Math AI HL

A nswers 593

ii b i 2 2.5 t

yx 1x
60 1.25 t

45 1 2 2.5
0.75

30 0.5

15 0.25

00 50 100 150 200 x 00 0.5 1 1.5
t ii y 0.5
11 a i 2.5 1 1.5
2
x 1.5
1.5 1
0.5
1

0.5

0

− 0.50 0.5 1 1.5 2 2.5 00
12 a i x
ii y
2.5 10
2
1.5 5
1
0.5 0
0x
−0.5 −5
−1
−1.−50.5 0 0.5 1 1.5 −100 0.5

594 Answers

ii y 3, 5; 21 , 2113 a −
10 ⎛ ⎞⎛ ⎞
⎝⎜ − ⎠⎟ ⎝⎜ ⎟⎠
0
x ⎛⎞ −3t ⎛ ⎞ 5t ⎛ ⎞
−10 10 ⎝⎜ − ⎟⎠ + ⎝⎜ ⎠⎟
x Ae 1 Be 1b ⎜ ⎟=
−20 t y 2 2⎝ ⎠
2.5
−3−010 −5 0 5 c A = 1, B = 3 x = e−3t + 3e5t, y = −2e−3t + 6e5t
bi x x 14 a  b x (t) = 3e−t + e4t
2
2.5 Hint: You do not need to find the eigenvectors
2
1.5 of the matrix.
1
0.5 15 a ±0.6i
0
−0.5 b i Increasing
−1
−1.5 ii
−2
−2.50 0.5 1 1.5 2 y
ii y 5
5 4
4 3
3 2
2 1
1 0x
0 −1
−1 −2
−2 −3
−3 −4
−4 − 5− 5 − 4 − 3 − 2 − 1 0 1 2 3 4 5
−5 −2 −1 0 1
1, 6; 23 , 1116 a
⎛ ⎞⎛ ⎞ y

⎜ ⎟⎜ ⎟
⎝− ⎠ ⎝ ⎠

b 10

8

6

4

2
0x

−2

−4

−6

−8

−10−10 −8 −6 −4 −2 0 2 4 6 8 10

A nswers 595

⎛⎞ t⎛ ⎞ x
⎜⎝ − ⎠⎟+
x Ae 2 Be 1 ;c ⎜ ⎟=
y 3 1⎝ ⎠
6t ⎛ ⎞ 8
⎝⎜ ⎠⎟

x = 2et + 4e6t, y = −3Aet + 4e6t

d Not suitable, as it predicts indefinite growth. 6

17 a 3.7 b (2, 2)

18 Options b and c y 4

19 a 10

2

5 0t

0x 012345

c Both oscillate with a decreasing amplitude.

−5 20 a 10, −5, ⎛ 1 ,⎞ ⎛ 2 ⎞
⎜⎝ 3 ⎝⎜ 1 ⎠⎟
⎟⎠

−10−10 −5 0 5 10 b x (t) = 0.2e10t + 14.8e−5t, y (t) = 0.6e10t + 7.4e−5t
bx
cy
50

4 40
1
30 3
2 2

0 t 20 1

− 2 10

− 4 00 10 20 30 40 50 x

012345 d 1:3

21 a 23
±5

b 200 birds, 8050 insects; increasing

596 A nswers
c
y 23 a x stays fairly constant, y increases.
2
b Option iii
1
24 a 200 sharks, 500 fish
0
b A = ⎛ −3 4⎞ c −1 ± 2i
⎜⎝ −2 1⎟⎠

dy

x 10

− 1 0.5 8

− 2− 2 − 1 0 1 2 6
d 8200
4
8100
2
8000
stcesni 00 2 4 6 8 10 x
7900
e The number of sharks oscillates around and
78000 100 200 300 tends to 200, the number of fish oscillates
birds around and tends to 500.

Both population sizes oscillate/change Exercise 13D
periodically.
22 a The number of spiders tends to 600 and the 1 a Unstable b Unstable
number of flies tends to 300.
2 a Stable b Stable
bx
10 3 a Unstable b Unstable

8 4 a Stable b Stable

6 5 a Unstable b Unstable

4 6 a 9.42 b 0.121

7 a  b 

8 a 0.346 b 0.144

9 b i 699 000 ii 39 400

c b i is more accurate because it has a smaller

step length.

10 b 

11 b 0.107

2
Example: x 0( ) = 1, y 0( ) = 5

00 5 10 15 20 25 t

A nswers 597

cx br
0.5 12
10
0.4 8
6
0.3 4
2
0.2 0t
−2
0.1 −40 1 2 3 4 5 6 7 8

00 0.5 1 1.5 2 2.5 t Includes a region with a negative population,
which is impossible.
d 0.23
14 a i
12 a 0.3 years b 1.5 years c $1.9
f
d $2 2
1.5
e For example, not taking into account random 1
variation. Price seems too stable in the long 0.5
0t
term. − 0.5
−1
13 a f − 1.5
− 20 1 2 3 4 5 6 7 8
2

1.75

1.5

1.25

1

0.75

0.5 ii 1.1 mm, 6.3 ms

0.25 b i 3.7 mm ii 0.4 mm
ii 1.61
00 1 2 3 4 5 6 7 8 t 15 a i 1.64
ii 0.753%
Oscillates with decreasing amplitude, levelling b y = 2e0.3t − e0.4t ii 0.964
off around 500 foxes. ii 6.12%
c i 2.57%

16 a i 0.875

c i 4.02%

598 Answers

Chapter 13 Mixed Practice 7y

1 a y = Ae23 sin2x b y = 5e23sin2x 2
2 y = Aex3−2x
1
3 2.45
0 x
4 a, b y

4

3.5

3 −1

2.5 −2

2 −2 −1 0 1 2

1.5

1 8 b V = (3t + c)2 c 11.7 seconds

0.5 ⎛⎞t⎛ ⎞ 5t ⎛ ⎞
00 0.5 1 1.5 2 2.5 3 3.5 4 x ⎜⎝ − ⎠⎟+ ⎜⎝ ⎟⎠
x Ae 1 Be 19 b ⎜ ⎟=
y 2 2⎝ ⎠

c 1.89 3x2 247 10 x = −13.6, y = 25.4

5 a 5.058 b y = 3 + 11 y

c 0.0256% 2 3

6 a y = (x2 + c)2 2

by 1

0x

−1

−2
4

− 3− 3 − 2 − 1 0 1 2 3

a 1.3 b0

x 12 a 2.3 b 1.7

c Tends towards x = 1, y = 1

A nswers 599

13 y b − ln(e−x + e − 1); 0.0178
1
c Decrease the step length.
0.8
N 2e18 0.2⎜⎛⎝t−1π2 1⎛ − πt⎛ ⎞⎞⎞
0.6 cos 6⎝⎜ ⎟⎠⎠⎟⎟⎠
= ⎝⎜
0.4
19 a 3.92 b y = 2e3− 3 c 1.90%
x

20 b i $1.31

ii
P

0.4

0.2

00 0.5 1 1.5 2 2.5 3 x 0.2

Hint: The phase plane is actually showing 0t

dx = 2 − x, dy = 1− xy  0.2
dt dt

a 0.7 b2  0.4

14 a 1.57 0

b Approximation is less than true value; within 5 10 15 20 25

each step the gradient increases, but Euler’s

method uses a constant gradient. iii 17 years

15 a R = R0e−kt b ln 2 m A21 a = e−kt b 0.0990 c 23 years

k = 21(x−1)2 −

c The time taken to go from 1 to 1 and from 1 y Ae 222 a b 49.7
1 etc. will be the same. 2 4 4
to 8 23 y = ln(3sin 2x + c)

16 a y 24 (9.46, 3.71)

10 25 a f(x, t) = e−t − 4x b −0.427
9 26 b 69.7° b 0.994ms−1

8 27 a dv = 0.05(10 − 0.2v2)
7 dt
6

5
4

3
2
1
00 1 2 3 4 5 6 7 8 9 10 x

b 0.6

17 a 

600 A nswers

28 a Y 32 b p
5 1
4
3 0.8

0.6

2 0.4

1 0.2

0t 00 0.5 1 1.5 2 2.5 t
012345
ci p = e2t ii ln3 ≈ 1.10 weeks
y eb = (2 − 2e−t) 9 + e2t

c ln ⎛ 2 ⎞ ≈ 0.426 years 33 a dx = y
− ln dt
⎝2 2⎠

d 738 000 dy = x
29 a 2.7701 dt
y
by b i4
2.82

2.81 2
2.80
2.79 0 x

2.78
−2

2.77

2.760 0.02 0.04 0.06 0.08 0.1 0.12 x −4−4 −2 0 2 4

d 2.814 ii Saddle
30 0.904
31 b 3.96 c i t values are not explicitly shown.

ii 6.08

iii Because the gradient is positive and
increases with x, so the estimate of
gradient at the beginning of the interval
will be below the average value for the
interval.

A nswers 601

x td i ( ) = ex − e−x ii 16.1% dI S =2I
4
iii Smaller step length or better method 3
e.g. Runge–Kutte. 2
1
Y34 a i

3

2

1 0S
01234
0 1 2 t
0 3 e i S = 10 + 4e−6t

iii 2000 Hint: You could do this by finding eigenvalues
and eigenvectors of a matrix, or you could
b y = 2 − 2e−3t
ci Y substitute I = 15 − S into the first differential

3 equation. Which is easier?

ii 5 million

f e.g. Population stays constant, no death, no
immune resistance, more infected people does
not increase the infection rate.

2 Applications and
interpretation HL:
1 Practice Paper 1

1 a 1 b 5
3 16

0 t 2 a 2+ x + 2y b x
01 3 y
2
3 a i $464 ii $632
ii Y = 2 − 2e1.5(e−2t−1)
b No, predicts unlimited growth

iii 1550 4 a   b 11

35 a dS = −0.2S + 0.4I 5 a 0 < f(t)  16 b g (t) = − 1 ln ⎛t ⎞
dt 3
c 0.231 years ⎝16⎠

dI = 0.2S − 0.4I (4 + 3i) p 2 , q 1
dt 5 5 −5
6 a b = =
b ii 
c S = 2I 7 a 1 x − 3 b 1 x− 5
3 2 2 2
−

c 2 x − 1 c
3 2
− +

602 A nswers

8 a un + 1 = 1.04un + 100 Applications and
interpretation HL:
b 32 c 25.1% Practice Paper 2

9 15.7cm3 b 2.85
−1 ± i 47
10 a 6 b 63

11 a 18°C d 30°C 1 a 320 m2 b 28.7 m
d 45.4 m
c 0.11 c 151°
b 120
12 a 18°C b (5.8, 2.2) e 5290 m2 d 0.520

13 a 5.3623 b6 c 10.6% 2 a x = 11.25, σ = 2.08

14 v = t t + ln(t + 1), a = (t 1 + t 1 c 0.64
+1 + 1)2 +1
e 0.6

15 a A = 9.4, B = 1.8, C = 5.8 3 a ⎛ 0.72 0.16 ⎞ b 0.300
⎝⎜ 0.28 0.84 ⎟⎠
b 8°C
1, 14
16 a 13 b 43.7 c λ = 25

17 a 74 , 11v1⎛ ⎞ ⎛− ⎞
⎝⎜ ⎟⎠ ⎜⎝ ⎟⎠
y = v2 =
3
⎛1 0⎞
2.5 d P ⎛4 −1 ,⎞ D 0⎜ 14 ⎟
= 7⎝⎜ 1 =
⎟⎠ ⎜
⎝ 25 ⎟
⎠

2 4 4 ⎛14⎞n
11 11⎝25⎠
1.5 e −

1 f 4
11

0.5 4 a Breakdowns occur at a constant rate and
independently of each other.

0 0 0.5 1 1.5 2 2.5 3 x b i 0.449 ii 0.953 iii 0.0357
c 0.00728
d E(X) = Var(X) = 4.8
H0: λ = =4.80.;0H551:8 > 4.8
b 1.47 e λ 0.05
18 a p-value >
y = 5 10e+ −0.03(t−1)2
Don’t reject H0.
b Decreases to 500.
f 0.0251
19 V = 28.2sin(30t − 0.748)
g 0.869

5 a 3 km per minute

4b ⎛ ⎞
10⎜ ⎟ at t = 3 for A and t = 6 for B .
10⎜ ⎟

⎜⎝ ⎟⎠

c (6 − 0.5t)2 + (3 + 0.5t)2 + (−18 + 4t)2

d No – closest distance is 6.45 km.

A nswers 603

6 a dx = y, dy = 3x − 2 y The population being studied is not all
dt dt children, but only those in a specific age
b i 1.548 range.

ii x c i Experts' opinion

6 ii See if the results correlate with a more
widely accepted test (concurrent validity)
or future important measures of speech
development (predictive validity).

4 d i 5.83, 4.29

ii χ2 = 31.3, p = 2.61 × 10−6 < 0.05
iii χ2 = 4.33, p = 0.115 > 0.05

2 iv It supports the validity, since the data needs
to be drawn from a normal distribution.
However, it also needs to have equal
variance which has not been tested.

e i It is not the same situation – there may have

00 1 2 3t been natural improvement with increasing

age.

x 1 et 3 e−3t ii orsf = 0.885 > 0.649, so there is evidence
4 4 the test is
c = + correlation, suggesting that

Unstable as λ1 > 0, λ2 < 0 reliable.

d 16.3% 2ai AB CDE F

1 1 0 1 ⎛ 1 −1⎞ A ⎛0 1 0 1 0 1⎞⎜
2 ⎛ 0 1 ⎞⎠⎟, 2 ⎜⎝ 1 1⎠⎟ B ⎜ 1 0 1 0 0 1 ⎟
7a ⎜⎝ C ⎜ 0 1 0 1 1 0 ⎟
⎟
D ⎜ 1 0 1 0 1 1 ⎟
b 1 ⎛ 1 −1⎞ c 1 E ⎜ 0 0 1 1 0 1 ⎟
2 ⎜⎝ 1 1⎠⎟ 2 F ⎜⎜⎝ 1 1 0 1 1 0 ⎟⎟⎠

d 1 e1 ii 73
4
b i No, as some vertices have odd degrees.
Applications and
interpretation HL: ii 75 minutes (repeat AB and CE )
Practice Paper 3
iii 126; all vertices have even degrees.

c i a = 12, b = 9 ii 37 (FADECBF)
iv 34  T  37
iii 32

1 a To reduce variation due to different ages. d F (p = 0.191)

b i H0: μ1 = μ2; H1: μ1 ≠ μ2 Be the Examiner answers
ii t = 0.69, p = 0.512 > 0.05. No significant
1.1 Solution 3
evidence of difference between the two 1.2 Solution 2
groups. 2.1 Solution 3
2.2 Solution 2
iii e.g. Too strong a statement – no significant 2.3 Solution 3
evidence of a difference rather than no
difference.

The result in the test is not the same as
speech ability.

604 A nswers

3.1 Solution 2
4.1 Solution 1
4.2 Solution 2
5.1 Solution 3
5.2 Solution 2
5.3 Solution 2
6.1 Solution 1
8.1 Solution 3
8.2 Solution 3
8.3 Solution 2
9.1 Solution 2
10.1 Solution 2
10.2All three solutions are correct
11.1 All three solutions are correct
13.1 Solution 1

Glossary

Acceleration The rate of change of velocity Critical region The set of all values which would lead to
rej ecting the null hypothesis
Adjacency matrix A matrix that shows the number of
edges between each pair of vertices Critical value(s) The value(s) on the boundary of the
critical region
Adjacent Two vertices are called adj acent if they are
j oined by an edge. Two edges are called adj acent if Cross product Another term for vector product
they have a common vertex.
Cycle A walk that starts and ends at the same vertex
Amplitude Half the distance between the maximum and has no other repeated vertices (so it is a closed
and minimum values of a periodic function path)

Argand diagram Another term for the complex plane Definite integral An integral with limits. This results in
a numerical answer (or an answer dependent on the
Argument (of a complex number) The angle a given limits) and no constant of integration
number in the complex plane makes with the real axis,
measured anticlockwise Degree (of vertex) The number of edges coming out
of a vertex
Base vectors The vectors i, j and k, which are of
magnitude 1 and parallel to the x, y and z axes Determinant A numerical value calculated from the
respectively. elements of the matrix, similar to the magnitude of
a vector except that a determinant can be positive or
Cartesian form (of a complex number) A way of negative
writing a complex number, z, in terms of its real and
Diagonalization The process of expressing a matrix M
imaginary parts: z = x + iy , where x, y ∈
M PDPin the form = −1, where D is a diagonal matrix
Central limit theorem (CLT) A result in statistics which
states that, for a large sample, the sample mean and P is a matrix whose columns are the eigenvectors
approximately follows a normal distribution of M.

Chain rule A rule for differentiating composite functions Digraph Another term for ‘directed graph’

Characteristic equation The equation det(M − λI) = 0, Directed graph A graph that has arrows on the edges
to indicate the allowed direction
where λ is an eigenvalue of the matrix M
Direction vector (of a line) A vector parallel to a given
Circuit A walk that starts and ends at the same vertex line
and has no repeated edges (so it is a closed trail)
Displacement Distance in a certain direction
Column vector A representation of a vector that lists its
components parallel to the coordinate axes one above Displacement vector A vector from one point to
the other in a 2 × 1 or 3 × 1 matrix another point

Complete graph A graph in which every vertex is Dot product Another term for scalar product
connected to every other vertex by exactly one edge
Edges The lines that connect vertices in a graph
z x yComplex conjugate If = + i , then the complex
conjugate of z is z* = x − iy Mv vEigenvalue A scalar λ that satisfies = λ for a
matrix M and vector v
Complex number A number that can be written in the
Mv vEigenvector A vector v that satisfies = λ for a
form x + iy , where x, y ∈ and i = −1
matrix M and scalar λ
Complex plane A Cartesian plane where the x-axis
represents the real part of a complex number and the Elements The entries in a matrix
y-axis the imaginary part Equilibrium point A point (a, b) Such that if

Components The magnitude of a vector in a given x 0( ) = a, y 0( ) = b then x t( ) = a, y t( ) = b for all times t.
direction, often parallel to the coordinate axes.
Eulerian circuit A circuit that traverses each edge
Concave-down The part(s) of a curve where the second exactly once
derivative is negative
Eulerian graph A graph that has an Eulerian circuit
Concave-up The part(s) of a curve where the second
derivative is positive Eulerian trail A walk that uses each edge exactly once
(without returning to the starting point)
Confidence interval An interval that has a certain
probability of containing the true mean Exponential (Euler) form A way of writing a complex
number, z, in terms of its modulus, r, and argument,
Connected (graph) A graph in which every two vertices
are connected (directly or indirectly) θ: z = reiθ

Focus In a phase portrait, an equilibrium point with
complex eigenvalues

606 Glossary

Fractals Shapes formed by repeated application of a Path A walk with no repeated vertices
transformation Period The smallest value of x after which a function

General solution (of differential equation) The repeats
solution containing an unknown constant Phase portrait A diagram showing how x and y values

Graph A set of vertices and edges change over time

Hamiltonian cycle A cycle visits each vertex exactly Phase shift The horizontal shift between two periodic
once graphs

Hamiltonian graph A graph that has a Hamiltonian Point of inflection A point on a curve where the
cycle concavity changes

Hamiltonian path A path that visits each vertex exactly Poisson distribution A probability distribution used to
once model independent random events which happen at a
constant average rate
Identity matrix A square matrix with 1 as each element
of the diagonal from top left to bottom right and 0 as Position vector A vector from the origin to a point
every other element
Prim’s algorithm An algorithm for finding a minimum
Imaginary part If z = x + iy , then the imaginary part of spanning tree that starts with a vertex and then adds
z is the real number y the closest possible vertex at each stage

In-degree In a digraph the number of edges directed Product rule A rule for differentiating a product of two
into a vertex functions

Inverse matrix The inverse of a square matrix M is the Quotient rule A rule for differentiating a quotient of
two functions
matrix M–1 such that MM−1 = M−1 M = I
Radians An alternative measure of angle to degrees:
Kruskal’s algorithm An algorithm for finding a
minimum spanning tree that adds edges, starting with 2π radians = 360°
the shortest, until the tree is connected
Random walk Movement around a graph where at any
Limits (of integration) The lower and upper values vertex the decision of which edge to travel on next is
used for a definite integral made at random

Linearizing The process of writing the equation of a Real part If z = x + iy , then the real part of z is the real
number x
graph of the form y = axn Or y = kax In the form of a Reflection in the y-axis (or x-axis) Multiplication of

straight line graph by taking logs the x-values (or y-values) of all points on a curve by −1
Log–log graph A graph of log y against log x
Reliable The conclusions of a test are reliable if similar
Markov chain A system consisting of two or more conclusions would be reached on each occasion the
states in which the probability of being in any given test is conducted in similar circumstances
state depends only on the previous state
Resultant vector The vector that results from the sum
Matrix A rectangular array of elements, which may be of two or more vectors
numerical or algebraic
Saddle point In a phase portrait, an equilibrium point
Minimum spanning tree A tree of minimum total length with real eigenvalues of opposite signs
(weight) which includes every vertex of the graph
Scalar A quantity that has only magnitude but no
Modulus (of complex number) The distance of a direction
number from the origin in the complex plane
Scalar product A scalar value given by a b cosθ,
Modulus–argument (polar) form A way of writing
a complex number, z, in terms of its modulus, r, and where θis the angle between a and b

argument, θ: z = r(cosθ + isinθ) Semi-Eulerian A graph that has an Eulerian trail
Semi-log graph A graph of log y against x
Node In a phase portrait, an equilibrium point with
real eigenvalues of the same sign (can be stable or Simple graph A graph that has no multiple edges and
unstable) no vertex j oined to itself.

Order (of a matrix) A matrix with m rows and n Singular A matrix with zero determinant
Slope field A plot of the tangents at all points (x, y)
columns has order m × n
Solid of revolution A 3D shape formed by rotating part
Out-degree In a digraph the number of edges directed
out of a vertex of a curve 360° around the x-axis (or y-axis)

Parametric form (of equation of line) A form of the Square matrix A matrix with the same number of rows
equation where x, y and z are expressed in terms of a
parameter and columns (of order n × n)

Glossary 607

Stable equilibrium point A point towards which Unbiased estimator In statistics, an estimator
solution curves move with time calculated from a sample, whose expected value
equals the population parameter
Steady state The state s of a Markov chain such that
Unit square The square with vertices (0, 0), (1, 0), (1, 1)
Ts = s for a transition matrix T and (0, 1)

Stretch Multiplication of the x-values (or y-values) of all Unit vector A vector of magnitude 1
points on a curve by a given scale factor
Unstable equilibrium point An equilibrium point that
Strongly connected A digraph in which any two is not stable
vertices are connected in both directions
Upper and lower bounds (travelling salesman
Subgraph A new graph formed by using only some of problem) The largest and smallest possible values for
the edges of the original graph the length of the shortest Hamiltonian cycle

Subtends An arc subtends the angle at the centre of a Valid A process is valid if it is measuring what it is
circle formed between the two radii extending from intended to measure
each end of the arc to the centre
Vector A quantity that has both magnitude and
Sum of square residuals The sum of the squared direction
differences between each data value and the
corresponding prediction a regression model makes Vector product A vector perpendicular to the two
for that data value
given vectors with magnitude a b sinθ, where θis
Sum to infinity The sum of infinitely many terms of a
geometric sequence the angle between a and b

Survey Any method for collecting data for analysis Velocity Speed in a certain direction

The Chinese postman problem To find the shortest Velocity vector A vector that gives the direction of
path around the graph which uses each edge at least motion of an obj ect and whose magnitude gives the
once and returns to the starting point speed with which the obj ect is moving

The travelling salesman problem To find the shortest Vertices The points of a graph
route around a graph which visits each vertex at least
once and returns to the starting point Volume of revolution The volume of a solid of
revolution
Trail A walk with no repeated edges
Walk Any sequence of adj acent edges in a graph
Transition matrix (graph) A matrix of the probabilities
of moving from one vertex to another in a graph Weight The number associated with each edge in a
weighted graph
Transition matrix (Markov chain) A matrix representing
the probabilities of changing from one state to another, Weighted adjacency table A table that shows the
or staying in the same state, in a Markov chain weights of edges between each pair of vertices

Translation The addition of a constant to the x-values Weighted graph A graph in which each edge has a
(or y-values) of all points on a curve number associated with it

Tree A connected graph in which there are no closed Zero matrix A matrix whose elements are all 0
paths (cycles)
z-interval A confidence interval for the mean when you
Type I error Rej ecting the null hypothesis when it was
true do know the population variance
t-interval A confidence interval for the mean when you
Type II error Failing to rej ect the null hypothesis when
it was false do not know the population variance

Index

acceleration 40, 446–7 connected graph 221–2 equilibrium point 482
acute angles 129 coordinate systems 164 equivalences 8, 120
adjacency matrix 224–7, 233–5, 240–2, cosine rule 59, 120, 125–30, 205 Euler, L. 235, 446, 475
data Eulerian graphs 235–7, 260
254 Euler’s method 473–5, 485–6, 492–3
algorithms collection 321–2 exponential (Euler) form 202–4
linearizing 16–17 exponential equations 3, 9–10
Dj ikstra’s algorithm 256 representing 15 exponential models 176–8
greedy 256 decimalization 116 exponents
Kruskal’s algorithm 250–2, 256 definite integral 418–22
minimum spanning tree 250–6 degrees of freedom 327–9 exponential (Euler) form 202–4
PageRank 244–5 De Moivre’s theorem 204 laws of 4–5, 17, 384
Prim’s algorithm 252–6 derivatives 385, 397–402, 446–7 rational 4–5
angles 31, 114–15 Descartes, R. 164 representation 4
acute 129 differential equations 467–94
approximations 126 coupled systems 475–81 see also logarithms
radians 116–27, 178 Euler’s method 473–5, 485–6,
in triangles 128 fractals 139
Argand diagram 195, 199, 209–10 492–3 functions 157
base vectors 33 phase portraits 481–7
binomial distribution 295–6, 356–8 second order 491–2 chain rule 389–93, 395–6
Brin, S. 244 separation of variables 468–71 composite 158–9, 170, 389–90
calculus 382, 418, 466 slope elds 472–3 differentiation 384–402
differentiation 384–402 exponential 176–8, 385
see also differentiation; integration chain rule 389–93, 395–6 integration 412–37
composite functions 389–90 inverse 161–2
Cardano, G. 197 concave-down 398 logistic models 179–80
Cartesian coordinates 164 concave-up 398 one-to-one 162
Cartesian form 192–9, 204 derivatives 385, 397–402 piecewise models 180
Cauchy distribution 292 logarithmic functions 385–7 product rule 391–3
chain rule 389–93, 395–6, 447 product rule 391–3 quotient rule 392–4
Chinese postman problem 260–1 quotient rule 392–4 reections 169
chi-squared test 326 rates of change 395–6 sinusoidal models 178–9, 205
circle second derivative 397–401 stretches 167–8
trigonometric functions 385 transformations of graphs 164–73
area 119, 396 translations 165–6
length of an arc 118 see also integration fundamental theorem of calculus 418
radius 118–19, 396 Gabriel’s horn 430
unit 116–17, 125–8 directed graph 221 generalization 2
direction geometric designs 114–15
see also radians geometric representation 209–12
in graphs 221 geometric sequences 12–13
circular motion 456–7 vectors 32, 47, 61–5, 68–9, 141–2 Google, PageRank algorithm 244–5
coding 85 gradian 116
coefficient of determination 336 see also kinematics graphs
column vectors 32, 85 adj acency matrix 224–7, 233–5,
complete graph 221 displacement 444–9, 452–3, 455–7
complex conjugate 193–4 displacement vector 40, 452, 455 240–2
complex numbers 190–1 Djikstra’s algorithm 256 complete 221
Eadie–Hofstee plot 25 concavity 398–402
Cartesian form 192–9, 204 eigenvalues 104–8, 141–2, 479–81 connected 221–2
complex plane 194–5 eigenvectors 104–8, 141–2 coordinate systems 164
exponential (Euler) form 202–4 encryption 85 differentiation 398–402
geometric representation 209–12 enzymatic reaction 25 directed 221
modulus–argument form 199–202 equations Eulerian 235–7, 260
multiplication 209–10 Hamilton cycles 237–8, 265–8
quadratic equations 195–7 differential 467–94 integration 413–14
sinusoidal functions 205 equation of a line 45–9 linearizing data 16–17
composite functions 158–9 matrices 101–2 logarithmic scales 15–19
concave-down 398 quadratic 195–7
concave-up 398 simultaneous 101
confidence intervals 340–4 trigonometric 131

Index 609

moving around 232–3 linearizing data 16–17 period 178
PageRank algorithm 244–5 log equations 8–9 phase portraits 481–7
planar 222 natural logarithmic models 177–8 piecewise models 180
reections 169 planar graphs 222
simple 221 see also exponents point of inflection 401–2
stretches 167–8 Poisson distribution 295–8, 356, 358–9
subgraphs 222 logistic models 179–80 populations 320
transformations 164–73 log–log graph 17–19 position vector 40, 209
transition matrix 242–5 lower bound 267–8 powers 6, 107
translations 165–6 magnitude, vectors 32, 39–41 Prim’s algorithm 252–6
trees 222–3 Mann-Whitney test 323 probability
types of 221 Markov chains 301–5
upper and lower bounds 265–7 matrices binomial distribution 295–6
vertices 220 central limit theory 291–2
weighted 240–1 addition 86–7
graph theory 219–20 algebra 86–7 distribution of X 290
algorithms 244–56 characteristic equations 104
Chinese postman problem 260–1 denition 86 long term 304
travelling salesman problem determinant 95–7 Markov chains 301–5
diagonalization 105–8 normal distribution 291–2
265–70 eigenvalues 104–8, 141–2, 479–81 Poisson distribution 295–8
eigenvectors 104–8, 141–2 steady state 304–5
see also graphs identity matrix 90–1 transforming variables 284–7
inverse 95–8, 101 transition matrix 301–2
greedy algorithms 256 multiplication 87–90, 95–6 vectors 302–3
half-life 176–7 notation 95 product rule 391–3
Hamilton graphs 237–8, 265–8 order of 86–7 proj ectile motion 454–7
harmonic series 12 powers 107 Pythagoras’ theorem 39, 126
i 192–3 square matrix 86, 90, 95 Pythagorean identity 130
imaginary numbers 192–3 subtraction 86–7 quadratic equations 195–7
integration 411–37 systems of equations 101–2 quantum mechanics 85
as transformations 134–42 questionnaires 321–3
denite integral 418–22 zero matrix 90–1 quotient rule 392–4
by inspection 414–15 mean 286–7, 296 radians 116–27, 178, 385
in kinematics 448 Michaelis–Menten kinetics 25 random variables 283–7
by substitution 437 minimum spanning tree algorithms rates of change 395–6
trigonometric functions 412–13 250–6 rational exponents 4–5
volume of revolution 428–30 modelling reflections 169
inverse functions 161–2 exponential models 176–8 regression models 334–5
j erk 446 logistic models 179–80 reliability 287, 322–3
jounce 446 piecewise models 180 research questions 320
kinematics 52–5, 444–57 sinusoidal models 178–9 residuals 335–6
circular motion 456–7 modulus–argument form 199–202 resultant vector 34
derivatives 446–7 Runge-Kutta methods 475
displacement 446–9 motion see kinematics; vectors scalars 32
modelling time-shift 456–7 multiplication 37
proj ectile motion 454–7 natural logarithmic models 177–8 scalar product 58–9, 63
Koch Snowflake 139 Newton, I. 418, 446 scalar quantities 31
Königsberg bridges problem 235–6 Noether, E. 446
Kruskal’s algorithm 250–2, 256 non-linear regression 334–6 see also vectors
Lagrange, J. 446 non-parametric tests 323
laws of exponents 4–5, 17, 384 normal distribution 291–2 second derivative 397–401
laws of logarithms 7–10 number line 191 second order differential equations
Leibniz, G. 418, 446 Page, L. 244
linear transformations 284–7 PageRank algorithm 244–5 491–2
Lineweaver–Burk plot 25 parallel form measures 322 semi-log graph 17–19
logarithmic scales 15–19 parallelogram 35, 38, 67 sequences
logarithms parallel vectors 35, 38, 61
differentiation 385–7 parametric form 47–8 geometric 12–13
exponential equations 9–10 patterns 7, 114–15, 139, 195 harmonic series 12
laws of 7–10 Pearson’s product-moment shapes, properties of 30
Sierpinski Triangle 139
correlation 336 simple graph 221
simple harmonic motion (SHM) 494

610 Index

simultaneous equations 101 stretches 167–8 variables
sine rule subgraphs 222 discrete vs continuous 356
sum of square residuals 335–6 random 283–7
triangles 128–9 sum to infinity 12 selecting 321
unit circle 125–8 tangent rule 129–30 separation of 468–71
sinusoidal models 178–9, 205 test-retest 322
slope fields 472–3 time 446 see also probability; variance
solid of revolution 428
space 446 see also kinematics variance 284–7, 296
speed 449, 453 t-interval 342 vectors 31–2
square bracket notation 418
square matrix 86, 90, 95 Torricelli’s trumpet 430 addition 34–5
statistics transformations 134–42 angles between 59–64
binomial distribution 295–6, 356–8 base vectors 33
chi-squared test 326 of graphs 164–73 column vectors 32, 85
coefcient of determination 336 transforming variables 284–7 components of 68–9
condence intervals 340–4 transition matrix 242–5, 301–2 direction vector 32, 47
correlation 359–60 translations 165–6 displacement 40, 53
data collection 321–2 travelling salesman problem eigenvectors 104–8, 141–2
degrees of freedom 327–9 equation of a line 45–9
distribution 326–9, 342–4, 356–9 265–70 equations 31
errors 364–7 triangles 128 kinematics 52–5
investigation cycle 320–1 trigonometry 114 magnitude 32, 39–41
non-linear regression 334–6 parallel 35, 38, 61
paired samples 350–1 area of a sector 119–20 parametric form 47–8
Poisson distribution 295–8, 356, cosine rule 125–30 perpendicular 61
length of an arc 116–18 points of intersection 48–9
358–9 matrices as transformations position 40, 209
populations 320 representing 32–3
population variance 324–5 134–42 scalar product 58–9, 63
reliability 322–3 modulus–argument form subtraction 35
selecting variables 321 unit vectors 39
signicance 327, 364–5 199–202 vector product 64–8
statistical design 320–5 radians 116–27 vector quantities 31
sum of square residuals 335–6 sine rule 125–30 velocity vectors 52–5
testing 322 tangent rule 129–30
tests for the mean 347–51 trigonometric functions 114–15, see also kinematics; scalars

t-test 347–8 131, 385, 412–13 velocity 31, 52–5, 446, 449, 452–6
vertices 220–1
unbiased estimators 324–5, 328–9 t-test 323, 347–8 volume of revolution 428–30
unpaired samples 350 wave equation 178
validity 322–3 Type I errors 364–5 wavelength 178
variables 356 Type II errors 366–7 weighted graphs 240–1
unbiased estimators 324–5, 328–9
z-test 347–50 unit circle 116, 125 z-interval 343
unit square 86, 135 z-test 347–50
unit vectors 39
upper bounds 265–6
validity 322–3