12 October, 2014 St Mungo's Academy 1 ADVANCED HIGHER MATHS REVISION AND FORMULAE UNIT 1

11 April 2023 St Mungo's Academy 1

ADVANCED HIGHER MATHS

REVISION AND FORMULAE

UNIT 1

11 April 2023 St Joseph’s College 2

Unit 1 Outcome 1 BINOMIAL and PARTIAL FRACTIONS

Expand (a − b)5 a5 − 5a4b + 10a3b² − 10a²b3 + 5ab4 − b5.

Expand (2x − 1)3 8x3 − 12x² + 6x − 1

rr

r r

4(2 5 )x yExpand 4 3 2 2

3 4

16 160 600

1000 625

x x y x y

xy y

5(3 2 )a bExpand 5 4 3 2

2 3 4 5

243 810 1080

720 240 32

a a b a b

a b ab b



61( )x

x

6 6 6 6 6 6 6

0 1 2 3 4 5 6

2 36 5 4 3

4 5 62

1 1 16 15 20

1 1 115 6

x x x xx x x

x xx x x

6 4 22 4 6

15 6 16 15 20x x x

x x x



12x

45x

3 2

2

4 9 2 62

x x xx x

4 1x 2

4 62

xx x

3 74 1

2x

x x


f(x) functionc

x

2x

xn

(ax + b)n

sin x

cos x

sin(ax + b)

cos(ax + b)

0

f’(x) derivative

1

2

nxn−1

an(ax+b)n−1

cos x

−sin x

acos(ax + b)

-asin(ax + b)

These are your standard derivatives

for now!

All of these were covered

in the Higher course

Unit 1 Outcome 2 DIFFERENTIATION


New Trigonometric Functions

AND




Product Rule

Quotient Rule


1 - Derivative of sin x.The derivative of f(x) = sin x is given by

f '(x) = cos x 2 - Derivative of cos x.

The derivative of f(x) = cos x is given by

f '(x) = - sin x 3 - Derivative of tan x.

The derivative of f(x) = tan x is given by

f '(x) = sec 2 x

The six basic trigonometric derivatives



4 - Derivative of cot x.The derivative of f(x) = cot x is given by

f '(x) = - cosec 2 x

5 - Derivative of sec x.The derivative of f(x) = sec x is given by

f '(x) = sec x tan x6 - Derivative of cosec x.

The derivative of f(x) = cosec x is given by

f '(x) = - cosec x cot x

The six basic trigonometric derivatives




Higher derivatives



Motion

v = dx/dt a = dv/dt = d2x/dt2

These are used in the example over the page


Find velocity and acceleration after (a) t secs and (b) 4 secs forparticles travelling along a straight

line if:

(i) x = 2t3 – t2 +2

(ii) x = t2 +8/t

(iii) x = 8t + et

88 m/s

6t2 – 2t 12t – 2

46 m/s2

2t – 8/t2

7.5 m/s

2+ 16/t3

2.25 m/s2

e4 m/s2

et 8 + et

8+ e4 m/s

63 m/s 55 m/s2



Unit 1 Outcome 3 INTEGRATION



+




Example of rotating the region about x-axis

In the picture shown, a solid is formed by revolving the curve y = x about the x-axis, between x = 0

and x = 3. FIND THE VOLUME

Example 1



Find the volume of the solid obtained by the region bounded by y=x3, y=8, and x=0 around the y-axis.

o

8

x

y

3 yx

Example of rotating the region about y-axisExample 2

28

3

0

V y dy

85

3

0

3

5

yV

5

33(8)0

5V

96

5V



Sketch the graph of 2

32

xx

xy

.[You need not find the coordinates of any stationary points.]

0x2

3

2

3

y

2

3,0

Solutiony-axis: When

The curve cuts the y-axis at

02

32

xx

x03 x 3x

0yx-axis: When

,

The curve cuts the x-axis at (3, 0).

Unit 1 Outcome 4 PROPERTIES OF FUNCTIONS


This means that is a non-vertical asymptote.

022 xx0)1)(2( xxVertical Asymptotes:

2x 1x or

x –21 –2 –19 09 1 11

y

2

32

xx

xy

Non-Vertical Asymptote:

x 0yAs

,

(since the degree of the denominator is higher than the degree of the numerator).

0y










Non-Vertical Asymptote











f(-x)





Past Paper 2002 Q1 5 marks

Use Gaussian elimination to solve the following system of equations

x + y + 3z = 22x + y + z = 23x + 2y + 5z = 5

1 1 3 2

2 1 1 2

3 2 5 5

R2 –2R1

R3 -3R1

1 1 3 2

0 1 5 2

0 1 4 1

R3 –R2

1 1 3 2

0 1 5 2

0 0 1 1

z = 1

-y –5z = -2

-y = -2 +5

y = -3

x - 3 + 3 = 2

x = 2

(x,y,z) = (2, -3, 1)

Unit 1 Outcome 5 GAUSSIAN ELIMINATION


Past Paper 2003 Q6 6 marksUse elementary row operations to reduce the following system of Equations to upper triangle form

x + y + 3z = 13x + ay + z = 1x + y + z = -1

1 1 3 1

3 1 1

1 1 1 1

a

R2 –3R1

R3 -R1

1 1 3 1

0 3 8 2

0 0 2 2

a

R3/-2

z = 1

(a-3)y –8 = -2

(a-3)y = 6

y = 6/(a-3)

x + 6/( a –3) + 3 = 1

x = -2 –6/(a-3)

a=3 gives z = ¼ from R2 and z = 1 from R3. Inconsistent equations!

1 1 3 1

0 3 8 2

0 0 1 1

a

Hence express x, y and z in terms of parameter a.Explain what happens when a = 3




x + y + 2z = 12x + y + z = 03x + 3y + 9z = 5

1 1 2 1

2 1 0

3 3 9 5

R2 –2R1

R3 -3R1

1 1 2 1

0 2 3 2

0 0 3 2

R3/3

z = 2/3

(-2)y –2 = -2

y = 0

x = 1 –4/3

x = -1/3

=2 gives R2 = R3. Infinite number of solutions!

Explain what happens when = 2

Use Gaussian elimination to solve the system of equations below when . 2




2x - y + 2z = 1x + y - 2z = 2x - 2y + 4z = -1

1 1 2 2

1 2 4 1

2 1 2 1

R2 –R1

R3 -2R1

1 1 2 2

0 3 6 3

0 3 6 3

R3-R2

z = t

-3y +6t = -3

y = 2t+1

x +2t+1 –2t = 2

x = 1

x,y,z:x=1, y=2t+1 and z = t)

Use Gaussian elimination to obtain solutions of the equations

1 1 2 2

0 3 6 3

0 0 0 0


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12 October, 2014 St Mungo's Academy 1 ADVANCED HIGHER MATHS REVISION AND FORMULAE UNIT 1