Pure Flashcards
(128 cards)
1
Q
A
2
Q
What is a^x = n in log form?
A
log a n = x
3
Q
Ln x + ln y =
A
Ln xy
4
Q
Ln x - Ln y =
A
Ln x/y
5
Q
Ln x^y =
A
Y Ln x
6
Q
Ln 1/x =
A
-Ln x
7
Q
Ln e =
A
1
8
Q
Ln 1
A
0
9
Q
What is the factor theorem?
A
If f (a) = 0, then (x - a) is a factor and vice versa
10
Q
When do you flip an inequality sign?
A
When multiplying or dividing by a negative number.
11
Q
A
12
Q
Question
A
Answer
13
Q
What is the cosine rule?
A
a^2 = b^2 + c^2 - 2bc cos A
Or
Cos A = (b^2 + c^2 - a^2) / 2bc
14
Q
Sine rule?
A
Sin A / a = Sin B / b = Sin C / c
15
Q
Area of a triangle formula?
A
1/2 ab Sin C
16
Q
How do you find arc length with radians?
A
r θ = arc length
17
Q
How do you find sector area with radians?
A
1/2 r^2 θ = sector area
18
Q
How do you find segment area with radians?
A
1/2 r^2 (θ - sin θ) = segment area
19
Q
A
20
Q
Question
A
Answer
21
Q
Definition of a function?
A
Many - to - one OR one - to - one
22
Q
What is the domain?
A
Allowed input (x) values
23
Q
What is range?
A
Allowed output (y) values
24
Q
How do you execute composite functions (fg(x))
A
Work from the inside out, find g(x) first then sub into f(x)
25
How to find domain and range for inverse functions?
Domain becomes range, range becomes domain
26
Graphically, what is an inverse function?
Function reflected in line y = x
27
How do you find an inverse function?
Rearrange to make x the subject, rewrite with x in place of y and y in place of x
28
29
Question
Answer
30
How do you find coordinates of midpoint of a line?
[ (x*1* - x*2*) / 2 , (y*1* - y*2*) / 2]
31
How find gradient of line?
How find normal gradient?
Gradient = Δy / Δx
Normal gradient = -Δx / Δy
32
Formula for equation of a line?
Y - y*1* = m (x - x*1*)
33
What is the discriminant?
What does its values tell us?
b^2 - 4ac
Greater than 0; 2 real roots
Less than 0; no real roots
Equal to 0; 1 repeated root
34
What go reciprocal graphs look like?
Asymptotes with the coordinate axes
35
Equation for a circle?
(x -a)^2 + (y - b) = r^2
36
What is transformation f(x) + a?
Translation vector 0i + aj,
37
Transformation f(x + a)?
Translation vector -ai + 0j
38
Transformation a f(x)?
Stretch in y direction X a
39
Transformation f(ax)
Stretch in x direction X 1/a
40
Transformation -f(x)?
Reflection in x axis (y = 0)
41
Transformation f(-x)
Reflection in y axis (x = 0)
42
Transformation |f(x)|
Parts below x axis are reflected up in the x axis
43
Transformation f(|x|)
**Right** of y axis reflects **left** in y axis
44
45
Question
Answer
46
AB in terms of a and b?
AB = b - a
a = OA
b = OB
47
When are vectors a and b parallel?
If a = λb an λis constant
When they are scalar multiples
48
How do you find magnitude if a vector; |AB|?
If vector = x**i** + y**j** then magnitude is root (x^2 + y^2)
Using Pythagoras
49
What is a unit vector?
â = a/|a|
50
How do you find direction of a vector/ angles with axes?
Draw a diagram, use trigonometry
To find angles between vectors, use sin or cosine rules
51
52
Question
Answer
53
Formula for binomial coefficient?
(n**C** r)
n! / r!(n-r)!
54
Binomial expansion formula for values greater than 1?
(a + bx)^n = a^n + n**C**r • a^(n-r) • b^r
55
Binomial expansion formula for values less than 1?
(a + bx)^n =
a^n[1 + nx + n(n-1)…/r! • (bx/a)^r]
Valid for |x| less than 1 or |bx/a|
56
57
Question
Answer
58
Arithmetic series formula for **U***n*?
**U***n* = a + (n-1)d
59
Arithmetic series sum of formulas?
**S***n* = n/2 (2a + (n-1)d)
**S***n* = n/2(a + L)
60
Geometric formula for **U***n*?
**U***n* = ar^(n-1)
61
Geometric series sums formulae?
**S***n* = [a (1-r^n)] / (1-r)
Sum to infinity = a/(1-r)
62
What are properties of increasing, decreasing and periodic series?
Increasing: next term greater than prev.
Decreasing: next term less than prev.
Periodic: if **U***n+k* = **U***n* and its period/order is k
63
64
Question
Answer
65
How convert from degrees to radians?
Divide by 360, times by 2π
66
What are the small angle approximations?
Sin θ = θ
Cos θ = 1- θ^2/2
Tan θ = θ
67
Definition of tangent (trig. )?
Sin x/Cos x = Tan x
68
How do you find more solutions for Sin x, Cos x and Tan x?
Sin x: sin(180-x) then add 360 to each solution within interval
Cos x: cos(360-x) then add 360 to each solution within interval
Tan x: tan x then add 180 to answer until interval fulfilled
69
What are the inverse trig functions?
Cosec x = 1/sin x
Sec x = 1/cos x
Cot x = 1/tan x = cos x/sin x
70
How do sin x and cos x relate?
Sin x = Cos (90 - x), Cos x = sin (90 - x)
71
Sin^2 x + Cos^2 x =?
1
72
1+ tan^2 x =?
Sec^2 x
73
1 + Cot^2 x =?
Cosec^2 x
74
Sin (A+B) =?
Sin A Cos B + Sin B Cos A
75
Cos (A+B)
Cos A Cos B - Sin A Sin B
76
Tan (A+B) =?
(Tan A + Tan B) / 1 - Tan A Tan B
77
Sin 2x =
2sin x cos x
78
Cos 2x =
(All 3 versions)
Cos^2 x - Sin^2 x
2Cos^2x - 1
1 - 2Sin^2x
79
Tan 2x =?
2Tan x / (1 - Tan^2 x)
80
Sin^2 x =?
(Rearranged double angle)
1/2 -1/2 Cos 2x
81
Cos^2 x =?
1/2 + 1/2 Cos 2x
(Rearranged double angle)
82
What is the harmonic identity?
(R Cos (x+a))
If R Cos α = a, R Sin α = b
R = rt. a^2 + b^2
Tan α = b/a
83
84
Question
Answer
85
What is differentiation from first principles?
(Expression)
F’(x) = lim f (x+h) - f (x)
———————
h
86
What do different values of derivatives tell us?
F’(x) < 0: decreasing
F’(x) = 0: stationary point
F’(x) > 0: increasing
87
What do different values of second derivatives tell us?
F”(x) < 0: concave/max
F”(x) = 0: point of inflection
F”(x) > 0: convex/min
88
Derivative of ax^n
anx^(n-1)
89
Derivative of a^x
ln a • a^x
90
Derivative of e^x
e^x
91
Derivative of ln x
1/x
92
Derivative of Sin kx
K Cos kx
93
Derivative of Cos kx
-k Sin kx
94
Derivative of Tan x
Sec^2 x
95
Derivative of Sec x
Sec x Tan x
96
Derivative of Cot x
—Cosec^2 x
97
Derivative of Cosec x
Cosec x Cot x
98
What is the chain rule?
f (x) ~> f ‘(x) • x’
99
What is quotient rule?
u/v ~> vu’-uv’
———
v^2
100
What is 1/(dx/dy) ?
Dy/dx
101
How to find dy/dx when given dy/dt and dx/dt?
Divide dy/dt by dx/dt, this will give
dy/dt X dt/dx so the dts cancel leaving dy/dx
102
What is product rule?
uv ~> uv’ + u’v
103
What is the implicit differentiation rule?
F(y) ~> f ‘(y) X dy/dx
104
What is rate of change?
How do you connect them?
Rate means over dt
dv = dv X …
dt … dt
105
106
Question
Answer
107
What is a numerical method for finding a root?
Sign change method - if function is continuous and there is a sign change between to input values, you must equal 0 between these points hence there is a root
108
How do you draw a staircase/cobweb diagram?
Draw a line y = x. Draw from the x axis a straight line up through to the graph from the input value. Draw a horizontal line from the point of intersection with the graph to the line y = x. Draw a line up from here vertically to the graph and repeat.
109
What is the newton-raphson method?
Iterative formula used for finding roots
**x***(n+1)* = **x***n* - f (**x***n*) / f ‘(**x***n*)
110
ax^n integrates to
a / (n+1) •x^(n+1)
111
e^x integrates to
e^x
112
1/x integrates to
ln |x|
113
Cos kx integrates to
1/k sin x
114
Sin kx integrates to
-1/k Cos kx
115
116
Question
Answer
117
What is the trapezium rule?
∫ y dx = 1/2 h [( first + last ) + 2( sum of terms in between)]
118
Where do you use reverse chain rule?
When the original function could have been differentiated by chain rule to get this. Eg. F ‘(g (x)) ~> g ‘(x) f ‘(g(x))
(So in diff X out diff (orig. in))
119
How do you integrate by reverse chain rule?
Integrate outside the brackets, integrate the whole bracket as an x^a term and then divide all by inside the bracket **differentiated**
So, int outside • 1 / (n+1)( og inside) • 1/ diff inside
120
∫ ax^n integrates to
a/(n+1) • x^(n+1) + c
121
e^x integrates to
e^x + c
122
1/x integrates to
Ln |x| + c
123
Cos kx integrates to
1/k Sin kx + c
124
Sin kx integrates to
-1/k Cos kx + c
125
Sec^2 x integrates to
Tan x + c
126
Sec x Tan x integrates to
Sec x + c
127
Cosec^2 x integrates to
—Cot x + c
128
What is the change of base formula?
(**log***k*x) / (**log***k*b)
