alia ysia ica Italy India monash.edu/scienceusers.monash.edu/~leo/moodle/eng1091/labs/2up-online.pdf · ENG1091 ing ercises Campus Campus alia ysia ica Italy India monash.edu/science

ENG1091

Mathematics for Engineering

Laboratory class exercises

Clayton Campus2014 Campus

Australia Malaysia South Africa Italy India monash.edu/science

School of Mathematical Sciences Monash University

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

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

26-Jul-2014 2


Supplementary exercises

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

Please note that the exercises provided by James for Improper Integrals is rather thin(just one question). So you are encouraged to complete the supplementary exercises onImproper Integrals. This will be sufficient study for questions of this kind should theyappear on the final exam.

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SCHOOL OF MATHEMATICAL SCIENCES

ENG1091


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) in thedirection of v

˜.

3. For each pair of vectors given below, calculate the vector dot product and the angle θbetween 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 perpendicularto 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. Assuming thatthe 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 the para-metric 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 parametricform 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 dot product.)

12. Find the distance between the parallel planes defined by the equations 2x−y+ 3z = −4and 2x− y + 3z = 24. (Hint: Use the cross product to construct a line normal to bothplanes, 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: Usethe 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, definethe lines within parallel planes and then go back to problem 12.)

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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

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ENG1091


Laboratory class 2

Row operations and linear systems

Solve each of the following system of equations using Gaussian elimination with back-substitution. Be sure to record the details of each row-operation (for example, as a noteon 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 for thefollowing 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 two freeparameters).

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

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

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 questions 1and 5.

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Matrix determinants

14. Compute the determinant for the coefficient matrices in questions 7 and 8. What doyou 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 second byexpanding 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).

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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 62 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.

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ENG1091


Laboratory class 3

Matrices and Determinants Pt 2.

1. Compute the following determinants using expansions about any suitable row or col-umn.

(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 a multipleof some other row.

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

(c) If any two columns of a square matrix are equal then the determinant of that matrixis 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 inverse of 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 matrix equationis 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 .

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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 the action ofR(θ).

2. You may have recognised the two vectors in the previous question to be the familar basisvectors 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, what doyou 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, the inverseof 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 a rowvector.

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 the followingmatrices.

(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 that theeigenvectors of part (b) are mutually orthogonal (i.e., 0 = vT1 v2, 0 = vT1 v3 and 0 = vT2 v3).

10. Suppose the matrix A has eigenvectors v with corresponding eigenvalues λ. Show thatv 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 eigenvectorof A.

12. Suppose the matrix A has eigenvectors v with corresponding eigenvalues λ. Deduce theeigenvectors 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.

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ENG1091



Matrix operations

1. Each of the vectors will have been rotated about the origin by the angle θ in a counter-clockwise direction.

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

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

3. Any object rotated first by θ and then by φ could equally have been subject to a singlerotation 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 rotation willbe the same. Thus R(θ)R(φ) = R(φ)R(θ). Equally, you could have started by writingθ + 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 linearly independenteigenvectors. Your may answers may appear different from those given here, you willneed to check that your eigenvectors are linear combinations of those given 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.

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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∫u dv = uv −

∫v du 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 will convergeand which will diverge.

(a)

∫ 1

0

ex

xdx (b)

∫ 1

0

1

1− x1/4 dx

(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

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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 ofa 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− x2 dx 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 will convergeand which will diverge.

(a)

∫ 1

0

ex

xdx diverges, use

1

x<ex

xover 0 < x < 1

(b)

∫ 1

0

1

1− x1/4 dx 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, use1

x<

1

x(1− x2) over 0 < x < 1

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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 + 1n+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 the limit.

(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 integral test.

(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 suggested seriesfor 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−nn+1

compare with∑∞

n=0

(13

)n

The Ratio Test

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

26-Jul-2014 25


(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 Kirk andhis 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 have travelleda further 5 light years. And when they travel those 5 light years we willhave moved ahead by a further 2.5 light years, and so on forever. 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?

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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 after eachcycle (a cycle ends when the fast ship just passes the location occupied by the slow shipat the start of the cycle). Each cycle takes a finite amount of time. The total elapsedtime is the sum of the times for each cycle. Kirk’s error was to assume that the timetaken for an infinite number of cycles must be infinite. We know that this is wrong – aninfinite series may well converge to a finite number.


Given the information in the question we can see that the fast ship is initially 10 lightyears behind the slow ship and that it is traveling twice as fast as the slow ship. Supposethe fast ship is traveling at v light years per year. The distance traveled by the fast shipdecreases by a factor of 2 in each cycle. Hence the time interval for each cycle alsodecreases by a factor of 2 in each cycle. The total time taken will then 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. Thus thefast ship is approaching the slow ship at a speed v/2 and it is initially 10 light yearsbehind. Hence it will take the Klingon’s 10/(v/2) light years to catch Kirk’s starship.

26-Jul-2014 28


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 series forthe 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 do thisquestion using Scientific Notebook.


Taylor Series

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

(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| <∞.

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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+ 213x3 + 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)

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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, thatsatisfies 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)

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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−3xe3 − 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

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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 solutions bysubstituting 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 the domainfor 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 transformation as inthe 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 – havefun!

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 at thestated 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)

26-Jul-2014 39


11. Use the result from the previous question to estimate the function at the stated points.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 is thusnot examinable.

Consider a function f = f(x, y) and its tangent plane approximation f̃ at some pointP . Both of these may be drawn as surfaces in 3-dimensional space. You might ask –How can I compute the normal vector to the surface for f at the point P? And that isexactly what we will do in this question.

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

Given your equation for the plane, write down a 3-vector normal to this plane. Hencededuce 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 need tocharactise 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)

26-Jul-2014 40


ENG1091



Limits

1. At which points are the following functions discontinuous (if any)? Assume the domainfor 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.

26-Jul-2014 42


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)

26-Jul-2014 43

Documents

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