The following formulas are given in the formula booklet: - Arithmetic sequences and series - Geometric sequences and series - Trapezium rule - Cosine rule (in both common forms)
You will need to remember the following: - Area of a sector - Arc length of a sector - Area of a triangle - Sine rule - Laws of logarithms - The two trig identities in this module
For example, 102 = 100. Therefore, log10 100 = 2. This is the same as solving for a like this: 10a = 100
Equally, any other base can be used. For example, log2(64) = 6 (because 26 = 64)
Log (without a base specified) usually refers to a base-10 logarithm and is present on many calculators (use the log■▯ button for any other base)
a2 - 3 The laws of logarithms:
log 1 = 0 (because a0 = 1)
log xy = log x + log y
log (x/y) = log x - log y
log xn = n log x
log (1/y) = - log y
log n√x = 1/n log x
loga a = 1
a5 Graphs of y = ax:
A graph of the curve y = ax has the following properties: - The y-intercept is always (0, 1). This is because a0 is always equal to 1 regardless of the value of a - The line y = 0 is an asymptote (the curve never touches this line, but gets closer for smaller values of x)
a6 Solving an equation of the from ax = b:
This can be solved for x by finding logab
If a is not 10, this is equal to log b ÷ log a
For example, solve 2x = 250: - x = log2250 = log 250 ÷ log 2 = 7.97 to 3 significant figures
To check your answer, substitute your value of x into the original equation
a7 Reducing to linear form:
Example: show that the following pairs of values satisfy the relationship R = kV n
V
1.0
2.0
3.0
4.0
5.0
R
0.02
0.32
1.62
5.12
12.5
- The relationship is in the form R = kVn. Therefore, taking logarithms gives log R = log k + log Vn - This is in the form y = mx + c. Therefore, it can be plotted onto a graph (after replacing R with y, k with c and Vn with x):
log V
0.00
0.30
0.48
0.60
0.70
log R
-1.70
-0.49
0.21
0.71
1.10
Because this graph is linear, the values satisfy the relationship R = kVn. To find the values of k and n, we can find the gradient of the line. The gradient is 4 and y-intercept is -1.7. Therefore, the relationship can be written as R = 1/50V4 (because 10-1.7 is approximately equal to 0.02, the error is due to rounding in the above tables)
Sequences and series
s1 - s2 Sequences:
A sequence is a set of numbers - Each number is a term - The value for a specific term is denoted with subscript - a1 is the first term, a10 is the tenth, etc - The general term is ak - A formula is given in terms of ak, for example ak = 5 + 2k. This is a deductive definition - With nth term notation, the above would be written as 2n + 5 - Alternatively, this can also be written like ak+1 = ak + 2 where a1 = 5. This is an inductive definition - This shows that the next term can be calculated by adding 2 to the current term
A sequence can be infinite or finite
s3 - s4 Series:
A series is the sum of all of the terms of a sequence, also known as summation
The notation is to use the ∑ sign (uppercase sigma). ∑ak where ak is a sequence formula, is equal to the sum of all terms added together, to infinity
To make the series finite, limits can be added. The upper limit is written above the ∑ and the lower limit is written below - The limits are included - if summing ak = k2 from k=1 to k=4, the sum would equal 12 + 22 + 32 + 42
s5 Periodicity:
A periodic sequence repeats itself. For example, the sequence 1, 2, 3, 1, 2, 3, 1, 2, 3 is periodic, with a period of 4
s6 Convergent and divergent sequences:
A converging sequence gradually gets closer to a number but never reaches it, for example ak = 1 ÷ k
A divergent sequence gets further and further from the starting point with each term by a large amount - e.g. ak = k3
s7 - s8 Arithmetic sequences:
Arithmetic sequences are sequences where the term increase by a fixed amount. This fixed amount is the common difference, or d
The first term can also be written as a, or a1
n is the number of terms (if the sequence is finite)
The last term, or an, is written as l (lowercase L)
The sum of an arithmetic sequence is also known as an arithmetic progression (AP)
ak = a + (k - 1) x d
l = a + (n - 1) x d
Summing:
The sum of an arithmetic series is found as follows: - S = ½ n (2a + (n - 1) d), or S = ½ n (a + l)
s9 - s10 Geometric sequences:
Geometric sequences are sequences where each term is found by multiplying the previous term by a fixed number
This fixed number is called the common ratio - For example, in the sequence 1, 2, 4, 8, 16, the common ratio, or r, is 2
The general formula is ak = ar k-1
When r is between -1 and 1, the sum of a finite geometric series is found with:
If r is greater than 1 or less than -1, then it is an infinite sequence (divergent)
For an infinite series, the above can be rewritten to:
Differentiation and integration
c1 - 4 Gradient of a curve:
The gradient at a point on a curve can be found by calculating the gradient of a tangent at this point
A slightly more accurate approach is to find the gradients of several chords of the curve - By using points on the curve very close to the point, a relatively accurate gradient can be calculated - However, the exact gradient cannot be determined: - If the point is (2, 4), then the gradient of the chord connecting (2, 4) and (2, 4) is undefined (division by zero is not possible) - Therefore, points close should be used - This is the concept of limit
The exact gradient can be found with the gradient function, dy/dx or f'(x)
The general pattern is that for y = xn, dy/dx = nxn-1 - For example: - y = x5→ dy/dx = 5x4 - y = x → dy/dx = 1x0 = 1
Additionally, if x has a coefficient, y = kxn→ dy/dx = knxn-1 - For example: - y = 5x3→ dy/dx = 15x2
For a constant term, dy/dx = 0 - For example: - y = 55 → dy/dx = 0 - This is because a graph of y = c is simply a horizontal, zero-gradient line
To differentiate a function, differentiate each component separately
To find the gradient at a point, differentiate the function of the line and then substitute in the x value of the point - For example: Find the gradient of y = x3 + 10x + 5 at point (2, 33) - Differentiate: dy/dx = 3x2 + 10 - Substitute in the value for x at the point (2); = 3x22 + 10 = 22 - Therefore, the gradient at (2, 33) is exactly 22
The gradient of a curve is sometimes referred to as the rate of change of y with respect to x
Differentiating fractions and roots:
To find dy/dx for a fraction/root (etc y = √x or y = 1/x2), they need to be converted into indices
√x = x0.5, so dy/dx(√x) = 1/2 x -0.5
A fraction a/b can also be written as ab-1. This is now in a form which can be differentiated, and the same rules apply
c5 - 7 The second derivative:
Stationary points are points on a curve where the gradient is zero
If it has a negative gradient just to the left and positive gradient just to the right, it is a minimum
If it is positive on the left and negative on the right, it is a maximum
If the curve both sides of the line have positive, or negative gradients, it is a stationary point of inflection
The second derivative is found by differentiating again. It is shown with f''(x) or d2y/dx2 - If, at a stationary point, the second derivative is negative, it is a maximum - If the second derivative is positive, it is a minimum point - If it is 0, it could be either a minimum, maximum or point of inflection. In this case, the gradient each side should be found to determine the nature of the point
When a tangent has a negative gradient, it is a decreasing function. If it has a positive gradient, it is an increasing function
c9 - 10, c13 Integration:
Integration is the inverse of differentiation
To integrate kxn, use k x xn+1÷ n+1. +c must always be added to the end because there could be a constant (constants are removed with differentiation). These constants are called arbitrary constants - For example, to integrate 5x4, add 1 to the power and divide by this new power - y = 5x5÷ 5 = x5
If given a coordinate on the curve, the value of c can be found by substituting the x and y values of the coordinate into the equation and solving for c
c11 - 12 Indefinite and definite integrals:
The integral symbol is ∫ - For example, ∫ 5x4 dx = x5 + c - This is known as an indefinite integral
A definite integral contains limits, at the top right and bottom right of the integral sign - First, integrate. Next, find the value of y at the upper limit and subtract the value for the lower limit - Example: - This is also the area under the curve between x = 1 and x = 4
To calculate the area of a curve which is partly below the x axis and partly above it, calculate the area in two parts - one for the part under the axis and one for the part above it. Find the roots of the curve so that the limits can be found
To find the area enclosed by two lines, either find the area under each and subtract (a sketch may help), or integrate the bottom curve subtracted from the top curve with the required limits
To find the area between a curve and the y-axis, integrate x in terms of y (e.g. to find the area between y = x - 1 and the y-axis, find: - Therefore, the area between the line and the y-axis is 12 square units
c13 - 16 Numerical integration:
These methods could be used when algebraic integration is not possible, but are only approximations
The area of a trapezium is A = ½h(a + b)
By splitting the area under a curve into several trapeziums, calculating the area of each and summing them, the area can be approximated
The more trapeziums used, the more accurate the area becomes
This method is called the trapezium rule and the following formula can be used: A ≈ ½ h ( y0 + yn + 2 (y1 + y2 + y3 + ... + yn-1 ) ) - h is the width of each vertical strip the curve is divided into - y0 is the height of the first strip - yn is the height of the last strip - y1 + y2 + y3 + ... + yn-1 is the sum of all heights of the strips (except from the first and last)
Sometimes this over-estimates the area and sometimes it under-estimates it
Determining if a trapezium rule estimate is over or under estimating:
If the curve is concave, it is an overestimate
If the curve is convex, it is an underestimate
The below diagram shows this: (top two are overestimates, bottom two are underestimates)
t1 - 4 Basic trigonometry:
Bearings: 0° is north, 90° is east, 180° is south, 270° is west
The angle of elevation is the angle between the horizontal and a point above it (e.g. for a bird from the ground, the angle of elevation is the angle between the viewer's eye-line when looking straight forward and the bird)
The angle of depression is the angle between the horizontal and a point below it
Unless in bearing form, on a graph 0° is the positive x direction from the origin
Graphs:
A sine graph can be created by measuring the y-coordinate of a circle every few degrees and plotting these on a graph with degrees on the x-axis
A cosine graph is a sine graph translated 90° to the left (cosθ = sinθ + 90)
A tan graph is undefined for θ = 90 as there is an asymptote every 180° (-90°, 90°, 270° etc)
t5 - 7 Identities:
For any angle, tanθ = sinθ ÷ cosθ
sin2θ + cos2θ = 1 - sin2θ means sin(θ)2 and not sin(θ2) - Therefore, you can factorise a squared trig function equation
t8 - 9 Area, sin and cosine rules:
In these formulas, vertices are labelled with uppercase letters and sides opposite to these are labelled with the lowercase letter
a/sinA = b/sinB = c/sinC - This is the sine rule and is best for finding values for sides
sinA/a = sinB/b = sinC/c is a rearranged form, best for finding angles
cosA = b2 + c2 - a2÷ 2bc and a2 = b2 + c2 - 2bc cos(A) are two forms of the cosine rule. The cosine rule is useful for finding an angle from 3 known side lengths
Area of a right-angled triangle is found with A = ½bh, and A = ½bc sinA for triangles with no right-angles
A major sector has a larger angle than π radians (180°), and a minor sector has less than π radians
Area of a sector is calculated with A = ½r2θ where r is the radius and θ is the angle in radians
Arc length is found with s = rθ
Curve sketching and transformations
Translation parallel to the y-axis by a units: y = f(x) + a - e.g. f(x) + 5 moves the entire curve up by 5 units
Translation parallel to the x-axis by a units: y = f(x - a) - e.g. f(x - 3) moves the entire curve right by 3 units - f(x + 3) moves the entire curve left by 3 units
Reflection in the x axis (y = 0): y = -f(x)
Reflection in the y-axis (x = 0): y = f(-x)
Stretch in the y direction: y = kf(x) where k is the scale factor
Stretch in the x direction: y = f(kx) where 1/k is the scale factor
