx+ 10y = 0

(e)d2y

dx2− 4

dy

dx+ 4y = 0 (f)

d2y

dx2+ 6

dy

dx+ 9y = 0

6. Find the particular solution, for the corresponding ODE in the previous question,that satisfies the following boundary conditions.

(a) y(0) = 1 and y(1) = 0 (b) y(0) = 0 and y(1) = 1

(c) y(0) = −1 and y(+π/2) = +1 (d) y(0) = −1 anddy

dx= 0 at x = 0

(e) y(0) = 1 anddy

dx= 0 at x = 1 (f)

dy

dx= 0 at x = 0 and

dy

dx=

1 at x = 1

Second order non-homogenous ODEs

7. Find the general solution for each of the following ODEs.

(a)d2y

dx2+dy

dx− 2y = 1 + x (b)

d2y

dx2− 9y = e3x

(c)d2y

dx2+ 2

dy

dx+ 2y = sin(x) (d)

d2y

dx2+ 6

dy

dx+ 10y = e2x cos(x)

(e)d2y

dx2− 4

dy

dx+ 4y = 2x (f)

d2y

dx2+ 6

dy

dx+ 9y = cos(x)


ENG1091



Separable first order ODEs

1. (a) y = Cex2

(b) y = ±√

2 cos(x) + C

(c) y = 2 +C

sin(x)(d) y =

C

x

Non-separable first order ODEs

2. (a) y = 1 (b) y =1

4+

3x

2

(c) y = e2x (d) y = xex

(e) y =1

4cos(2x) +

1

4sin(2x) (f) y = −1− x+

1

5cos(x) +

2

5sin(x)

3. (a) y = 1 + Ce−x (b) y =1

4+

3x

2+ Ce−2x

(c) y = e2x + Cex (d) y = xex + Cex

(e) y =1

4cos(2x)+

1

4sin(2x)+Ce−2x (f) y = −1−x+

1

5cos(x)+

2

5sin(x)+

Ce2x

Integrating factor

4. (a) y = x− 1

2+ Ce−2x (b) y =

x

3+C

x2

(c) y = 3 + Ce− sin(x) (d) y =C − loge(cos(x))

sin(x)

Second order homogenous ODEs

5. (a) y = Aex +Be−2x (b) y = Ae3x +Be−3x

(c) y = (A cos(x) +B sin(x)) e−x (d) y = (A cos(x) +B sin(x)) e−3x

(e) y = (A+Bx) e2x (f) y = (A+Bx) e−3x

6. (a) y(x) =1

e3 − 1

(e3−2x − ex

)(b) y(x) =

e3x − e−3x

e3 − e−3(c) y(x) =(

− cos(x) + eπ/2 sin(x))e−x

(d) y(x) = − ((3 sin(x) + cos(x)) e−3x

(e) y(x) =

(1− 2x

3

)e2x (f) y(x) = −1

9(1 + 3x) e3−3x

Second order non-homogenous ODEs

7. (a) y = −3

4− x

2+ Aex +Be−2x

(b) y =(A+

x

6

)e3x +Be−3x

(c) y =1

5(−2 cos(x) + sin(x)) + (A cos(x) +B sin(x))e−x

(d) y =1

145(5 cos(x) + 2 sin(x)) e2x + (A cos(x) +B sin(x))e−3x

(e) y =1 + x

2+ (A+Bx)e2x

(f) y =1

50(4 cos(x) + 3 sin(x)) + (A+Bx)e−3x


ENG1091


Laboratory class 9

l’Hopital’s rule

1. Use l’Hopital’s rule to verify the following limits

(a) −2 = limx→−1

x2 − 1

x+ 1(b)

4

5= lim

x→0

sin(4x)

sin(5x)

(c)−1

π2= lim

x→1

1− x+ log(x)

1 + cos(πx)(d) 0 = lim

x→∞

log(log(x))

x

(e)1

4= lim

x→0

x

tan−1(4x)(f) 0 = lim

x→∞e−x log(x)

2. Prove that for any n > 00 = lim

x→∞xne−x

3. Prove that for any n > 00 = lim

x→∞x−n log(x)

Coupled first order ODEs

4. Solve each of the following coupled ODEs by first differentiating each equation andthen making suitable combinations to de-couple the equations. Verify your solutionsby substituting back into the original ODEs.

(a)du

dx= 5u+ 3v

dv

dx= u+ 7v

(b)du

dx= 6u+ 3v

dv

dx= −4u− v

(c)du

dx= 4u− 2v

dv

dx= −u+ 3v

(d)du

dx= 8u+ 4v

dv

dx= −7u− 3v

5. Solve each of the coupled ODEs of the previous question by way of eigenvectors andeigenvalues.


ENG1091


Laboratory class 10

Limits

1. At which points are the following functions discontinuous (if any)? Assume thedomain for each function to be R or R2.

(a) f(x) = sin(x) (b) g(x) = (2− x)/(2 + x)

(c) h(x) = log x (d) p(x) = (1+2x−x2)/(1+2x+x2)

(e) r(x, y) = tan(x+ y) (f) s(x, y) = (x− y)2/(x+ y)2

(g) t(u, v) = (1 +u+u2)/(1 + v+ v2) (h) w(u, v) = exp(−u2 − v2)

2. Use your calculator to estimate the following limits.

(a) limx→0

sin(x)

x(b) lim

x→1

1 + x

1− x

(c) lim(x,y)→(0,0)

sin(x+ y)

x+ y(d) lim

(x,y)→(1,1)

(x+ y − 1)2

(x− y + 1)2

(e) lim(x,y)→(1,0)

x2 − y2 − 1

x2 + y2 − 1(f) lim

(x,y)→(0,0)

1− exp(−x2y2)xy

Partial Derivatives

3. Evaluate the first partial derivatives for each of the following functions

(a) f(x, y) = cos(x) cos(y) (b) f(x, y) = sin(xy)

(c) f(x, y) = log(1 + x)/ log(1 + y) (d) f(x, y) = (x+ y)/(x− y)

(e) f(x, y) = xy (f) f(u, v) = uv(1− u2 − v2)

4. For the function f(x, y) = y2 sin(x) verify that

∂

∂x

(∂f

∂y

)=∂

∂y

(∂f

∂x

)

Chain Rule

5. Given f(x, y) = 2x2 + 4y − 2 and x(s) = 3s, y(s) = 2s2 compute df/ds by directsubstitution (i.e. first construct f(s)) and also by the chain rule.

6. Given f(x, y) = 2xy and x(r, θ) = r cos θ, y(r, θ) = r sin θ compute ∂f/∂x, ∂f/∂y,∂f/∂r and ∂f/∂θ,

7. Let f = f(x, y) be an arbitrary function of (x, y). Using the same transformationas in the previous question express

∂2f

∂x2+∂2f

∂y2

in terms of partial derivatives of f in r and θ. This is a long and tedious question– have fun!

Directional derivatives

8. Compute df/ds for the function f(x, y) = xy + x + y along the curve x(s) =r cos(s/r), y(s) = r sin(s/r). Also, verify that (dx/s) i

˜+ (dy/ds)j

˜is a unit vector.

9. Compute the directional derivative for each for the following functions in the stateddirection. Be sure that you use a unit vector!

(a) f(x, y) = 2x+ 3y at (1, 2), t˜

= (3 i˜

+ 4j

˜)/5

(b) g(x, y) = sin(x) cos(y) at (π/4, π/4), t˜

= ( i˜

+ j

˜)/√

2

(c) h(x, y, z) = log(x2 + y2 + z2) at (1, 0, 1), t˜

= i˜

+ j

˜− k˜

(d) q(x, y, z) = 4x2 − 3y3 + 2z2 at (0, 1, 2), t˜

= 2 i˜− 3j

˜+ k˜

(e) r(x, y, z) = z exp(−2xy) at (1, 1,−1), t˜

= i˜− 3j

˜+ 2k

˜(f) w(x, y, z) =

√1− x2 − y2 − z2 at (0.5, 0.5, 0.5), t

˜= 2 i

˜− j

˜+ k˜

Tangent planes

10. Compute the tangent plane f̃ approximation for each of the following functions atthe stated point.

(a) f(x, y) = 2x+ 3y at (1, 2)

(b) g(x, y) = sin(x) cos(y) at (π/4, π/4)

(c) h(x, y, z) = log(x2 + y2 + z2) at (1, 0, 1)

(d) q(x, y, z) = 4x2 − 3y3 + 2z2 at (0, 1, 2)

(e) r(x, y, z) = z exp(−2xy) at (1, 1,−1)

(f) w(x, y, z) =√

1− x2 − y2 − z2 at (0.5, 0.5, 0.5)

11. Use the result from the previous question to estimate the function at the statedpoints. Compare your estimate with that given by a calculator.

(a) f(x, y) at (1.1, 1.9) (b) g(x, y) at (3π/16, 5π/16)

(c) h(x, y, z) at (0.8, 0.1, 0.9) (d) q(x, y, z) at (0.1, 1.1, 1.9)

(e) r(x, y, z) at (0.8, 1.2,−1.1) (f) w(x, y, z) at (0.6, 0.4, 0.6)

12. This is more a question on theory rather than being a pure number question. It isthus not examinable.

Consider a function f = f(x, y) and its tangent plane approximation f̃ at some

point P . Both of these may be drawn as surfaces in 3-dimensional space. You mightask – How can I compute the normal vector to the surface for f at the point P?And that is exactly what we will do in this question.

Construct f̃ at P (i.e write down the standard formula for f̃). Draw this as a surfacein the 3-dimensional space. This surface is a flat plane tangent to the surface for fat P (hence the name, tangent plane).

Given your equation for the plane, write down a 3-vector normal to this plane.Hence deduce the normal to the surface for the function f = f(x, y) at P .

13. Generalise your result from the previous question to surfaces of the form 0 =g(x, y, z). This question is also a non-examinable extension. But it is fun! (agreed?).

Maxima and Minima

14. Find all of the extrema (if any) for each of the following functions (you do not needto charactise the extrema).

(a) f(x, y) = 4− x2 − y2 (b) g(x, y) = xy exp(−x2 − y2)

(c) h(x, y) = x− x3 + y2 (d) p(x, y) = (2− x2) exp(−y)

(e) q(x, y, z) = 4x2 + 3y2 + z2 (f) r(x, y, z) = arctan((x−1)2 + y2 +z2)


ENG1091



Limits

1. At which points are the following functions discontinuous (if any)? Assume thedomain for each function to be R or R2.

(a) None (b) x = −2

(c) x = 0 (d) x = −1

(e) x+ y = ±π/2,±3π/2,±5π/2 · · · (f) x+ y = 0

(g) None (h) None

2. Use your calculator to estimate the following limits.

(a) 1 (b) ∞

(c) 1 (d) 1

(e) No unique limit, try limits alongthe axes.

(f) 0

Partial Derivatives

3. Evaluate the first partial derivatives for each of the following functions

(a)∂f

∂x= − sin(x) cos(y)

∂f

∂y= − cos(x) sin(y)

(b)∂f

∂x= y cos(xy)

∂f

∂y= x cos(xy)

(c)∂f

∂x=

1

(1 + x) log(1 + y)

∂f

∂y=

− log(1 + x)

(1 + y) log2(1 + y)

(d)∂f

∂x=−2y

(x− y)2∂f

∂y=

2x

(x− y)2

(e)∂f

∂x= y

∂f

∂y= x

(f)∂f

∂u= v(1− 3u2 − v2) ∂f

∂v= u(1− u2 − 3v2)

Chain Rule

5. df/ds = 52s

6. ∂f/∂x = 2y, ∂f/∂y = 2x, ∂f/∂r = 4r cos θ sin θ, ∂f/∂θ = 2r2(cos2 θ − sin2 θ),

7. This is not an easy question, two chocolate frogs if you got it right!

∂2f

∂x2+∂2f

∂y2=∂2f

∂r2+

1

r

∂f

∂θ+

1

r2∂2f

∂θ2

Directional derivatives

8. df/ds = r(cos2(s/r)− sin2(s/r)

)− sin(s/r) + cos(s/r).

9. (a) 18/5 (b) 0

(c) 0 (d) 35/√

14

(e) −2 exp(−2)/√

14 (f) −2/√

6

Tangent planes

10. (a) f̃(x, y) = 8 + 2(x− 1) + 3(y − 2)

(b) f̃(x, y) = (1/2) + (1/2)(x− π/4)− (1/2)(y − π/4)

(c) f̃(x, y, z) = log 2 + (x− 1) + (z − 1)

(d) f̃(x, y, z) = 5− 9(y − 1) + 8(z − 2)

(e) f̃(x, y, z) = exp(−2)(−1 + 2(x− 1) + 2(y − 1) + (z + 1))

(f) f̃(x, y, z) = (1/2)− (x− (1/2))− (y − (1/2))− (z − (1/2))

11. The calculator’s answer is in brackets.

(a) 7.9 (7.900) (b) 0.304 (0.2397)

(c) 0.393 (0.3784) (d) 3.7 (3.267)

(e) -0.149 (-0.1613) (f) 0.4 (0.3464)

12. This question is not examinable.

For a surface written in the form z = f(x, y) the vector

N =

(∂f

∂x

)i˜

+

(∂f

∂y

)j

˜− k˜

is normal to the surface.

13. This question is not examinable.

For a surface written in the form 0 = g(x, y, z) the vector

N = ∇g =

(∂g

∂x

)i˜

+

(∂g

∂y

)j

˜+

(∂g

∂z

)k˜

is normal to the surface.

Maxima and Minima

14. (a) (0, 0) (b) (0, 0) and the four points(±1/

√2,±1/

√2)

(c) (±1/√

3, 0) (d) None

(e) (0, 0, 0) (f) (1, 0, 0)

modern engineering mathematics, 4th ed. - monash...

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