Cambridge International AS and A Level Mathematics Pure Mathematics 1

Differentiation

P1^

5

10  In his book Mathematician’s Delight, W.W. Sawyer observes that the arch of

Victoria Falls Bridge appears to agree with the curve

y=^11612021 x

–^2

taking the origin as the point mid-way between the feet of the arch, and

taking the distance between its feet as 4.7 units.

(i) Find d

d

y

x

.

(ii) Evaluate d

d

y

x

when x = −2.35 and when x = 2.35.

(iii) Find the value of x for which d

d

y

x

= −0.5.

11  (i) Use your knowledge of the shape of the curve y = x^1 to sketch the curve

y = (^1) x + 2.
(ii) Write down the co-ordinates of the point where the curve crosses the x
axis.
(iii) Differentiate y =^1 x + 2.
(iv) Find the gradient of the curve at the point where it crosses the x axis.
12  The sketch shows the graph of y = (^) x^42 + x.
x
y
–2.35 O +2.35
x
y
O