1. No category

C1
1
Worksheet G
DIFFERENTIATION
A curve has the equation y = 2 + 3x + kx2 − x3 where k is a constant.
Given that the gradient of the curve is −6 at the point P where x = −1,
a find the value of k.
(4)
Given also that the tangent to the curve at the point Q is parallel to the tangent at P,
b find the length PQ, giving your answer in the form k 5 .
2
3
The point A lies on the curve y =
12
x2
(5)
and the x-coordinate of A is 2.
a Find an equation of the tangent to the curve at A. Give your answer in the form
ax + by + c = 0, where a, b and c are integers.
(5)
b Verify that the point where the tangent at A intersects the curve again has the
coordinates (−1, 12).
(3)
Given that y =
x2 − 6 x − 3
3x
1
2
, show that
dy
( x + a) 2
can be expressed in the form
, where
3
dx
bx 2
a and b are integers to be found.
4
(6)
The curve with equation y = x3 + ax2 − 24x + b, where a and b are constants, passes
through the point P (−2, 30).
a Show that 4a + b + 10 = 0.
(2)
Given also that P is a stationary point of the curve,
5
b find the values of a and b,
(4)
c find the coordinates of the other stationary point on the curve.
(3)
y
1
y = x 2 − 4 + 3x
− 12
O A
B
x
C
1
−1
The diagram shows the curve with equation y = x 2 − 4 + 3x 2 .
The curve crosses the x-axis at the points A and B and has a minimum point at C.
6
a Find the coordinates of A and B.
(5)
b Find the coordinates of C, giving its y-coordinate in the form a 3 + b, where a and b
are integers.
(5)
f(x) ≡ x3 + 4x2 + kx + 1.
a Find the set of values of the constant k for which the curve y = f(x) has two
stationary points.
(5)
Given that k = −3,
b find the coordinates of the stationary points of the curve y = f(x).
 Solomon Press
(4)
C1
7
DIFFERENTIATION
Worksheet G continued
f(x) ≡ 2x3 + 5x2 − 1.
a Find the set of values of x for which f(x) is increasing.
(5)
b Find an equation of the normal to the curve y = f(x) at the point A (−1, 2).
(3)
The normal to the curve y = f(x) at the point B is parallel to the normal at the point A.
c Find the exact coordinates of the point B.
8
(4)
y
l
y = x3 − 3x2 − 8x + 4
P
m
O
x
Q
The diagram shows the curve with equation y = x3 − 3x2 − 8x + 4.
The straight line l is the tangent to the curve at the point P (−1, 8).
a Find an equation of line l.
(4)
The straight line m is parallel to l and is the tangent to the curve at the point Q.
9
b Find an equation of line m.
(4)
c Find an equation of the normal to the curve at P.
(2)
d Hence, or otherwise, show that the distance between lines l and m is 16 2 .
(4)
A curve has the equation y =
x (k − x), where k is a constant.
Given that the gradient of the curve is
2 at the point P where x = 2,
a find the value of k,
(5)
b show that the normal to the curve at P has the equation
x + 2 y = c,
where c is an integer to be found.
10
11
(5)
f(x) = x3 − 3x2 + 4
a Show that (x + 1)(x − 2)2 ≡ x3 − 3x2 + 4.
(2)
b Hence state, with a reason, the coordinates of one of the stationary points of the curve
with equation y = f(x).
(2)
c Using differentiation, find the coordinates of the other stationary point of the curve.
(5)
The curve C has the equation
3
y = 2x − x 2 , x ≥ 0.
a Find the coordinates of any points where C meets the x-axis.
(3)
b Find the x-coordinate of the stationary point on C and determine whether it is a
maximum or a minimum point.
(6)
c Sketch the curve C.
(2)
 Solomon Press