P1^
6
Exercise(^) 6A
EXERCISE 6A 1 Given that d
d
y
x
(^) = 6 x^2 + 5
(i) find the general solution of the differential equation
(ii) find the equation of the curve with gradient function d
d
y
x
(^) and which passes
through (1, 9)
(iii) hence show that (−1, −5) also lies on the curve.
2 The gradient function of a curve is d
d
y
x
(^) = 4 x and the curve passes through the
point (1, 5).
(i) Find the equation of the curve.
(ii) Find the value of y when x = −1.
3 The curve C passes through the point (2, 10) and its gradient at any point is
given by d
d
y
x
(^) = 6 x^2.
(i) Find the equation of the curve C.
(ii) Show that the point (1, −4) lies on the curve.
4 A stone is thrown upwards out of a window, and the rate of change of its
height (h metres) is given by d
d
h
t
(^) = 15 − 10 t where t is the time (in seconds).
When t = 0, h = 20.
(i) Show that the solution of the differential equation, under the given
conditions, is h = 20 + 15 t − 5 t^2.
(ii) For what value of t does h = 0? (Assume t 0.)
5 (i) Find the general solution of the differential equation d
d
y
x
(^) = 5.
(ii) Find the particular solution which passes through the point (1, 8).
(iii) Sketch the graph of this particular solution.
6 The gradient function of a curve is 3 x^2 − 3. The curve has two stationary
points. One is a maximum with a y value of 5 and the other is a minimum
with a y value of 1.
(i) Find the value of x at each stationary point. Make it clear in your solution
how you know which corresponds to the maximum and which to the
minimum.
(ii) Use the gradient function and one of your points from part (i) to find the
equation of the curve.
(iii) Sketch the curve.