department of mathematics - caedmon … resources and links to video tutorials; exam papers and...
TRANSCRIPT
DEPARTMENT OF MATHEMATICS
A2 level Mathematics
Core 3 course workbook
2015-2016
Name: _____________________________________
Welcome to Core 3 (C3) Mathematics. We hope that you will use this workbook to give you an organised set of notes for revision and that it will help you enjoy and benefit from the course more. ORGANISATION Keep this workbook, worksheets, homework and assessments in a file + BOOK SCRUTINY You will be asked to hand this in/bring it along to a progress interview with
your teacher or Mr Gower during the year REVISION BOOKLET We will produce revision a revision programme once you have completed TRIAL EXAM all or most of the Core 3 course. As last year there will be an expectation that
you do one chapter a week/fortnight to prepare for trial and actual exams. There will be one opportunity to resit the trial exam
You may also find the following useful: Free Electronic copy of C3 textbook: (linked from the department website)
Dropbox.com For resources and links to video tutorials; exam papers and solutions, formula booklet: Use the Mathematics department website:
ccwmaths.wordpress.com
And CCW Maths on twitter: @ccwmaths
For exam practice questions with solutions or exam past papers use (linked from the department website)
Examsolutions.net Physicsandmathstutor.com
Mathspapers.co.uk
Mr J Gower Head of KS5 Mathematics
FORMULA SHEET
Exponential and natural logarithm functions:
In general: ex = y and so: x = ln y
The usual laws of logarithms apply, so:
ln a + ln b = ln ab ln a – ln b = ln
b
a ln ak = k ln a
Trigonometric Identities
You need to know the trig identities from C2:
tan
cos
sin and 1cossin 22
You can easily use the second of these to derive these C3 identities:
xx 22 tan1sec and cosec 2 x x2cot1
There are a few other identities which you need to be able to use (but the next few are given in the
formula book):
sin(A ± B) = sin A cos B ± cos A sin B
cos(A ± B) = cos A cos B ∓ sin A sin B
BA
BABA
tantan1
tantan)tan(
By making A = B in the above identities, you can easily derive these double angle formulae (these
are not given in the formula book):
sin 2A = 2sin A cos A
cos 2A = cos2A – sin2A = 2 cos2 𝑥 − 1 = 1 − 2 sin2 𝑥
A
AA
2tan1
tan22tan
Reciprocal Trig functions
𝑐𝑜𝑠𝑒𝑐 𝑥 = 1
sin 𝑥, sec 𝑥 =
1
cos 𝑥, cot 𝑥 =
1
tan 𝑥,
Inverse Trig functions
𝑎𝑟𝑐𝑠𝑖𝑛 𝑥 = sin−1 𝑥 , 𝑎𝑟𝑐𝑐𝑜𝑠 𝑥 = cos−1 𝑥 , 𝑎𝑟𝑐𝑡𝑎𝑛 𝑥 = tan−1 𝑥,
Differentiation:
The Chain Rule (for differentiating a function of a function)
Very simply: dx
du
du
dy
dx
dy (where y is a function of u, and u is a function of x)
The Product Rule (for differentiating products of functions)
If u and v are functions of x, and y = uv, then:
dx
duv
dx
dvu
dx
dy (this is not given in the formula book, so learn it!)
The Quotient Rule (for differentiating quotients of functions)
If u and v are functions of x, and y =v
u, then:
2v
dx
dvu
dx
duv
dx
dy
(this is in the formula book, but in a different form)
Trig differentiation
𝑓(𝑥) 𝑓′(𝑥)
sin 𝑘𝑥 k cos 𝑘𝑥
cos 𝑘𝑥 − k sin 𝑘𝑥
tan 𝑘𝑥 k sec2 𝑘𝑥
cosec 𝑥 − cosec 𝑥 cot 𝑥
sec 𝑥 sec 𝑥 tan 𝑥
cot 𝑥 −cosec2 𝑥
Contents
1. Algebra and Functions
2. Exponential and logarithm functions
3. Trigonometry
4. Differentiation
5. Numerical methods
6. Revision
C3WB: Algebra and Functions
page
Algebra Notes • BAT find equivalent expressions by dividing by the denominator
• BAT add, subtract, multiply and divide rational expressions
• BAT simplify rational expressions
C3WB: Algebra and Functions
page
WB1 Express 𝑥
(𝑥+3)−
6
(𝑥−1) as a single fraction in its simplest form.
WB2
Common
factor in the
denominator
Express ...
2
3
4
32
xx as a single fraction in its simplest form
C3WB: Algebra and Functions
page
WB3
Exam Q Express
4𝑥−4
2(𝑥−4)−
36
2(𝑥−4)(2𝑥−5) as a single fraction in its simplest form
Given that
𝑓(𝑥) = 4𝑥 − 4
2(𝑥 − 4)−
36
2(𝑥 − 4)(2𝑥 − 5)− 2
b) Show that
𝑓(𝑥) = 12
(2𝑥 − 5)
C3WB: Algebra and Functions
page
WB4
Simplify: 12𝑥−96
𝑥(𝑥−8)(7𝑥+7)÷
3
(7𝑥+7)2
C3WB: Algebra and Functions
page
Top heavy fractions Notes
BAT Manipulate algebra fluently and efficiently
BAT Simplify top heavy algebraic fractions
C3WB: Algebra and Functions
page
WB5
Top heavy
fraction
Simplify 1
225 2
x
xx
WB6
Top heavy
fraction (evil)
Show that 4127
3251422
23
x
CBAx
xx
xxx
C3WB: Algebra and Functions
page
Functions Introduction notes
BAT understand definition of a function as a one to one mapping
BAT recognise odd and even functions
BAT state the domain and range of functions
C3WB: Algebra and Functions
page
WB7 Sketch the graph of y = f(x) where f(x) = x2 + 1, x ≥ -2
What is the domain of f ?
What is the range of f ?
C3WB: Algebra and Functions
page
WB8 The function h(x) is defined by ℎ(𝑥) = 1
𝑥 + 2, 𝑥𝜖ℝ 𝑥 ≠ 0
Solve these equations
h(x) = 3 h(x) = 4 h(x) = 1
Explain why the equation h(x) = 2 has no solution
Sketch a graph of h(x)
C3WB: Algebra and Functions
page
WB9 Draw a sketch of the function defined by:
𝑓: 𝑥 ⟼ (2𝑥 + 1, − 3 < 𝑥 < 4
13 − 𝑥, 4 ≤ 𝑥 < 10
)
and state the range of f(x)
C3WB: Algebra and Functions
page
WB10 Draw a sketch of the function defined by g(t) = 3t + 2,
and state it’s domain and range
C3WB: Algebra and Functions
page
WB11 Draw a sketch of the function defined by 𝑓(𝑥) = 6
𝑥+1, x 1
State it’s domain and range
Is f(x) an odd function? Give a reason for your answer
C3WB: Algebra and Functions
page
Inverse Functions notes
BAT Find the inverse of functions
BAT Find the domain and range of inverse functions
BAT understand the inverse of a function graphically
C3WB: Algebra and Functions
page
WB12 Find the Inverse function: x
xxf ,5
3)(
2
WB13
Find the Inverse function: 𝑓(𝑥) = 6
3𝑥−2, 𝑥 ≠
2
3
C3WB: Algebra and Functions
page
WB14 Find the Inverse function: 𝑓(𝑥) =
3𝑥+2
2𝑥−5
WB15
Find the Inverse function:
xx
xf ,53
12sin)(
C3WB: Algebra and Functions
page
WB16
Inverse function graphically
Sketch on the same axes f(x) and its inverse
Sketch on the same axes g(x) and its inverse
C3WB: Algebra and Functions
page
Composite Functions notes
BAT Find composite functions
BAT Find the domain and range of composite functions
C3WB: Algebra and Functions
page
WB 17 A gas meter indicates the amount of gas in cubic feet used by a consumer. The
number of therms of heat from x cubic feet of gas is given by the function f
where f(x) = 1.034x, x > 0
A particular gas company’s charge in £ for t therms is given by the function g
where g(t) = 15 + 0.4t
(i) Find the cost of using 100 cubic feet
(ii) Find a single rule for working out the cost given the number of cubic
feet of gas used
WB 18
The functions f and g are defined by: 12:
,: 2
xxg
xxxf
Find i) fg(x) ii) fg(0) iii) gf(x) iv) gf(1) v) ff(x) vi) ff(-2)
vii) gg(x) viii) gg(-7)
C3WB: Algebra and Functions
page
WB 19 The function f has domain [−∞, ∞] and is defined by 𝑓(𝑥) = 4𝑒𝑥
The function g has domain [1, ∞] and is defined by g(𝑥) = 3 ln 𝑥
a) Explain why 𝑔𝑓(−3) does not exist
b) Find in its simplest form an expression for 𝑓𝑔(𝑥) stating its domain and
range
C3WB: Algebra and Functions
page
Modulus Functions and transformations notes
BAT Understand and draw graphs of modulus functions
BAT Find intersections of graphs, including modulus graphs
BAT Solve inequalities – three methods
BAT apply transformations to graphs
C3WB: Algebra and Functions
page
Review. Transformations of graphs
Learn these and practice
them
f(x) = x3
sketch f(3x)
sketch f(x+2)
sketch 4 f(x)
f(x) = 2x2 + 3
sketch f(3x)
sketch f(x+2)
sketch 4 f(x)
f(x) = 2x3 + 4x
sketch f(2x)
sketch f(x – 1)
sketch 3f(x)
f(x) = Sin x
sketch f(2x)
sketch f(x + 90)
sketch 3 f(x)
f(x) = Cos x
sketch – f(x)
sketch f(x – 30)
sketch 2 f(x)
f(x) = Tan x
sketch f(2x)
sketch f(x + 180)
sketch f (– x)
(i) Shifts
(ii) Stretches
(iii) Reflections
+A
(x, y) (x + A, y)
+A
(x, y)
(x , y + A)
×
(x, y)
( x , y)
×A
(x, y)
(x, Ay)
(x, – y)
(x, y)
(x, y) (-x, y)
Axf )( is a shift
in the y direction
)( Axf is a shift
in the x direction
)(Axf is a stretch by scale
factor A
1 in the x direction
)(xAf is a stretch by scale
factor A in the y direction
)( xf is a reflection of
the graph in the y axis
)(xf is a reflection of
the graph in the x axis
C3WB: Algebra and Functions
page
WB 20 The functions f and g are defined by: 𝑓(𝑥) = 𝑥 + 3
𝑔(𝑥) = |4𝑥 + 6| Solve the inequality 𝑓𝑔(𝑥) > 12
C3WB: Algebra and Functions
page
WB21
Solve the Inequality |𝑥 − 12| > |3𝑥|
C3WB: Algebra and Functions
page
WB22 Given that 𝑓(𝑥) = |𝑥| and g(𝑥) = 𝑥 + 3
Sketch the graphs of the composite functions 𝑓𝑔(𝑥) 𝑎𝑛𝑑 𝑔𝑓(𝑥)
Indicating clearly which is which
C3WB: Algebra and Functions
page
WB23
Solve the inequality |3𝑥 − 1| < |6𝑥 − 1|
C3WB: Algebra and Functions
page
WB24
a) Solve the equation |9𝑥2 − 61| = 60
b) Hence, or otherwise, solve the inequality |9𝑥2 − 61| ≥ 60
C3WB: Algebra and Functions
page
WB25 𝑓(𝑥) = 𝑥4 − 4𝑥 − 240
a) Show that there is a root of f(x) = 0 in the interval [-4, -3]
b) Find the coordinates of the turning point on graph of y = f(x)
Given that 𝑓(𝑥) = (𝑥 − 4)(𝑥3 + 𝑎𝑥2 + 𝑏𝑥 + 𝑐)
find the values of a,b and c
d) Sketch the graph of y = f(x)
e) Hence sketch the graph of y = f(x)
C3 WB: Exponential & natural Logarithm
page
Exponential and Natural logarithm Graphs - Notes BAT Understand and sketch transformations of exponential and natural
logarithm graphs
C3 WB: Exponential & natural Logarithm
page
WB1
Sketch the graph of: y = 10e-x
WB2
Sketch the graph of: y = 3 + 4e0.5x
C3 WB: Exponential & natural Logarithm
page
WB3
Sketch the graph of: y = 3 + ln(2x)
WB4
Sketch the graph of: y = ln(3 - x)
C3 WB: Exponential & natural Logarithm
page
Exp and ln - Solving equations - Notes BAT solve equations and manipulate algebra using rules of indices and
logarithms
C3 WB: Exponential & natural Logarithm
page
WB5 Solve
i) 𝑒𝑥 = 3
ii) 𝑒𝑥+2 = 7
iii) ln 𝑥 = 4
iv) ln(3𝑥 − 2) = 3
WB6 Make x the subject of:
i) 𝑙𝑛𝑦 − 𝑙𝑛𝑥 = 4𝑡
ii) 𝑦 = 5𝑒2𝑥
C3 WB: Exponential & natural Logarithm
page
WB7 Solve
i) 𝑙𝑛𝑥 −16
𝑙𝑛𝑥= 6
ii) 𝑒2𝑥 − 8𝑒𝑥 + 12 = 0
WB8
Exam Q
Find the exact solutions to these equations
a) ln(3𝑥 − 8) = 2
b) 3𝑥𝑒8𝑥+3 = 18
C3 WB: Exponential & natural Logarithm
page
Growth Functions - Notes BAT Solve real life problems involving growth functions of the
form y = Aebx+c
C3 WB: Exponential & natural Logarithm
page
WB9 The Price of a used car is given by the formula: 𝑃 = 1600𝑒−
𝑡
10
a) Calculate the value of the car when it is new
b) Calculate the value after 5 years
c) What is the implied value of the car in the long run (ie – what value does it
tend towards?)
d) Sketch the Graph of P against t
C3 WB: Exponential & natural Logarithm
page
WB10 The number of elephants in a herd can be represented by the equation: 𝑁 = 150 − 80𝑒−
1
40
Where n is the number of elephants and t is the time in years after 2003.
a) Calculate the number of elephants in the herd in 2003
b) Calculate the number of elephants in the herd in 2007
c) Calculate the year when the population will first exceed 100 elephants
d) What is the implied maximum number in the herd?
C3WB: Trigonometry
page
Trig Reciprocal Functions - Notes BAT understand sec, cosec and cot and their graphs
BAT solve equations involving sec, cosec and cot
C3WB: Trigonometry
page
WB1
a) Will cosec 200 be positive or negative?
b) Find the value of sec 280 to 2 decimal places
WB2 Solve these equations: ( 0 < 2π )
a) 𝒔𝒆𝒄 𝜽 = 𝟏𝟎
b) 𝟓 𝒄𝒐𝒔𝒆𝒄 𝜽 = 𝟐𝟓
c) 𝒄𝒐𝒕 𝟑𝜽 + 𝟕 = 𝟔. 𝟓
C3WB: Trigonometry
page
Graphs of Reciprocal functions - sec x, cosec x, cot x
C3WB: Trigonometry
page
WB3 Sketch, in the interval 0 ≤ θ ≤ 360,
the graph of 𝑦 = 1 + sec 2𝜃
C3WB: Trigonometry
page
Trig Identities and Equations - Notes BAT prove identities using reciprocal functions
BAT know and use new trig identities
C3WB: Trigonometry
page
WB4
Simplify 𝑠𝑖𝑛𝜃 𝑐𝑜𝑡𝜃 𝑠𝑒𝑐 𝜃
WB5
Simplify sin 𝜃 cos 𝜃 (sec 𝜃 + 𝑐𝑜𝑠𝑒𝑐 𝜃)
C3WB: Trigonometry
page
WB6
Show that cot𝜃 𝑐𝑜𝑠𝑒𝑐 𝜃
𝑠𝑒𝑐2𝜃+ 𝑐𝑜𝑠𝑒𝑐2𝜃≡ 𝑐𝑜𝑠3𝜃
WB7 Prove that 𝑐𝑜𝑠𝑒𝑐4𝜃 − 𝑐𝑜𝑡4𝜃 ≡
1+𝑐𝑜𝑠2𝜃
1−𝑐𝑜𝑠2𝜃
C3WB: Trigonometry
page
WB8
Prove that 𝑠𝑒𝑐2𝜃 − 𝑐𝑜𝑠2𝜃 ≡ 𝑠𝑖𝑛2𝜃 (1 + 𝑠𝑒𝑐2𝜃)
WB9
Solve the equation 4𝑐𝑜𝑠𝑒𝑐 𝜃 − 9 = cot 𝜃 in the interval 0 ≤𝜃 ≤ 360
C3WB: Trigonometry
page
WB10
Solve these equations:
a) 4𝑐𝑜𝑠2𝜃 + 5 sin 𝜃 = 3 in the interval −𝜋 ≤ 𝜃 ≤ 𝜋
b) 2𝑡𝑎𝑛2𝜃 + sec 𝜃 = 1 in the interval −𝜋 ≤ 𝜃 ≤ 𝜋
C3WB: Trigonometry
page
Inverse trig questions - Notes BAT use and understand applications and graphs of the inverse Trig functions
C3WB: Trigonometry
page
Graphs of arcsin x, arcos x, arctan x
C3WB: Trigonometry
page
WB11
a) Work out in degrees, the value of arcsin (−√2
2)
b) Work out in radians, the value of cos [arcsin (−1)]
WB12
a) Work out, in radians, the value of arcsin(0.5)
b) Work out, in radians, the value of arctan(√3)
C3WB: Trigonometry
page
WB13
a) Work out, in radians, the value of arcsin(0.5)
b) Work out, in radians, the value of arctan(√3)
C3WB: Trigonometry
page
Addition Formulae - Notes BAT understand where the addition formula come from and know the Identities for
sin 2x, cos 2x and tan 2x
BAT use the addition formulae to solve ‘show that’ problems
BAT use the addition formulae to solve equations
C3WB: Trigonometry
page
WB14
Show that sin 15 =√6 − √2
4
WB15 Given that sin 𝐴 = −
3
5 in the range 180 < 𝐴 < 270 and cos𝐵 =
− 12
13 where B is Obtuse . Find the value of tan(𝐴 + 𝐵)
C3WB: Trigonometry
page
WB16
Rewrite the following as a single Trigonometric function: 2 sin𝜃
2cos
𝜃
2cos 𝜃
WB17
Show that 1 + cos 4𝜃 can be written as 2𝑐𝑜𝑠22𝜃
C3WB: Trigonometry
page
WB18
Given that cos 𝑥 =3
4 in the range [180, 360] find the exact value of 2 sin 2𝑥
WB19 Given that x = 3 sin 𝜃 and 𝑦 = 3 − 4 cos 2𝜃
Eliminate and express y in terms of x
C3WB: Trigonometry
page
WB20
Solve the equation 3 cos 2𝑥 − 𝑐𝑜𝑠 𝑥 + 2 = 0 in the range 0 ≤ 𝑥 ≤ 360
C3WB: Trigonometry
page
R cos ( x + alpha) - Notes BAT write expressions of the form acosθ + bsinθ, where a and b are constants, as a
sine or cosine function only
BAT solve equations of the form acosθ + bsinθ, or find their max/min value
C3WB: Trigonometry
page
WB21
Show that 3 sin 𝑥 + 4 cos 𝑥 can be expressed in the form 𝑅 sin(𝑥+∝)
where 𝑅 > 0, 0 < 𝛼 < 90
WB22
a) Show that you can express sin 𝑥 − √3 cos 𝑥 in the form 𝑅 sin(𝑥+∝)
where 𝑅 > 0, 0 < 𝛼 < 90
b) Hence, sketch the graph of y = sin 𝑥 − √3 cos 𝑥
C3WB: Trigonometry
page
WB23
a) Express 2 cos 𝜃 + 5 sin 𝜃 can be expressed in the form 𝑅 cos(𝜃−∝)
where 𝑅 > 0, 0 < 𝛼 < 90
b) Hence, solve 2 cos 𝜃 + 5 sin 𝜃 = 3 in the range 0 < 𝜃 < 360
WB24
a) Express 12 cos 𝜃 + 5 sin 𝜃 in the form 𝑅 cos(𝜃−∝)
where 𝑅 > 0, 0 < 𝛼 < 90
b) Hence, find the maximum value of 12 cos 𝜃 + 5 sin 𝜃 and the smallest positive
value of at which it arises
C3WB: Trigonometry
page
WB23
a) Express 2 cos 𝜃 + 5 sin 𝜃 can be expressed in the form 𝑅 cos(𝜃−∝)
where 𝑅 > 0, 0 < 𝛼 < 90
b) Hence, solve 2 cos 𝜃 + 5 sin 𝜃 = 3 in the range 0 < 𝜃 < 360
WB24
a) Express 12 cos 𝜃 + 5 sin 𝜃 in the form 𝑅 cos(𝜃−∝)
where 𝑅 > 0, 0 < 𝛼 < 90
b) Hence, find the maximum value of 12 cos 𝜃 + 5 sin 𝜃 and the smallest positive
value of at which it arises
C3WB: Differentiation
page
Chain rule - Notes BAT differentiate a function of a function, using the chain rule
BAT apply 𝒅𝒚
𝒅𝒙=
𝟏𝒅𝒙
𝒅𝒚
to solve a problem
C3WB: Differentiation
page
WB1
Differentiate 𝑦 = (3𝑥4 + 𝑥)5
WB2
Given that 𝑦 = √5𝑥2 + 1 Find the value of f’(x) at (4,9)
C3WB: Differentiation
page
WB3 Given that 𝑦 = (𝑥2 − 7𝑥)4 Calculate
𝑑𝑦
𝑑𝑥 using the chain rule
WB4 Given that 𝑦 =
1
√6𝑥−3
Calculate 𝑑𝑦
𝑑𝑥 using the chain rule
C3WB: Differentiation
page
WB5 Calculate 𝑑𝑦
𝑑𝑥 for the following equation 𝑦3 + 𝑦 = 𝑥
and find the gradient at the point (2,1)
C3WB: Differentiation
page
Product rule - Notes BAT differentiate a product of two functions
C3WB: Differentiation
page
WB6
differentiate 𝑦 = 𝑥2(3𝑥 − 9)
WB7
Differentiate 𝑦 = (𝑥2 + 3)(4𝑥 + 1)
C3WB: Differentiation
page
WB8
Given that: 𝑓(𝑥) = 𝑥(2𝑥 + 1)3, find f’(x)
WB9
Given that: 𝑓(𝑥) = 𝑥2√3𝑥 − 1, find f’(x)
C3WB: Differentiation
page
Chain, Product and Quotient rules - Notes BAT differentiate a quotient of two functions
BAT differentiate complex functions using the chain rule, product rule
and quotient rules efficiently
C3WB: Differentiation
page
WB10
Given that 𝑦 =𝑥
2𝑥+5 Calculate
𝑑𝑦
𝑑𝑥
WB11
Given that 𝑦 =3𝑥
(𝑥+4)2 Calculate
𝑑𝑦
𝑑𝑥
C3WB: Differentiation
page
Exponential and natural logarithm functions- Notes BAT differentiate the exponential and natural logarithm functions
BAT apply the chain. Product and Quotient rules
BAT solve geometry problems using what we have learned so far
C3WB: Differentiation
page
WB12
Chain rule
a) Find the gradient of 𝑦 = 5 ln 𝑥 + 𝑒−𝑥 − 4 ln 2𝑥 when 𝑥 = 4
b) Find the gradient of 𝑦 = 3 ln(3 𝑥2 + 6𝑥) when 𝑥 =1
2
C3WB: Differentiation
page
WB13
Product rule
Differentiate the following: a) 𝑦 = 𝑥 𝑒𝑥 b) 𝑦 = 𝑥3 ln 𝑥
C3WB: Differentiation
page
WB14
Quotient rule
Differentiate the following: a) 𝑦 = 𝑒3𝑥−1
𝑥 b) 𝑦 =
𝑙𝑛5𝑥
𝑥
C3WB: Differentiation
page
WB15
Geometry
Find the equation of the tangents to the curve 𝑦 = 3𝑒−5𝑥
at the point where 𝑥 = 0
WB16
Geometry
Find the equation of the normal to the curve 𝑦 = −1
4ln 𝑥
at the point where 𝑥 = 1
C3WB: Differentiation
page
Trig functions - Notes BAT differentiate trig functions using what we have learned so far
BAT differentiate mixed trig, exponential, logarithm and other functions
C3WB: Differentiation
page
WB17
Differentiate: a) 𝑦 = sin 3𝑥 b) 𝑦 = sin2
3𝑥 c) 𝑦 = 𝑠𝑖𝑛2𝑥
WB18
Differentiate: a) 𝑦 = 𝑐𝑜𝑠(4𝑥 − 3) b) 𝑦 = 𝑐𝑜𝑠3𝑥 c) 𝑦 = 3 cos 𝑥
C3WB: Differentiation
page
WB19
Find the derivative of 𝑦 = tan 𝑥
WB20
Find the derivative of 𝑦 = tan 4𝑥
C3WB: Differentiation
page
WB21
Find the derivative of 𝑦 = 𝑥 tan 2𝑥
WB22
Find the derivative of 𝑦 =ln 𝑥
sin𝑥
C3WB: Differentiation
page
WB23
Find the derivative of 𝑦 = 𝑒𝑥 sin 𝑥
C3WB: Differentiation
page
Reciprocal Trig functions - Notes BAT differentiate reciprocal trig functions using what we have learned so far
BAT differentiate mixed trig, exponential, logarithm and other functions
C3WB: Differentiation
page
WB24 Find the derivative of 𝑦 = cot 𝑥
Find the derivative of 𝑦 = sec 𝑥
Find the derivative of 𝑦 = cosec 𝑥
C3WB: Differentiation
page
WB25
Find the derivative of 𝑦 = sec 3𝑥
WB26
Find the derivative of 𝑦 =𝑐𝑜𝑠𝑒𝑐 2𝑥
𝑥2
C3WB: Numerical methods
page
Notes BAT Show that a root of an equation lies between two values
BAT Approximate solutions using an iterative formula
BAT check solutions with an appropriate method
C3WB: Numerical methods
page
WB1
Show that the equation 𝑓(𝑥) = ln 𝑥 − 1 Has a root between 2 and 3
WB2 Find a root of the equation 𝑥3 − 12𝑥 + 12 = 0
C3WB: Numerical methods
page
WB3
Find a root of the equation 𝑒−𝑥 − 𝑥 = 0
WB4 α is the positive root of the equation 𝑥3 −1
𝑥− 1 = 0
Show that to 3 decimal places α = 1.221
C3WB: Numerical methods
page
WB5
Exam Q The curve 𝑦 = 5𝑥 intersects the line 𝑦 = 𝑥 + 2 at the point where 𝑥 = 𝛼
a) Show that α lies between 0 and 1
b) Show that the equation 5𝑥 = 𝑥 + 2 can be rearranged into x = ln(𝑥+2)
ln 5
c) Use the iteration 𝑥𝑛+1 = ln(𝑥𝑛+2)
ln 5 with 𝑥1 = 0 to find 𝑥3 to two sf
WB6
Exam Q 𝑓(𝑥) = −𝑥3 + 7𝑥2 − 5
a) Show that the equation f(x) = 0 can be rewritten as 𝑥 = √5
7−𝑥
b) Starting with 𝑥1 = 0.9 , use the iteration 𝑥𝑛+1 = √5
7−𝑥𝑛 to calculate the values
of 𝑥2, 𝑥3, and 𝑥4 giving your answers to 4 decimal places
c) Show that 𝑥 = 0.906 is a root of f(x) = 0 correct to 3 decimal places