DifferentiationP1^
5
EXERCISE 5D 1 The graph of y = 6 x − x^2 is shown below.The marked point, P, is (1, 5).
(i) Find the gradient function
d
dy
x.
(ii) Find the gradient of the curve at P.
(iii) Find the equation of the tangent at P.
2 (i) Sketch the curve y = 4 x − x^2.
(i) Differentiate y = 4 x − x^2.
(iii) Find the gradient of y = 4 x − x^2 at the point (1, 3).
(iv) Find the equation of the tangent to the curve y = 4 x − x^2 at the point (1, 3).
3 (i) Differentiate y = x^3 − 4 x^2.
(ii) Find the gradient of y = x^3 − 4 x^2 at the point (2, −8).
(iii) Find the equation of the tangent to the curve y = x^3 − 4 x^2 at the point
(2, −8).
(iv) Find the co-ordinates of the other point at which this tangent meets the
curve.
4 (i) Sketch the curve y = 6 − x^2.
(ii) Find the gradient of the curve at the points (−1, 5) and (1, 5).
(iii) Find the equations of the tangents to the curve at these points.
(iv) Find the co-ordinates of the point of intersection of these two tangents.
5 (i) Sketch the curve y = x^2 + 4 and the straight line y = 4 x on the same axes.
(ii) Show that both y = x^2 + 4 and y = 4 x pass through the point (2, 8).
(iii) Show that y = x^2 + 4 and y = 4 x have the same gradient at (2, 8), and state
what you conclude from this result and that in part (ii).
6 (i) Find the equation of the tangent to the curve y = 2 x^3 − 15 x^2 + 42 x at (2, 40).
(ii) Using your expression for d
dy
x
, find the co-ordinates of another point on
the curve at which the tangent is parallel to the one at (2, 40).
(iii) Find the equation of the normal at this point.yx5 Py = 6x – x^2O 1 6