DifferentiationP1^
5
6 Given that y = x^3 + 4 x
(i) find d
dy
x
(ii) show that y = x^3 + 4 x is an increasing function for all values of x.
7 Given that y = x^3 + 3 x^2 − 9 x + 6
(i) find d
dy
x
and factorise the quadratic expression you obtain
(ii) write down the values of x for which d
dy
x= 0
(iii) show that one of the points corresponding to these x values is a
minimum and the other a maximum
(iv) show that the corresponding y values are 1 and 33 respectively
(v) sketch the curve.
8 Given that y = 9 x + 3 x^2 − x^3
(i) find d
dy
x
and factorise the quadratic expression you obtain
(ii) find the values of x for which the curve has stationary points, and classify
these stationary points
(iii) find the corresponding y values
(iv) sketch the curve.
9 (i) Find the co-ordinates and nature of each of the stationary points of
y = x^3 − 2 x^2 − 4 x + 3.
(ii) Sketch the curve.
10 (i) Find the co-ordinates and nature of each of the stationary points of the
curve with equation y = x^4 + 4 x^3 − 36 x^2 + 300.
(ii) Sketch the curve.
11 (i) Differentiate y = x^3 + 3 x.
(ii) What does this tell you about the number of stationary points of the
curve with equation y = x^3 + 3 x?
(iii) Find the values of y corresponding to x = −3, −2, −1, 0, 1, 2 and 3.
(iv) Hence sketch the curve and explain your answer to part (ii).
12 You are given that y = 2 x^3 + 3 x^2 − 72 x + 130.(i) Find d
dy
x.
P is the point on the curve where x = 4.
(ii) Calculate the y co-ordinate of P.
(iii) Calculate the gradient at P and hence find the equation of the tangent to
the curve at P.
(iv) There are two stationary points on the curve. Find their co-ordinates.
[MEI]