pure knowledge Flashcards
l (112 cards)
1
Q
Describe a transformation that transforms the curve y=x3 to y=-x3
A
Reflection in y or x axis
( x is to the power of an odd number, so it can also be a reflection in x-axis)
2
Q
If transformation is inside brackets, does it effect x or y value?
A
x value
3
Q
for translation, instead of saying right or left say positive x direction
A
:)
4
Q
What are mappings?
A
A mapping takes an ‘input’ from one set of values to an ‘output’ in another
5
Q
What can mappings be?
A
‘many-to-one’ (many ‘input’ values go to one ‘output’ value)‘one-to-many’‘many-to-many’‘one-to-one’
6
Q
y = x2
What sort of mapping is this?
A
Many to one:e.g. 22 and -22 both equal 4
7
Q
y = √x
What type of mapping is this?
A
one to many
e.g. √4 is equal to both -2 and 2
8
Q
y = ±√x2
What type of mapping is this?
A
many to many
e.g.
±√-4 and ±√4 both equal 4 and -4
9
Q
y = x3
What sort of mapping is this?
A
one to one
e.g. -23 is only equal to 8
10
Q
What is the difference between a mapping and a function?
A
A function is a mapping where every ‘input’ value maps to a single ‘output’Many-to-one and one-to-one mappings are functionsMappings which have many possible outputs are not functions
11
Q
What does ℕ mean?
A
natural number (positive whole number)
12
Q
What does ℤ mean?
A
integer (any whole number, including negative and positive numbers and zero)
13
Q
What does ℚ mean?
A
Quotient (fraction)
14
Q
What does ℝ mean?
A
real number
15
Q
Domain
A
The domain of a function is the set of values that are allowed to be the ‘input’
16
Q
Range
A
The range of a function is the set of values of all possible ‘outputs’
17
Q
How do you graph an inverse function?
A
reflect in the line y = x
18
Q
sketch the graph of y = √x
A
19
Q
if it touches the axis that counts as intersect the axis
A
:)
20
Q
What type of function must it be for an inverse to be possible?
A
inverse function only exists if it is one to one
21
Q
What are the quadrants of a graph?
A
22
Q
When doing integration by parts…
A
DON’T change limits around
23
Q
Write the inequality 3 is bigger than x, and x is less than and equal to 5 in set notation
A
x belongs to the real numbers, such that, x is bigger than 3 and less than or equal to 5)
24
Q
Write the inequality 3 is bigger than x, and less than and equal to 5 in interval notation
A
25
Write less than or equal to -3 in interval notation
26
Write x is less than or equal to -3 or bigger than 6 in set notation
27
Write x is less than or equal to -3 or bigger than 6 in interval notation
28
a2=
b2 + c2 - 2bcCosA
29
cotangent = tan
cosecant = cosec
secant = sec
30
cosecant = cosec
secant = sec
cotangent = tan
31
sketch a cot diagram
32
sketch a cot diagram
33
sketch the graph of y = sec(x)
34
sketch the graph of y = sec(x)
35
sketch the graph of y = cosec(x)
36
sketch the graph of y = cosec(x)
37
How can we rewrite y= sin(x) to show the inverse, x=
x= sin-1(y)x= arcsin(y)
38
How can we rewrite y= cos(x) to show the inverse, x=
x= cos-1(y)x= arccos(y)
39
How can we rewrite y= tan(x) to show the inverse, x=
x= tan-1(y)x= arctan(y)
40
How do you get the inverse of y=cos ?write the constrictions for both and the constricted graphs of both the function and the inverse of the function?
Only one-to-one functions have inverses and so the domain of sin, cos and tan have to be constricted to make them one-to-one functions.
41
How do you get the inverse of y=sin ?write the constrictions for both and the constricted graphs of both the function and the inverse of the function?
Only one-to-one functions have inverses and so the domain of sin, cos and tan have to be constricted to make them one-to-one functions.
42
How do you get the inverse of y=tan ?write the constrictions for both and the constricted graphs of both the function and the inverse of the function?
Only one-to-one functions have inverses and so the domain of sin, cos and tan have to be constricted to make them one-to-one functions.
43
Area of triangle
0.5abSinC
44
The domain of f(x) is x>8
The range of f(x) is f(x)>12
What is the range and domain of f'(x)
The range of a function will be the domain of its inverse function
The domain of a function will be the range of its inverse functionrange of f'(x) is f(x)>8domain of f'(x) is x>12
45
How do you get to the Newton Raphson method?
ankipro root is when y=0
46
Is 0 positive or negative?
Neither!!
47
When using the trapezium rule for integration, what is the formula for height of trapezium?
(Top limit - bottom limit)/number of strips
48
What are the formulas for exponential growth?
y = Ae^(kt)
k>0
y = ab^t
b>1
49
What are the formulas for exponential decay?
y = Ae^(-kt)
k>0
y = ab^t
1>b>0
50
How do you know what is decaying faster in exponential decay?
Compare half lifes
shorter half-life means faster decay.
51
How do you know what is increasing faster in exponential growth?
A smaller doubling time means faster growth.
52
Why for the general binomial expansion does IxI
ankipro
The restriction |x|
53
What is the nth term formula for a geometric sequences?
a is the first term
r is the common ratio
ar^(n-1)
54
What is the nth term formula for an arithmetic formula?
an = a1 + (n - 1)d
55
How do you know a sequence is arithmetic?
the difference between consecutive terms in the sequence is constant
56
How do you know a sequence is geometric?
there is a common ratio between consecutive terms in the sequence/ as multiplying by a common ratio each time
57
What is the difference between a series and a sequence?
A series is the sum of a sequence
58
This is one of the formulas for an arithatic series, what is d?
common difference
59
Derive the formula for an arithmetic series
sn=
sn (written backwards)=
2sn=
sn=
60
Derive the formula for a geometric series
61
How do you calculate the sum to infinity of a geometric series
given in formula book just need to explain r value
62
What is an obtuse angle?
angle between 90 and 180 degrees
63
What is an acute angle?
angle between 0 and 90 degrees
64
What is a straight line angle ?eflex (between 180 and 360 degrees),
180 degrees
65
What is a reflex angle?
between 180 and 360 degrees
66
What is a full rotation angle?
360
67
What does this mean ?
ankipro
68
What are the conditions for working out the sum use of an arithmetic and geometric series formulae
ankipro
69
what is a recurrence relation?
describes each term in a sequence as a function of the previous term – ie un+1 = f(un)
70
How can arithmetic sequences be defined using recurrence relations?
71
How can geometric sequences be defined using recurrence relations?
72
What is the sign method?
A sign change between f(a) to f(b) means there must be a root between a and b.
73
What sort of functions can you use the sign change method to find roots?
Using sign change to find a root is only appropriate for continuous functions in a small interval
74
What are the limitations of the sign change method?
Remember this will only work if the function is continuous and the interval is small enough
may fail if:
If the interval is too large there may be more than one root within it:
An even number of roots mean roots are missed entirely as a sign change is not identified
An odd number of roots ( > 1 ) may mean not ALL roots are identified
In a discontinuous function a sign change may occur which is not caused by a root but by an asymptote
If a function touches the x-axis but does not pass through it there would be a root but no sign change
75
For interval notation, what does ( or ) mean?
Does not include the number
76
For interval notation, what does [ or ] mean?
Does include the value
77
what does it mean by using the iterative method/ iterations to find an approximate for a root?
(include what the formula is)
Iteration is one way to do this, by repeatedly using each answer as the new starting value for a function, we can achieve an ever more accurate answer
Iterations are shown using the notation xn + 1 = g(xn)
78
How do you rearrange equations into an iterative formula?
rearranged in x = g(x)
which can be rewritten as the iterative formula
xn+1= g(xn)
79
in iterative formula what notation e.g. x9 does the starting value take
x0
80
in a iteration diagrams what are the two ways in which iterations can be shown?
iterations can be shown on diagrams called staircase or cobweb diagrams
81
what is plotted on an iteration diagram?
These can be drawn by plotting the graphs of y = x against y = g(x) from your iterative formula
82
on iteration diagram what is meant by converging and diverging
For converging diagrams each iteration from x0 will get closer to the rootFor diverging diagrams each iteration from x0 will get further from the root
83
on an iterative diagram where do you find the root
when the line y=x converges with iterative formula
84
in general how do you know if an iterative diagram is a cobweb or staircase
if the gradient g(x) is negative at the roots it will create a cobweb diagram
if the gradient g(x) is positive at the roots it will create a staircase diagram
85
for an iterative formula what can the gradient be
between -1 and 1
86
iterative, newton-Raphson and sign change
find approximations
87
What is the Newton-Raphson method?
Newton-Raphson method finds roots of equations in the form f(x) = 0
88
when can the newton- raphson method fail
the starting value x0 is too far away from the root leading to a divergent sequence or a different root
the tangent gradient is too small, where f’(x) close to 0 leading to a divergent sequence or one which converges very slowly
the tangent is horizontal, where f’(x) = 0 so the tangent will never meet the x‑axisthe equation cannot be differentiated (or is awkward and time-consuming to do)
89
what should you label on a sketch of a polynomial
y-axis intercept
x-axis intercepts (roots)
turning points (maximum and/or minimum)
90
what is a reciprocal graph?
Reciprocal graphs involve equations with an x term on the denominator
e.g. y = k/x or y=k/(x^2)
91
sketch y = 1/x
92
sketch y = 1/x2
93
Sketch a/x when a>0
94
Sketch a/x when a
95
Sketch a/x2 when a>0
96
Sketch a/x2 when a>0
97
For reciprocal graphs of a/x or a/x2, how does the size of a effect the graph
example use a/x
The closer a is to 0 the more L-shaped the curves are
e.g.
98
what is an asymptote
a line that the graph tends to but does not actually touch or cross
99
how do you show an asymptote for a graph
dotted line
100
write what this mean
y is inversely proportional to the square of x and so can be written as y=k/x2
ankipro
101
modulus value aka
absolute value
102
what is a vertex of a modulus function,
the point where the graph changes direction
103
How do I sketch the graph of the modulus of a function: y = |f(x)|?
Pencil in the graph of y = f(x)
Reflect anything below the x-axis, in the x-axis, to get y = |f(x)|
104
How do I sketch the graph of a function of a modulus: y = f(|x|)?
Sketch the graph of y = f(x) only for x ≥ 0
Reflect this in the y-axis
105
What is the difference between y = |f(x)| and y = f(|x|)?
The graph of y = |f(x)| never goes below the x-axis
It does not have to have any lines of symmetry
The graph of y = f(|x|) is always symmetrical about the y-axis
It can go below the x-axis
106
integration tip: if sine or cosine is to power of odd number
replace all but one using sin2 + cos2 = 1 rule
107
integration tip: if sine or cosine alone is to power of even number
use cos2x formula
108
What are the formulas we can derive from cos2x?
cos^2x = 0.5 + 0.5cos2x
sin^2x = 0.5 - 0.5cos2x
109
how do you get to cos2x = 0.5 + 0.5cos2x
from the cos 2A formula?
cos2x = cos2x - sin2x
cos2x = cos2x -1( 1-cos2x)
cos2x = cos2x - 1 + cos2x
cos2x = 2cos2x - 1
2cos2x = cos2x + 1
cos2x = 0.5 + 0.5cos2x
110
What are the three ways to finding the area under a curve?
exact:
definite integral
the limit of summing an infinite number of rectangles
estimation:
trapezium rule
sum of the rectangular areas
111
What is the formula for calculating the exact area under a curve by using Integration as the limit of a sum/ sum of infinite number of rectangles?
112
why does Integration as the Limit of a Sum work to calculate area under a curve?
If the number of rectangles increases and their width decreases, the estimate is more accurate
