Pure Flashcards
(147 cards)
1
Q
factorise x^2+y^2+xy
A
x^2 + 2xy + y^2 - xy
(x+y)^2 - xy
(x+y)^2 - (√xy)^2
using DOTS: (x+y+√xy)(x+y-√xy)
2
Q
probability A given B formula
A
P(A|B) = P(A∩B) / P(B)
3
Q
Make q the subject of 2pr = q2 + 2pq
A
q2 + 2pq = 2pr
(q + p)2 - p2 = 2pr
(q + p)2 = 2pr + p2
q + p = √(2pr + p2)
q = p√(2pr) - p
need to check this one
(q + p)2 = q2 + 2qp + p2. You don’t want the p2 though, so subtract it.
4
Q
Simplify 1/4x31/12
A
1/22x31/12
= 2-2x31/12
5
Q
Can you simplify 2-3 * x-3?
A
Yes; (2x)-3
6
Q
Rearrange to make x the subject:
1/x = 1/p + 1/q
A
1/x = q/pq + p/qp
1/x = (p+q) / pq
pq = x(p+q)
x = pq / (p + q)
7
Q
Rearrange to make x the subject:
(1-3x)2=t
A
+-√t = 1-3x
3x = 1+-√t
x = (1 +-√t) / 3
8
Q
Simplify (x+2) / x3
A
(x+2) * 1/x3
= (x+2) * x-3
= x-2 + 2x-3
9
Q
Simplify (1-x) / √x
A
(1-x) / x1/2
= (1/x1/2) - (x/x1/2)
= (1/2 x -1/2) - (1/2 x 1/2)
10
Q
area of a kite formula
A
area = p * q / 2
where p and q are the perpendicular diagonals
11
Q
When writing out inequalities, what do ( and [ mean?
A
( = not including
[ = including; equal to
e.g. (-∞, 6] => -∞ < x ≤ 6
12
Q
f(x) = x2 - (k+8)x + 8k + 1
Show that when k = 8, f(x) > 0 for all values of x. [3]
A
Sub k = 8 into f(x): [1]
f(x) = x2 - (8+8)x + 8(8) + 1 = x2 - 16x + 65
To show f(x) > 0, you must complete the square: [1]
(x - 8x)2 - 82 + 65
= (x - 8x)2 + 1
Concluding statement: [1]
When k = 8, (x - 8x)2 is a square so is always >= 0, so adding 1 means it’s always > 0
13
Q
A stone is thrown from the top of a cliff. The height, h, in metres, of the stone above the ground level after t seconds is modelled by the function h(t) = 3925/32 - 4.9(t-1.25)2.
Find, with justification:
a) the time taken after the stone is thrown for it to reach ground level and
b) the maximum height of the stone above the ground level and the time taken after which this maximum height is reached.
A
a) h(t) = 0, so solving the equation gives 4.9(t-1.25)2 = 3925/32, t = + and - (5√785)/28
t can’t be negative therefore time taken = (5√785)/28s
b) maximum height is when 4.9(t-1.25)2 = 0, therefore
maximum height = 3935/32m and
time taken = 1.25s
14
Q
real number (examples and non-examples)
A
Basically almost every number you can think of - points on a number line
Includes integers (positive and negative) rational and irrational numbers
Doesn’t include infinity or imaginary numbers
15
Q
rational number (examples and non-examples)
A
A number that can be expressed as a fraction of two integers, where the denominator is not zero
e.g. 7, 1.5
not e.g. √2, π, e
16
Q
irrational number (examples and non-examples)
A
Any real number that cannot be expressed as the quotient of two integers where the denominator isn’t 0
e.g. √2, π, e
17
Q
natural number
A
positive integer
18
Q
identity
A
An equation that is always true, no matter what values are substituted
19
Q
Why does completing the square give the turning point?
Consider both positive and negative quadratics.
A
Positive x2 graph:
* For the curve y = (x - a)2 + b, the only part that can vary is the (x - a)2 term. It’s squared so will always be positive - the minimum value of this will be 0.
* Therefore when (x - a)2 = 0, x - a = 0 so x = a.
* When x = a, the minimum value is y = 0 + b = b, so the turning point is (a, b)
Negative x2 graph:
* y = -(x - a)2 + b.
* The term -(x - a)2 will always be negative, so the maximum value occurs when -(x - a)2 = 0, so when x = a.
* When x = a, the minimum value is y = 0 + b = b, so the turning point is (a, b)
20
Q
A diver launches herself off a springboard. The height of the diver, in metres, above the pool t seconds after launch can be modelled by the following function:
h(x) = -10(t - 0.25)2 + 10.625
Find the maximum height of the diver and the time at which this maximum height is reached.
A
The maximum height is reached when t - 0.25 = 0
therefore at t = 0.25s and the maximum height = 10.625m
21
Q
Solve 3/x < 4
A
times by x2 so you’re not multiplying by a negative number:
3x < 4x2
4x2 - 3x > 0
if = 0:
x = 0, x = 3/4
x < 0, x > 3/4
22
Q
Prove the quadratic formula.
A
- ax2 + bx + c = 0
- a [x + (b/2a)x] + c = 0
- Completing the square:
a ([x + b/2a]2 - (b/2a))2 + c = 0 - a[x + b/2a]2 - b2/4a2 * a + c = 0
- a[x + b/2a]2 - b2/4a + c = 0
- a[x + b/2a]2 = b2/4a - c
- a[x + b/2a]2 = b2-4ac/4a
- [x + b/2a]2 = b2-4ac/4a2
- x + b/2a = +- √(b2-4ac) / 2a
- x = -b/2a +- √(b2-4ac) / 2a
- x = -b/2a +- √(b2-4ac) / 2a
- x = [-b +- √(b2-4ac)] / 2a
watch the negative for the c term
23
Q
domain
A
The set of possible inputs for a function (often x)
24
Q
range
A
The set of possible outputs of a function (often y or f(x))
25
f(x) is above g(x) when x < 2 and when x > 5. What does this mean about f(x) and g(x)?
The values x < 2 and x > 5 satisfy f(x) > g(x).
26
point of inflection meaning
A point of a curve where the direction of a curve changes e.g. the point in the graph of x3 equidistant of the turning points where the curve changes from being concave to convex
27
Describe how the graph changes from f(x) to f(2x).
Stretched parallel to the x-axis by the factor 1/2.
28
Describe how the graph changes from f(x) to f(1/3x).
Stretched parallel to the x-axis by the factor 3.
29
Describe how the graph changes from f(x) to 2f(x).
Stretched parallel to the y-axis by the factor 2.
30
Describe how the graph changes from f(x) to 1/3f(x).
Stretched parallel to the y-axis by the factor 1/3.
31
Describe how the graph changes from f(x) to -f(x).
Reflection in the x-axis.
| y-coordinates flip sign
you simply multiply the output by negative one
32
Describe how the graph changes from f(x) to f(-x).
Reflection in the y-axis.
| x-coordinates flip sign
33
Graph of 4/x^2
In quadrants 1 and 2, asymptotes of the x and y axes
34
Reflection of the graph y = -1/(x)2 in the x-axis
This is the same as the graph of y = 1/(x)2
35
Graph of y = cos(x)
'Like a bucket'
Starts at 1 at 0°, decreases to 0 at 90°, -1 at 180°, increases back to 0 at 270° and 1 at 360°
36
Graph of y = sin(x)
'Like a wave'
Starts at 0 at 0°, increases to 1 at 90°, decreases to 0 at 180° and -1 at 270°, increases back to 0 at 360°
37
Graph of y = tan(x)
Starts at 0 at 0°, gradient increases towards but doesn't touch 90° and -90° (vertical asymptotes at -90° and 90° - vertical asymptotes every 180°)
38
Point A has the coordinates (-3,4). L1 passes through point A. The point (1,2) What is its equation? Give your answer in the form ax + by + c = 0
m = (4 - [-3]) / (-3 - 1) = 7/-4 = - 7/4
y - y1 = m(x-x1)
y - 2 = - 7/4(x - 1)
y - 2 = - 7/4x + 7/4
y = - 7/4x + 15/4
4y = -7x + 15
7x + 4y - 15 = 0
39
f(x) = (x − 1)(x − 2)(x + 1)
a) State the coordinates of the point at which the graph y = f(x) intersects the y-axis.
b) The graph of y = af(x) intersects the y-axis at (0, −4). Find the value of a.
a) -1 * -2 * 1 = 2
(0,2)
b) 2a = -4
a = -2
40
What does the | mean in set notation?
"such that"; can also be written as :
41
{x: 5 < x ≤ 8, x ∈ ℤ} in words
All values of x such that x is greater than 5 but less than or equal to 8 and x is an element of the integers
42
Write x ≤ 5 or x > 7 in set notation.
{x: x ≤ 5, x ∈ ℝ} ∪ {x: x > 7, x ∈ ℝ}
43
f(x) = (x − 1)(x − 2)(x + 1).
The graph of y = f(x + b) passes through the origin. Find three possible values of b.
-1 + b = 0
b = 1
-2 + b = 0
b = 2
1 + b = 0
b = -1
44
What would the factorised form of 1 repeated root look like?
(x + a)2 = 0
45
formula to find the distance (d) between (x1, y1) and (x2, y2)
Use pythagoras theorem:
d2 = (x2 - x1)2 + (y2 - y1)2
d = = √[(x2 - x1)2 + (y2 - y1)2]
46
The curve C1 has equation y = −a/
x2 where a is a positive constant. The curve C2 has the
equation y = x2(3x + b) where b is a positive constant.
a Sketch C1 and C2 on the same set of axes then state the number of solutions to the equation x4(3x + b) + a = 0.
x2(3x + b) = −a/
x2
x2(x2(3x + b)) = -a
x4(3x + b) + a = 0
Sketch the graphs and you'll find the intersect once - there's one real solution,
47
A curve has the equation y = x3 − 6x2 + 9x
The point with coordinates (−4, 0) lies on the curve with equation y = (x − k)3 − 6(x − k)2 + 9(x − k) where k is a constant. Find the two possible values of k.
This is a translation by k units right.
For the point (-4, 0) to lie on the translated curve, either the point (0, 0) or (3, 0) has translated to the point (-4, 0).
For the coordinate (0, 0) to be translated to (-4, 0), k = -4 - 0 = -4.
For the coordinate (3, 0) to be translated to (-4, 0), k =-4 - 3 = -7.
so k = -4 or k = -7
48
Given that f(x) = 1/x, x ≠ 0,
a) Sketch the graph of y = f(x) – 2 and state the equations of the asymptotes
b) Find the coordinates of the point where the curve y = f(x) – 2 cuts a coordinate axis.
a) look online
asymptoses of x = 0, y = -2
b) Doesn't touch x-axis but does touch the y-axis, so x = 0
0 = 1/x - 2
x = 1/2
49
Find the centre and the radius of the circle with the equation x2 + y2 − 14x + 16y − 12 = 0
You must complete the square:
1) rearrange so x's and y's are separated
2) complete the square for x and y
3) centre = (-a, -b)
4) radius = square root of the number by itself on the right hand side
centre (7,-8) and radius = 5√5
| Example 7 of year 12 6C textbook (includes working)
50
secant
A line that intersects a circle at two distinct points
51
circumcircle
A circle which surrounds another shape with vertices that all touch but don't intersect the circle's circumference
52
circumcentre
Centre of circumcircle
53
inscribe
(of a shape inside another shape) to have all its vertices touching but not intersecting the outer shape's sides
54
circumscribe
A shape which surrounds another shape with vertices that all touch but don't intersect the outer shape's sides
55
Where does a perpendicular bisector of a chord always pass through?
The centre of the circle
56
You are given three points that lie on the circumference of a circle. How can you find the centre of the circle?
* Find the gradients of two different lines between two of the points
* Find the midpoints of the distances between these points
* Find the gradients of the perpendicular lines of these chords
* Use the gradients and midpoints to find the equations of the perpendicular bisectors
* Equate the perpendicular bisectors and solve to find the coordinates of the intersection - they will meet at the centre
57
What is the quotient and remainder of 3x2-5x+3 divided by 3x-1?
Polynomial division (e.g. using long division/grid method)
This gives a quotient of x+2 and **a remainder of -2**
58
dividend
The value that is divided by another value
59
divisor
The value that another value is divided by
60
quotient
The result from a division
61
polynomial
An algebraic expression that is made up of variables, constants, and exponents e.g. 5x3 - 2x2 + 8
The **exponents must be non-negative integers** and any **coefficients must be real**
| Many ("poly") terms ("nomial")
62
Is 5x3 - 2y2 + 8 a polynomial?
Yes - there can be many different variables in a polynomial.
63
binomial (+ examples)
An algebraic expression of the sum or the difference of two **terms** e.g. x + 2, 3x2 - 4
| A polynomial with only two ("bi") terms ("nomial")
64
the factor theorem
The factor theorem states that if f(x) is a polynomial then:
* If f(p) = 0, then (x – p) is a factor of f(x)
* If (x – p) is a factor of f(x), then f(p) = 0
65
When x =b/a, f(x) = 0. What is a factor of f(x)?
(ax – b) is a factor of f(x)
Think about it like rearranging to get 0:
if x = p, 0 = **x - p**
if x = b/a, 0 = **ax - b**
66
Fully factorise 2x3 + x2 – 18x – 9
* Manually try different values of x (usually between -4 and 4) OR use the table function in your calculator until the function equals 0
* Use the value of x and the factor theorem to put it into the form (ax - b)
* Perform polynomial division
* You may need to factorise the quotient if it's not fully factorised - 2x2 + 7x + 3 can be factorised further
(x − 3)(2x + 1)(x + 3)
(worked solution example 6a from chapter 7.3)
## Footnote
You MUST find one value of x, use the factor theorem and then do polynomial division to get all the marks
67
deduction proof
Starting from known factors or definitions, then using logical steps to reach the desired conclusion (often uses algebra)
68
exhaustion proof
Breaking the statement into smaller cases and proving each case separately (e.g. if x is even and if x is odd)
69
disproof by counter-example
Finding one example that does not work for the statement - this often needs some logic
70
What do you do if you want to prove something is always positive?
Complete the square
The squared term will always be greater than or equal to 0, so adding on a positive value will always make something positive
71
If you are trying to prove an identity, you must start from the left hand side. T/F and why?
False - in an identity you can start from the RHS or the LHS.
72
How do you find the value in Pascal's triangle that is in the 14th row and the 7th column?
nCr = 13C6
73
Explain what each part of the nCr formula means.
n! / r!(n-r!)
n! = factorial of the population
(n-r)! = gets rid of the tail end of the factorial of the population, leaving the factorial of the values you're looking for i.e. the different permutations
r! = order doesn't matter so this gets rid of the same combination in differing orders (e.g. ABC, ACB, BAC...)
74
Simplify (n!)/(n-3)!
n(n-1)(n-2)(n-3)! / (n-3)!
= n(n-1)(n-2)
75
Find the first three terms of (2 - 3x)5, simplifying the coefficients.
5C0 * 25 * (-3x)0 = 32
5C1 * 24 * (-3x)1 = -240x
5C2 * 23 * (-3x)2 = 720x2
(2 - 3x)5 = 32 - 240x + 720x2 + ...
## Footnote
Remember the coefficients should bounce between being positive and negative
76
Find the coefficient of x2 in (1 + 2x)8(2 − 5x)
7.
You need either:
* 2 2x's in the first bracket and 0 in the second
* 2 -5x's in the second bracket and 0 in the first
* 1 2x in the first bracket and 0 in the second
8C2 * 16 * (2x)2 = 112x2
112x2 * 27 = 14 336x2
7C2 * 25 * (-5x)2 = 16 800x2
16 800x2 * 18 = 16 800x2
8C1 * 17 * (2x)1 = 16x
7C1 * 26 * (-5x)1 = -2240x
16x * -2240x =-35 840x2
14 336 + 16 800 - 35 840 = -4 704
(**coefficient** of x2 is wanted)
77
nPr vs. nCr function on a calculator
nPr -> permutations - an arrangement of values in a particular order
nCr -> number of possible combinations, regardless of order
78
nPr vs. nCr formulae
nPr -> n! / (n-r)!
nCr -> n! / [r!(n-r)!]
where n is the total number of items and r is the number of items chosen
79
period of a wave
The amount of time it takes to complete one full wave cycle
80
How can you find a second value of sin(θ) from one value of sin(θ)?
sin(θ) = sin(180 - θ)
81
How can you find a second value of cos(θ) from one value of cos(θ)?
cos(θ) = cos(360 - θ)
82
How can you find a second value of cos(θ) or sin(θ) from another wave with one value of cos(θ) or sin(θ) respectively?
cos(θ) = cos(θ ± 360)
sin(θ) = sin(θ ± 360)
83
How can you write tan(θ) with the other trigonemtric ratios?
tan(θ) = O/A
sin(θ) = O/H
cos(θ) = A/H
tan(θ) = O/A = (O/H) / (A/H)
= sin(θ) / cos(θ)
## Footnote
"toads seek crows" -> t = s/c
84
What is the 'ambiguous case' in trigonometry?
Having multiple (e.g. two) potential answers as two angles can take the same sin() value.
85
unit circle
A circle with a radius of 1 unit, usually with a centre at the origin
86
What are a) cosθ , b) tanθ and c) sinθ in relation for a point P(x, y) on a unit circle?
Angles are measured anti-clockwise from the positive x-axis
a) **cosθ = A/H = x/1 = x-coordinate**
b) tanθ = O/A = y/x = gradient
c) **sinθ = O/H = y/1 = y-coordinate**
87
How can you use the quadrants to determine whether each of the trigonometric ratios is positive
or negative?
Use a CAST diagram, starting from quadrant 4 and working anticlockwise.
For an angle θ in each quadrant,
Quadrant 4: only **c**osθ is positive
Quadrant 1: **a**ll trig ratios are positive
Quadrant 2: only **s**inθ is positive
Quadrant 3: only **t**anθ is positive
88
Use a CAST diagram to determine whether sin(180 - θ) is positive or negative.
Use the corresponding **acute angle** made with the **x-axis**.
The angle/line ends up in the second quadrant, which is S in CAST - therefore sin(180 - θ) is positive and is equal to sin(θ)
89
Use a CAST diagram to determine whether cos(360° + θ) is positive or negative.
Use the corresponding **acute angle** made with the **x-axis**.
The angle/line ends up in the first quadrant, which is A in CAST - therefore cos(360° + θ) is positive and is equal to cos(θ)
90
Express tan500° in terms of trigonometric ratios of acute angles.
tan(500) = tan(500 - 360) = tan(140)
This lands in the second quadrant, so tan is negative (CAST). The line makes an angle of 40° with the x-axis, therefore tan(500) = -tan(40)
91
How can you prove the identitiy sin2θ + cos2θ ≡ 1?
Using a unit circle for a point (x,y), x2 + y2 = 1
BUT x = cosθ and y = sinθ, therefore sin2θ + cos2θ ≡ 1
92
How do you simplify
trigonometrical expressions and complete proofs (2)?
Change tan into sinθ/cosθ if appropriate
Any trig2 expressions can be converted using sin2θ + cos2θ ≡ 1
93
principal value
The value you get on a calculator when you use the inverse trigonometric functions
94
How can you solve 5sin x = -2 in the interval 0 ≤ x ≤ 360° using a CAST diagram?
| An alternative method is using a sin graph.
sinx = -0.4
-sinx = 0.4
Principal value of x ~ -23.6°
Draw this on a CAST diagram and the principal value lands in the fourth quadrant, where only cos is positive (sin here is negative).
When you draw x ~ -23.6° to the negative x-axis, only tan is positive (again, you're looking for where sin is negative).
If you calculate the angle going anticlockwise from the positive x-axis, the two solutions are x = 90 + 90 + 90 +90 -23.6 = 336.4° and x = 90 + 90 + 23.6 = 203.6°
| It's helpful to draw a (non-transformed) graph to show all the solutions
95
Solve the equation cos3θ = 0.766, in the interval 0 ≤ θ ≤ 360°.
3θ ~ 40.00
θ ~ 13.33
θ = 360 - 13.33 = 346.67
Wavelength = 360 / 3 = 120
Adding/subtracting 120:
θ = 13.33 + 120 = 133.33
θ = 133.33 + 120 = 253.33
253.33 + 120 -> out of range
θ = 346.67 - 120 = 226.67
θ = 226.67 - 120 = 106.67
106.67 - 120 -> out of range
therefore θ = 13.3, 107, 133, 227, 253. 347 (3 s.f.)
| It's helpful to draw a (non-transformed) graph to show all the solutions
96
The equation tan kx = -1/√3, where k is a constant and k > 0, has a solution at x = 60°. Find a value of k. Is this the only possible one?
tan 60k = -1/√3
60k = -30
k = -0.5 (this isn't greater than 0 so cannot be k)
However, 60k can also be equal to -30 + 180 = 150
so k = 2.5
60k could be equal to -30 + 260 = 330
where k = 5.5, so this isn't the only solution.
97
Solve the equation sin(3x - 45°) = 0.5 in the interval 0 ≤ x ≤ 180°.
3x - 45 = 30
3x = 75
x = 25
3x - 45 = 180 - 30
3x - 45 = 150
3x = 195
x = 65
Wavelength = 360 / 3 = 120°
Different wave:
x = 25 + 120 = 145
Adding/subtracting 120° to these angles gives values outside the interval, therefore x = 25°, x = 65° or x = 155°.
| It's helpful to draw 2 graphs (one translated) to show all the solutions
98
unit vector (+ most common examples)
a vector of length 1
i is the horizontal unit vector and j the vertical unit vector
99
Given that **c** = 3**i** + 4**j** and **d** = **i** − 2**j**, find t if **d** − t**c** is parallel to −2**i** + 3**j**.
**d** − t**c** = (**i** - 2**j**) - t(3**i** + 4**j**) = (1 - 3t) **i** + (-2 - 4t) **j**
k (1 - 3t) = -2
k = -2 / (1 - 3t)
k (-2 - 4t) = 3
k = 3 / (-2 - 4t)
-2 / (1 - 3t) = 3 / (-2 - 4t)
4 + 8t = 3 - 9t
17t = -1
t = -1/17
100
What is â?
The unit vector in the direction of **a**.
101
What is the unit vector of -7**i** + 24**j** in the direction of the vector?
Unit vector = vector / |vector|
|vector| = √(-7)2 + (24)2 = √(49 + 576) = 25
Unit vector = (-7**i** + 24**j**) / 25
= -0.28**i** + 0.96**j**
102
position vector
A vector that gives the position of a point, relative to the origin
103
The point A lies on the circle with equation x2 + y2 = 9. Given that the vector OA = 2k**i** + k**j**, find the exact value of k.
|vector OA| = 3
Using the position vector |OA| = √[(2k)2 + k2] = √5k2
√5k2 = 3
5k2 = 9
k2 = 9/5
k = 3/√5 (can rationalise if you want)
104
stationary point
a point where the derivative of f(x) is equal to 0 i.e. the gradient is 0; can be a turning point or point of inflection
105
derivative meaning
the rate of change of a function (with respect to a variable, often x or t)
106
differentiate 1/x from first principles
f'(x) = limh->0 [f(x+h) - f(x)] / [(x + h) - x]
= limh->0 [1/(x+h) - 1/x] / h
Be careful when dividing by h - multiply by 1/h
I'm too lazy just watch the vid :p
https://www.youtube.com/watch?v=GBOJl4fK5Hk
107
How can you decide what type of stationary point one is?
Check the points just before and after (e.g. ± 0.01)
OR
You can use the second derivative and see if f''(x) is positive, negative or 0
108
What three things can a stationary point be?
A **local** maximum, **local** minimum or a point of inflection
109
f''(a) is 0. What sort of stationary point is it?
You cannot tell - it could be a local maximum, local minimum or a point of inflection - you would have to check either side of the value of 'a' to find out
110
What does a point of inflection in the graph f(x) look like in the graph of f'(x)?
Touches the x-axis (at the x coordinate of the inflection)
111
What does a vertical asymptote in the graph f(x) look like in the graph of f'(x)?
A vertical asymptote (at the x coordinate of the asymptote)
112
What does a horizontal asymptote in the graph f(x) look like in the graph of f'(x)?
A horizontal asymptote at the x-axis
113
How do you solve differentiation optimisation problems?
Form an equation for the value given in terms of the model/diagram, and another with the values given in the worded question. Rearrange the first expression to get the unknown as the subject, then substitute this into the first equation to solve.
| This will probably not make any sense to anyone else lol
114
Find f(x) when f'(x) = (2√x - x2)([3 + x]/x5)
## Footnote
Check your answer (it is **very** easy to make silly division errors) and ensure your answer will pick up all the marks
Re-write to put in fractional and negative powers:
f'(x) = (2√x - x2)([3 + x]/x5)
= (2x1/2 - x2)([3 + x]x-5)
Expand the square bracket:
= (2x1/2 - x2)(3x-5 + x-4)
Expand the brackets:
6x-9/2 + 2x-7/2 -3x-3 - x-2
Integrate:
-12/7x-7/2 -4/5x-5/2 + 3/2x-2 + x-1 **+ c**
Then, to make Miss Lawlow-Kennedy happy, convert the expression to remove negative and fractional indices:
-12/(7√x7) -4/(5√x5) + 3/2x2 + 1/x + c
115
What things can you do to calculate the area under a section of a curve?
* Integrate the curve
* Find the area of a triangle and/or trapezium
* Combine these for the shaded area
116
When do you use modulus when integrating for an area bounded by a curve?
Any area bounded by the curve below the x-axis is negative, so you use the modulus when integrating for an area that is below the x-axis (otherwise the positive and negative values would cancel out)
117
What special mathematical property do exponential graphs of the form y = ax have? Explain why.
The graphs of their gradient functions are a similar shape to the graphs of the functions themselves.
At very negative values of x, the rate of increase in x is small but positive. As x increases, the rate of increase in x increases, so the gradient function is also exponential.
118
Why is e special?
Between b = 2 and b = 3 in y = bx, there is a value of b where the gradient function is exactly the same as the original function - this occurs when b is approximately equal to 2.71828
119
differentiate f(x) = e1.4x
f'(x) = 1.4e1.4x
120
easiest way to sketch a graph transformation e.g. of 1/x
Sketch the asymptote first before you draw the curve
121
Sketch the graphs of y = 2 + 10e-x. Give the coordinates of any points where the graph crosses the axes, and state the equations of any asymptotes.
Negative exponential graph (normal one reflected along the y-axis so that it's a decreasing graph)
Stretched by a scale factor of 10 in the vertical direction so the y-intercept of 1 becomes a 10
Translated by 2 units up so y-intercept of 12 -> **(0,10)**
**asymptote of x-axis gets translated up by 2 -> (y = 2)**
| Always check whether you need the *asymptote* or the *axis intercept*
122
Prove the multiplication law.
| For understanding's sake
Let f = logax and g = logay
x = af and y = ag
x * y = af * ag = af + g
logaxy = logaaf + g = f + g = logax + logay
123
Prove the division law.
| For understanding's sake
Let f = logax and g = logay
x = af and y = ag
x / y = af / ag = af - g
loga(x / y) = logaaf - g = f - g = logax - logay
124
Prove the power law.
| For understanding's sake
Let f = logax **(you only need one for this proof!)**
x = af
xb = (af)b = afb
logaxb = logaafb
logaxb = fb = logax * b
therefore logaxb = blogax
125
log3-9 = ?
Undefined - you can't log a negative number to get a real number as you cannot raise a base to a power to get a negative number.
126
Solve the equation 7x + 1 = 3x + 2
Take logs of both sides (can be any base)
log77x + 1 = log73x + 2
x + 1 = log73x + 2
x + 1 = (x + 2) log73
x + 1 = xlog73 + 2log73
Collecting like terms together:
x - xlog73 = 2log73 - 1
x (1 - log73) = 2log73 - 1
x = (2log73 - 1) / (1 - log73)
(then you can type this into your calculator and give your answer to e.g. 4 s.f.)
127
The graph of y = ln x is a reflection of the graph ____ in the line ____.
y = ex, y = x
128
intercept and asymptote of y = ln x
x=intercept of 1 -> (0, 1)
asymptote of the y-axis (x = 0)
129
Explain why ln x is only defined for positive values of x graphically.
ln x has a vertical asymptote at x = 0, so the curve never reaches negative values of x
130
ln (ex) = ?
x
| Inverse operation of raising e to the power x = taking natural logarithm
131
e (lnx) = ?
x
| Inverse operation of raising e to the power x = taking natural logarithm
132
Solve the equation 3x e4x - 1 = 5, giving your answer in the form (a + ln b) / (c + ln d).
3x e4x - 1 = 5
Using the multiplication log law:
ln (3x e4x - 1) = ln 5
ln (3x) **+** ln(e4x - 1) = ln 5
Rearranging, simplifying and factorising:
x ln 3 + (4x - 1) = ln 5
4x + x ln 3 - 1 = ln 5
x (4 + ln 3) - 1 = ln 5
Putting the equation formed into the form requested:
x = (1 + ln 5) / (4 + ln 3)
| Be very careful with the multiplication law
133
5e2x = ?
| Expand it out to show all the terms involved
5 * ex * ex
| It's a hidden square - e.g. can be used in hidden quadratics
134
Convert y = 4x3 into a linear equation.
y = 4x3
log y = log y (4 * x3)
Using the multiplication law:
log y = log y 4 + log y (4x3)
Using the power law and rearranging:
log y = 3 **log y (4x)** + *log y 4*
or y = m **x** + *c*
135
y = axb
What is the y-intercept of the straight line graph?
log a
(or whatever logarithm base you choose to use)
| This is the same as the y-intercept of y = abx
136
y = axb
What is the gradient of the straight line graph?
b
| This is different to the y-intercept of y = abx
137
When plotting a straight line graph y = axn, what do you label the axes?
**log y against log x**
(or whatever logarithm base you choose to use)
| Both axes need to be logs
138
Convert y = 4ax into a linear equation.
y = 4ax
log y = log (4 * ax)
Using the multiplication law:
log y = log 4 + log (ax)
Using the power law and rearranging:
log y = (log a) **x** + *log 4*
or y = m **x** + *c*
139
y = abx
What is the y-intercept of the straight line graph?
log a
(or whatever logarithm base you choose to use)
| This is the same as the y-intercept of y = axb
140
y = abx
What is the gradient of the straight line graph?
log b
(or whatever logarithm base you choose to use)
| This is different to the y-intercept of y = axb
141
When plotting a straight line graph y = abx, what do you label the axes?
log y against **x**
(or whatever logarithm base you choose to use)
| You wouldn't get a linear graph if you used log x
142
Rewrite 201.5t + 3 in powers of 10.
201.5t * 20 3
143
log P = 0.6t + 2 and P = abt. Find the values of a and b.
log P = 0.6t + 2
Make P the subject:
P = 100.6t + 2
Separate out the 0.6t + 2 bit:
P = 100.6t * 102
Simplifying and rearranging into the form given:
P = (100.6)t * 100
P = 100 * (100.6)t
a = 100, b = 100.6
(can't be the other way around because only b is to the power of t)
144
251/2
5 but not -5
Root powers don't give a positive and negative answer
145
volume of a cone
1/3 h pi r2
146
volume of a pyramid-adjacent shape
1/3 * base * height
147
formula for integrating between two curves (+ why this is the case)
integrate (top curve - bottom curve)
also written as integrate [f(x) - g(x)]
This works because they share limits, so you can just integrate them together
