Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

146

P1^

5

Find

(a) the x co-ordinates of A and B,

(b) the equation of the tangent to the curve at B,

(c) the acute angle, in degrees correct to 1 decimal place, between this

tangent and the line 2y = x + 5.

(ii) Determine the set of values of k for which the line 2y = x + k does not

intersect the curve y = x^2 – 4x + 7.

[Cambridge AS & A Level Mathematics 9709, Paper 12 Q10 November 2009]

20  The equation of a curve is y = 5 –^8

x

(i) Show that the equation of the normal to the curve at the point P(2, 1) is

2 y + x = 4.

This normal meets the curve again at the point Q.

(ii) Find the co-ordinates of Q.

(iii) Find the length of PQ.

[Cambridge AS & A Level Mathematics 9709, Paper 1 Q8 November 2008]

Maximum and minimum points

ACTIVITY 5.6 Plot the graph of y = x^4 − x^3 − 2 x^2 , taking values of x from −2.5 to +2.5 in steps of

0.5, and answer these questions.

(i) How many stationary points has the graph?

(ii) What is the gradient at a stationary point?

(iii) One of the stationary points is a maximum and the others are minima.

Which are of each type?

(iv) Is the maximum the highest point of the graph?

(v) Do the two minima occur exactly at the points you plotted?

(vi) Estimate the lowest value that y takes.

Gradient at a maximum or minimum point

Figure 5.14 shows the graph of y = −x^2 + 16. It has a maximum point at (0, 16).

y

O x



–4 4

Figure 5.14