Cambridge International AS and A Level Mathematics Pure Mathematics 1

Functions P1^ 4 Composite functions It is possible to combine functions in several different ways, and you have already met some ...

Composite functions 113 P1^ 4 In this case the composite function would be (to the nearest degree) C  34°C too cold 35°C  C  ...

Functions 114 P1^ 4 Notice the range of f must be completely contained within the domain of g. If this wasn’t the case you would ...

Inverse functions P1^ 4 Inverse functions Look at the mapping x  x + 2 with domain the set of integers. Domain Range ... ... .. ...

Functions 116 P1^ 4 (i) Temperature measured in Celsius  temperature measured in Fahrenheit. (ii) Marks in an examination  gra ...

Inverse functions P1^ 4 It is often helpful to define a function with a restricted domain so that its inverse is also a function ...

Functions 118 P1^ 4 This result can be used to obtain a sketch of the inverse function without having to find its equation, prov ...

Inverse (^) functions P1^ 4 ExaMPlE 4.5 Find f−^1 (x) when f(x) = 2 x − 3 and the domain of f is x  4. SOlUTION Domain Range Fu ...

Functions P1^ 4 The full definition of the inverse function is therefore: f−^1 (x) = x−^2 for x  2. The function and its invers ...

Exercise (^) 4B P1^ 4 6  (i) Show that x^2 + 4 x + 7 = (x + 2)^2 + a, where a is to be determined. (ii) Sketch the graph of y = ...

Functions P1^ 4 11  The function f is defined by f : x  2 x^2 – 8x + 11 for x ∈. (i)  Express f(x) in the form a(x + b)^2 + c, ...

P1^ 5 The gradient of a curve 123 Differentiation Hold infinity in the palm of your hand. William Blake This picture illustrates ...

Differentiation 124 P1^ 5 One method of finding the gradient of a curve is shown for point A in figure 5.2. ACTIVITY 5.1  Find t ...

P1^ 5 Finding the gradient of a curve 125 You have already seen that drawing the tangent at the point by hand provides only an a ...

Differentiation 126 P1^ 5 ACTIVITY 5.2  Take points X, Y, Z on the curve y = x^2 with x co-ordinates 3.1, 3.01 and 3.001 respect ...

P1^ 5 Finding the gradient from first principles ! Figure 5.6 shows Q in a position where h is positive, but negative values of ...

Differentiation P1^ 5 EXAMPLE 5.1 Find the gradient of the curve y = x^3 at the general point (x, y). SOLUTION Let P have the ge ...

P1^ 5 Exercise (^) 5A 129 EXERCISE 5A  1  Use the method in Example 5.1 to prove that the gradient of the curve y = x^2 at the p ...

Differentiation P1^ 5 Lim δ δ y x is written as d d y x . δx → 0 2 Using this notation, Wallis’s rule becomes y = xn ⇒ d d y x = ...

P1^ 5 Differentiating by using standard results Historical note The notation ddyx was first used by the German mathematician and ...