Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation P1^ 5 6  Given that y = x^3 + 4 x (i) find d d y x (ii) show that y = x^3 + 4 x is an increasing function for al ...

P1^ 5 Points of inflection 153 13  (i) Find the co-ordinates of the stationary points of the curve f(x) = 4x +^1 x . (ii) Find t ...

Differentiation P1^ 5 This is an example of a point of inflection. In general, a point of inflection occurs where the tangent to ...

P1^ 5 The (^) second (^) derivative ACTIVITY 5.7 Sketch the graph of the gradient of d d y x against x for the function illustra ...

Differentiation 156 P1^ 5 The next example illustrates the use of the second derivative to identify the nature of stationary poi ...

P1^ 5 The (^) second (^) derivative 157 (iv) !^ Remember that you are looking for the value of^ d d 2 2 y x at the stationary po ...

Differentiation P1^ 5 x y y = –x^4 O Figure 5.31 y = –x^4 d d y x = –4x (^3) : at (0, 0) d d y x^ = 0 d d 2 2 y x = –12x (^2) : ...

P1^ 5 Exercise (^) 5F 4  Given that y = (x − 1)^2 (x − 3) (i) multiply out the right-hand side and find d d y x (ii) find the po ...

Differentiation 160 P1^ 5 Applications There are many situations in which you need to find the maximum or minimum value of an ex ...

P1^ 5 Applications EXAMPLE 5.17 A stone is projected vertically upwards with a speed of 30 m s−^1. Its height, h m, above the gr ...

Differentiation P1^ 5 EXERCISE 5G  1  A farmer wants to construct a temporary rectangular enclosure of length x m and width y m ...

P1^ 5 Exercise (^) 5G 163 (i) Write down an equation linking x and y. (ii) Using your answer to part (i), form an expression for ...

Differentiation P1^ 5 7  A piece of wire 30 cm long is going to be made into two frames for blowing bubbles. The wire is to be c ...

Exercise (^) 5G P1^ 5 10  A fish tank with a square base and no top is to be made from a thin sheet of toughened glass. The dime ...

Differentiation P1^ 5 12  A garden is planned with a lawn area of 24 m^2 and a path around the edge. The dimensions of the lawn ...

The chain rule P1^ 5 The chain rule ●?^ Whatinformationisgivenbyd d andd d Whatinforma V h h t ? ttionisgivenbyd d d d V h h t × ...

Differentiation 168 P1^ 5 In the limit, as δx → 0, δ δ δ δ δ δ y x y x y u yu x u →→ → x d d d duand d , d and so the relationsh ...

The (^) chain (^) rule P1^ 5 EXAMPLE 5.19 Use the chain rule to find d d y x when y = (x^2 − 2)^4. SOLUTION Let u = x^2 − 2, the ...

Differentiation P1^ 5 !^ ‘Differentiate y^ =^ x^2 ’ could mean differentiation with respect to x, or t, or any other variable. I ...

Exercise (^) 5H P1^ 5 EXERCISE 5H    1  Use the chain rule to differentiate the following functions. (i) y = (x + 2)^3 (ii) y = ...