Key Facts for Core 3 (OCR)
There are 12 topics in C3
1. Successive Translations
From earlier modules
y = f(x – a) translation of a in the plus x direction
y = f(x) + a translation of a in the plus y direction
y = a f(x) stretch by factor a in the y direction
y = f(ax) stretch by a factor 1/a in the x direction
These can be combined to give more complicated transformations. You must apply them in the order implied by the question
You also have to be able to work backwards i.e. what is the transformation necessary in order to….
Look for the link equation that is a halfway house and defines the function after the first transformation.
e.g y = sin(x) to y = sin(0.5x – π/6)
The link equation is y = sin(x – π/6)
Then apply the stretch of a factor 2 in the x-axis to obtain the 0.5x term
For a quadratic the secret is your old friend – complete the square.
x2 – 4x – 7 = (x-2)2 – 11
So the transformation required on x2 is to shift right by 2 and then down by 11
2. Functions
A mapping is a rule (i.e. a recipe) connecting input values and output values. It can be many to one, one to one, one to many or many to many.
A many to one or a one to one mapping is called a function, i.e. a function only has one output value for any input value. This is why the √ function is only allowed to output positive values these days! Many input values are allowed to give the same output value e.g. y = x2. Sometimes the domain is restricted to make a mapping become a function or indeed a one-one function. For a one - one function each output value is defined by only one input value as well as each input defining only one output.
The domain of a function or a mapping is the set of input values. It is not necessarily all possible input values that the function could have but may just be a subset that is relevant for a particular case. The range is the set of output values for a given set of input values. In terms of y = f(x) then the domain is along the x axis and the range along the y axis.
The composition of two given functions means applying one then the other to the result of the first. The order does matter. Usually written as f(g(x)), meaning do g first and then f.
To find the inverse of a function (has to be one to one otherwise the inverse is not defined) swap over x and y and make y the subject. The inverse of a function is the reflection of f(x) in the line y = x.
The domain of f -1(x) is the range of f(x) and vica versa.
3. Exponential Growth and Decay
We already met this in C2. There it was called a geometric series.
un = a un-1 where a is a constant
If a > 0 then then un grows progressively bigger and if a < 1 then un decays exponentially.
Geometric series refer to discreet values but the relationship can equally be continuous, usually a function of time as in y = a t or, more generally, y = a bt
Examples are the rabbit population multiplying out of control and the temperature of something cooling down after being heated.
When t = 0 then the initial value is a. When t is very large then if b is negative then y tends to 0. If it is positive then y grows rapidly.
Remember that if y = a bt then logy = bt loga This can be used to find t for a given y.
Any logs can be used but remember that loga a = 1 which could make the calculation easier. This is only relevant if a is either 10 or the special value e (about 2.7). The latter occurs a lot in C3 because ex can be easily differentiated and integrated. loge is usually written as ln. They are on all scientific calculators.
4. Extending Calculus to f(ax + b)
The following results are easily obtained by the substitution z = ax + b. However, they are presented as things to learn in the book of the course.
If f(x) = (ax + b)n then f´(x) = a n(ax+b)n-1
More generally if f´(x) = g(x) then f´(ax + b) = a g(ax + b)
Similarly for integration
∫(ax + b)n = (1/a (n+1)) (ax + b)n+1 + k
or more generally
∫g(ax + b) = (1/a) f(ax + b) +k where f(x) is the simplest integral of g(x)
5. Calculus of logs and exponentials
f´(ex) = ex
∫ex = ex + k
f´(ln(x) = 1/x
∫(1/x) dx = ln(x) + k
6. Trigonometry
cosecθ = 1/sinθ
secθ = 1/cosθ
cotθ = 1/tanθ
These can be used with sin2θ + cos2θ = 1 to produce two new identities viz:
sec2θ = 1 + tan2θ and cosec2θ = 1 + cot2θ
You must know where to find the sum and difference angles in the data book and either learn or very quickly derive from them the double angle formula:
sin2θ = 2sinθcosθ
cos2θ = 2cos2θ –1 = 1 – 2sin2θ = cos2θ – sin2θ
tan2θ = 2tanθ / (1 – tan2θ)
Examiners are getting very keen on expressing asinθ + bcosθ in the form Rsin(θ+ α) and the like. Use the sum angle formula, then because it is an identity you will get a pair of equations for Rsinα and Rcosα. Square and add to get R. Divide to get tanα
MEI use arcsin etc rather than sin-1
7. The Modulus Function
x = x for x >= 0
x = -x for x < 0
To sketch y = f(x)
Sketch y = f(x) and reflect the part of the graph below the x axis in the x axis.
a x b = a x b
Similarly for division.
The distance between a and b on the number line is b – a
If a >= 0 and x – k <= a then k – a <= x <= k + a
Equations and inequalities involving the modulus function should be solved using sketches – much easier than critical values.
Finally x = a implies x2 = a2 and the reverse
x > a implies x2 > a2 and the reverse
and if a is not = 0
x < a implies x2 < a2 and the reverse
8. Numerical Solution of Equations
If f(x) is continuous between x = p and x = q then if f(p) and f(q) have opposite signs there is at least one root between q and p. Really you ought to sketch the function and make a comment about being continuous and it being obvious that there is only one root.
To find the root more accurately use linear interpolation. If f(p) = P and f(q) = Q and p < q then a better estimate of the root is given by x = p + P / ( P + Q ). This follows from similar triangles. This is a better way to do it. Don’t try to remember the formula.
Roots can also be found by iteration. If the sequence given by xr+1 = f(xr) with some initial value x0 converges to a limit l then l is a root of the equation x = f(x). You will be given an equation g(x) = 0 that you can easily put in the form x = f(x). Use this as the iteration formula. In fact you have to start with a value not too far from the root and the slope of the f(x) must be between about + and – 1. You only have to know that it might not converge.
9. The Chain Rule
If y = gf(x) so that u = f(x) and y = g(u) then dy/dx = (dy/du) . (du/dx)
This allows many functions to be differentiated by substitution.
It also allows rates of change problems to be solved e.g for a circle A = π r2
So dA/dt = dA/dr . dr/dt = 2πr dr/dt
This leads to the rule for differentiating a function of a function
y = f(g(x)) dy/dx = f′(g(x)).g′(x)
Which encompasses the rules earlier relating to functions of ax + b
10. Differentiating a product or a quotient
f´(uv) = u f´(v) + v f´(u)
and
f´(u/v) = (v f´(u) – u f´(v)) / v2
11. Volumes of Revolution
Volumes of revolution about the axes:
About the x axis = π ∫ y2dx with the relevant limits. Normally y = f(x) is given
About the y axis = π ∫ x2dy with the relevant limits. Normally x = f(y) is given
If the area is defined by the space between curves of revolution then replace f2(x) by (f(x) - g(x))2 and similarly for the volume about the y axis.
12. Simpson’s Rule
This is a more sophisticated version of the trapezium rule. Instead of a trapezium the shape is approximated by part of a parabola. The shape is divided into an even number of equal strips and then
For the simple case of just two strips the area is h/3 (ends + 4 x middles) where h is the strip width.
In the general case the area is h/3 (ends + 4 x odds + 2 x evens)
Remember there must be an even number of strips.
That’s it folks
Good luck
Richard Vincent
16th September 2005
Revised 11th June 2009