Download Exercise 2 - Elgin Academy

Elgin Academy
Mathematics Department
Higher
Unit 1 Revision
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Outcome 1
PC(a) Example
A line passes through the points (2,-6) and (-3, 4). Find the equation of this line.
m
y 2  y1
x 2  x1
46
32
10

5
 2

Equation is y  b  m( x  a)
y  6  2( x  2)
 y  2 x  2
Exercise 1
Find the equation of the lines passing through the following pairs of points:
1.
(-1, 3), (-2,-4)
2.
(-1,-2), (2, 4)
3.
(-1,-3), (4, 7)
PC(b) Example
o
A line makes an angle of 70 with the positive direction
of the x-axis. Find the gradient of this line.
m  tan    tan 70  2.75
Exercise 2
State the gradient of each line:
PC(c) Example
Write down the gradient of
(i)
a line parallel to y = 5x + 2
(ii) a line perpendicular to y = 5x + 2
i  for parallel lines m1  m 2 , so m parallel  5
(ii) for perpendicular lines m1  m 2  - 1, so m perpendicular  - 15
Exercise 3
State the gradient of the line (i) parallel to and (ii) perpendicular to each line:
1. y = 4x + 3
2. y = 6x – 1
3. y = 7x + 5
Outcome 2
PC(a) refer to note on graphs of related functions in text book.
Exercise 4
y
1. Copy the graph of y = f(x) twice and sketch
the graphs y = -f(x) and y = f(x – 2)
0
(2,-4)
y
3 x
(0, 4)
2. Copy the graph of y = f(x) twice and sketch
the graphs of y = f(-x) and y = f(x + 1)
1x
-2
y
3. Copy the graph of y = f(x) twice and sketch
the graphs of y = 1 + f(x) and y = f(x - 3)
2
1
x
( 2,-2)
4. The diagrams below show the graphs of y = sinxo or cosxo and related
functions. State the equation of each related graph.
y
y
3
y
1
3
180
y = sin xO
180
x
180
-1
x
y = cos xO
y = sin xO
-1
-3
PC(b) Example 1
y
Write down the equation of the graph
of the exponential function of the
form y = ax shown in the diagram.
y = ax
when x = 2, y = 9, so a2 = 9
a=3
(2, 9)
Equation is y = 3x.
x
Exercise 5
Each of the diagrams shows a function of the form y = ax. State the equation
of each function.
1.
y
2.
y
(2, 4)
(1, 4)
x
x
x
PC(b) Example 2
The diagram shows the graph of y = ex and its inverse function, y = loge x.
y
y = ex
1
y = loge x
x
1
Exercise 6
State the equation of the inverse of each of the following functions:
1.
y = 3x
2.
y = 5x
3.
y = 8x
4.
y = 10x
PC(c) Example
f(x) = 2x2 and g(x) = 3x – 1. Find an expression for f(g(x)).
f(g(x)) = f(3x – 1) = 2(3x – 1)2
Exercise 7
1.
f(x) = x2 + 3 and g(x) = x – 1. Find f(g(x)).
2.
h(x) = x3 and k(x) = 4x – 3. Find h(k(x)).
3.
f(x) = 5x2 – 1 and g(x) = 4x + 1. Find f(g(x)).
4.
h(x) = x + 2 and k(x) = cos x. Find k(h(x)).
5.
f(x) = 3x2 and g(x) = sin x. Find g(f(x)).
6.
h(x) = x2 + 3x and k(x) = cos x – 1. Find k(h(x)).
Outcome 3
PC(a) Example
dy
2 x 3  x 2  3x
, find
.
2
dx
x
2 x 3  x 2  3x 2 x 3 x 2 3x
y
 2  2  2  2 x  1  3x 1
2
x
x
x
x
dy
 2  0  1.3x  2  2  3x  2
dx
Given y 
Exercise 8
Find
1.
3.
dy
in each of the following examples:
dx
x 3  5x 2  2 x  1
2. y  x( x 2  3x  2 x 1  x 2 )
y
2
x
3
2 x  3x 2  x  2
5x 2  2 x  3
4.
y
y

x2
x3
PC(b) Example
The diagram shows the curve with
equation y  x 2  5x  14 , with a
tangent drawn at (3, 10).
Find the gradient of this tangent.
y
20
y  x 2  5 x  14
5
x
dy
 2x  5
dx
when x  3, m tangent  2  3  5  11.
-20
Exercise 9
1.
2.
3.
4.
A curve has equation y  x 2  5x  6 . Find the gradient of the tangent to
this curve at the point (3, 0).
A curve has equation y  x 3  x  2 . Find the gradient of the tangent to this
curve at the point (2, 8).
A curve has equation y  4 x 2  5x  3 . Find the gradient of the tangent to
this curve at the point (-2, 3).
A curve has equation y  x 3  2x 2 . Find the gradient of the tangent to this
curve at the point (-3, -9).
PC(c) Example
Find the coordinates of the stationary points of the curve with equation
y  x 3  3x 2  9 x  1 and use differentiation to determine their nature.
y  x 3  3x 2  9 x  1
dy
dy
 3 x 2  6 x  9 For stationary points
 0.
dx
dx
3x 2  6 x  9  0
3 x  3 x  1  0
x  3  0 or x  1  0
x  3 or x  1
x  3 
y   3  3 3  9 3  1  28
x 1
y  13  3.12  9.1  1  4
x
3


2


3
3 /1

 3 
 1

1

x3

0



x 1
dy
dx



0


0

0

slope
/
 3, 28 is a maximum
\
/
turning point and 1, - 4  is a minimum turning point.
Exercise 10
Find the coordinates of the stationary points of each curve and using
differentiation, determine their nature.
1. y  13 x 3  3x 2  8x  5
2. y  13 x 3  12 x 2  2 x  1
3. y  14 x 4  2 x 2  5
Outcome 4
Example
The local nursery had lost all their Christmas decorations. They can afford to
buy 30 new Christmas lights each year in November. Over the course of each
festive season, 40% of the lights fail.
a) Write down a recurrence relation for un+1, the number of lights working at
the start of the festive season.
b) Find the limit of the sequence generated by this recurrence relation and
explain what this limit means in the context of this question.
a) un+1 = 0.6 un + 30
b) Since –1 < 0.6 < 1 a limit exists as n gets very large.
for u n 1  au n  b , L 
b
30
30


 75
1  a 1  0.6 0.4
OR
as n  , u n 1  u n  L
L  0.6 L  30
0.4 L  30
L  75
The number of working lights at the start of each festive season will never
exceed 75.
Exercise 11
1. Before a public park opens at 9am each Monday, the cleansing department
removes 80% of the litter present. Each week, the public drops 20kg of
litter. There are un kg of litter at closing time one Sunday.
a) Write down the recurrence relation for un+1, the amount of litter at
closing time next Sunday.
b) Find the limit of the sequence generated by this recurrence relation and
explain what this limit means in the context of this question.
2. Before a public park opens at 9am each Monday, the cleansing department
removes 109 of the litter present. Each week, the public drops 25kg of litter.
There are un kg of litter at closing time one Sunday.
a) Write down the recurrence relation for un+1, the amount of litter at
closing time next Sunday.
b) Find the limit of the sequence generated by this recurrence relation and
explain what this limit means in the context of this question.
3. A car designer has calculated that coolant escapes from an engine’s cooling
system at a rate of 15 of the coolant present per month. 5 litres are added at
the end of each month.
a) If there are un litres at the beginning of a particular month, write down
the recurrence relation for un+1, the amount of coolant at the beginning
of the next month.
b) Find the limit of the sequence generated by this recurrence relation and
explain what this limit means in the context of this question.
Answers
Exercise 1
1. y – 3 = 7(x + 1)
2. y + 2 = 2(x + 1)
3. y + 3 = 2(x + 1)
Exercise 2
1. 0.577
2. -1.19
3. 0.839
Exercise 3
1. a) m = 4
b) m = 41
2. a) m = 6
b) m = 61
3. a) m = 7
b) m = 71
Exercise 4
1-3
Graphs
4. a) y = 3sin x
b) y = cos x + 2
c) y = sin 2x
Exercise 5
1. y = 2x
2. y = 4x
Exercise 6
1. y = log 3 x
2. y = log 5 x
Exercise 7
1. (x – 1)2 + 3
4. cos(x + 2)
2. (4x – 3)3
5. sin(3x2)
Exercise 8
1. 1 – 2x -2 + 2x -3
Exercise 9
1. 1
2. 3x2 + 6x + x -2
2. 13
3. -11
Exercise 10
1. max (4,  10 13 ), min (2,  11 23 )
3. min (2,  9), max (0,  5), min (2,  9)
3. y = log 8 x
4.
y = log 10 x
3. 5(4x + 1)2 – 1
6. cos(x2 + 3x) – 1
3. 2 – x –2 – 4x –3
4. -5x –2 + 4x –3 – 9x –4
4. 15
2. max (2, 4 13 ), min (1,  16 )
Exercise 11
1. a) un+1 = 0.2un + 20 2. a) un+1 = 0.1un + 25 3. a) un+1 = 0.8un + 5
b) L = 25,
b) L = 27.78,
b) L = 25,
the amount of litter
the amount of litter
the amount of coolant
settles at 25kg.
settles at 27.8kg
settles at 25litres