2024年4月1日发(作者：僧琬琰)
C1 D
IFFERENTIATION
Worksheet C
1
Find the gradient at the point with x-coordinate 3 on each of the following curves.
a y = x
3
b y = 4x − x
2
c y = 2x
2
− 8x + 3 d y = + 2
3
x
2
Find the gradient of each curve at the given point.
a y = 3x
2
+ x − 5
c y = x(2x − 3)
e y = x
2
+ 6x + 8
(1, −1)
(2, 2)
(−3, −1)
b y = x
4
+ 2x
3
f y = 4x + x
−2
(−2, 0)
(
1
, 6)
2
d y = x
2
− 2x
−1
(2, 3)
3 Evaluate f ′(4) when
4
a f(x) = (x + 1)
2
b f(x) = x
2
1
c f(x) = x − 4x
−2
d f(x) = 5 − 6x
2
3
The curve with equation y = x
3
− 4x
2
+ 3x crosses the x-axis at the points A, B and C.
a Find the coordinates of the points A, B and C.
b Find the gradient of the curve at each of the points A, B and C.
5
For the curve with equation y = 2x
2
− 5x + 1,
dy
a find ,
dx
b find the value of x for which
dy
= 7.
dx
6
7
Find the coordinates of the points on the curve with the equation y = x
3
− 8x at which the
gradient of the curve is 4.
A curve has the equation y = x
3
+ x
2
− 4x + 1.
a Find the gradient of the curve at the point P (−1, 5).
Given that the gradient at the point Q on the curve is the same as the gradient at the point P,
b find, as exact fractions, the coordinates of the point Q.
8
Find an equation of the tangent to each curve at the given point.
a y = x
2
(2, 4) b y = x
2
+ 3x + 4
c y = 2x
2
− 6x + 8 (1, 4) d y = x
3
− 4x
2
+ 2
(−1, 2)
(3, −7)
9 Find an equation of the tangent to each curve at the given point. Give your answers in the form
ax + by + c = 0, where a, b and c are integers.
a y = 3 − x
2
c y = 2x
2
+ 5x − 1
(−3, −6)
(
1
, 2)
2
b y = (2, 1)
d y = x − 3x (4, −2)
2
x
10
Find an equation of the normal to each curve at the given point. Give your answers in the form
ax + by + c = 0, where a, b and c are integers.
a y = x
2
− 4
c y = x
3
− 8x + 4
(1, −3)
(2, −4)
b y = 3x
2
+ 7x + 7
6
x
(−2, 5)
d y = x − (3, 1)
11
C1 D
IFFERENTIATION
Find, in the form y = mx + c, an equation of
Worksheet C continued
a the tangent to the curve y = 3x
2
− 5x + 2 at the point on the curve with x-coordinate 2,
b the normal to the curve y = x
3
+ 5x
2
− 12 at the point on the curve with x-coordinate −3.
A curve has the equation y = x
3
+ 3x
2
− 16x + 2.
a Find an equation of the tangent to the curve at the point P (2, −10).
12
The tangent to the curve at the point Q is parallel to the tangent at the point P.
b Find the coordinates of the point Q.
13
A curve has the equation y = x
2
− 3x + 4.
a Find an equation of the normal to the curve at the point A (2, 2).
The normal to the curve at A intersects the curve again at the point B.
b Find the coordinates of the point B.
14
f(x) ≡ x
3
+ 4x
2
− 18.
a Find f ′(x).
b Show that the tangent to the curve y = f(x) at the point on the curve with x-coordinate −3
passes through the origin.
15 The curve C has the equation y = 6 + x − x
2
.
a Find the coordinates of the point P, where C crosses the positive x-axis, and the point Q,
where C crosses the y-axis.
b Find an equation of the tangent to C at P.
c Find the coordinates of the point where the tangent to C at P meets the tangent to C at Q.
16
The straight line l is a tangent to the curve y = x
2
− 5x + 3 at the point A on the curve.
a find the coordinates of the point A,
Given that l is parallel to the line 3x + y = 0,
b find the equation of the line l in the form y = mx + c.
17
18
The line with equation y = 2x + k is a normal to the curve with equation y =
Find the value of the constant k.
16
x
2
.
A ball is thrown vertically downwards from the top of a cliff. The distance, s metres, of the ball
from the top of the cliff after t seconds is given by s = 3t + 5t
2
.
Find the rate at which the distance the ball has travelled is increasing when
a t = 0.6,
b s = 54.
19 Water is poured into a vase such that the depth, h cm, of the water in the vase after t seconds is
given by h = kt
3
, where k is a constant. Given that when t = 1, the depth of the water in the vase
is increasing at the rate of 3 cm per second,
1
a find the value of k,
b find the rate at which h is increasing when t = 8.
2024年4月1日发(作者：僧琬琰)
C1 D
IFFERENTIATION
Worksheet C
1
Find the gradient at the point with x-coordinate 3 on each of the following curves.
a y = x
3
b y = 4x − x
2
c y = 2x
2
− 8x + 3 d y = + 2
3
x
2
Find the gradient of each curve at the given point.
a y = 3x
2
+ x − 5
c y = x(2x − 3)
e y = x
2
+ 6x + 8
(1, −1)
(2, 2)
(−3, −1)
b y = x
4
+ 2x
3
f y = 4x + x
−2
(−2, 0)
(
1
, 6)
2
d y = x
2
− 2x
−1
(2, 3)
3 Evaluate f ′(4) when
4
a f(x) = (x + 1)
2
b f(x) = x
2
1
c f(x) = x − 4x
−2
d f(x) = 5 − 6x
2
3
The curve with equation y = x
3
− 4x
2
+ 3x crosses the x-axis at the points A, B and C.
a Find the coordinates of the points A, B and C.
b Find the gradient of the curve at each of the points A, B and C.
5
For the curve with equation y = 2x
2
− 5x + 1,
dy
a find ,
dx
b find the value of x for which
dy
= 7.
dx
6
7
Find the coordinates of the points on the curve with the equation y = x
3
− 8x at which the
gradient of the curve is 4.
A curve has the equation y = x
3
+ x
2
− 4x + 1.
a Find the gradient of the curve at the point P (−1, 5).
Given that the gradient at the point Q on the curve is the same as the gradient at the point P,
b find, as exact fractions, the coordinates of the point Q.
8
Find an equation of the tangent to each curve at the given point.
a y = x
2
(2, 4) b y = x
2
+ 3x + 4
c y = 2x
2
− 6x + 8 (1, 4) d y = x
3
− 4x
2
+ 2
(−1, 2)
(3, −7)
9 Find an equation of the tangent to each curve at the given point. Give your answers in the form
ax + by + c = 0, where a, b and c are integers.
a y = 3 − x
2
c y = 2x
2
+ 5x − 1
(−3, −6)
(
1
, 2)
2
b y = (2, 1)
d y = x − 3x (4, −2)
2
x
10
Find an equation of the normal to each curve at the given point. Give your answers in the form
ax + by + c = 0, where a, b and c are integers.
a y = x
2
− 4
c y = x
3
− 8x + 4
(1, −3)
(2, −4)
b y = 3x
2
+ 7x + 7
6
x
(−2, 5)
d y = x − (3, 1)
11
C1 D
IFFERENTIATION
Find, in the form y = mx + c, an equation of
Worksheet C continued
a the tangent to the curve y = 3x
2
− 5x + 2 at the point on the curve with x-coordinate 2,
b the normal to the curve y = x
3
+ 5x
2
− 12 at the point on the curve with x-coordinate −3.
A curve has the equation y = x
3
+ 3x
2
− 16x + 2.
a Find an equation of the tangent to the curve at the point P (2, −10).
12
The tangent to the curve at the point Q is parallel to the tangent at the point P.
b Find the coordinates of the point Q.
13
A curve has the equation y = x
2
− 3x + 4.
a Find an equation of the normal to the curve at the point A (2, 2).
The normal to the curve at A intersects the curve again at the point B.
b Find the coordinates of the point B.
14
f(x) ≡ x
3
+ 4x
2
− 18.
a Find f ′(x).
b Show that the tangent to the curve y = f(x) at the point on the curve with x-coordinate −3
passes through the origin.
15 The curve C has the equation y = 6 + x − x
2
.
a Find the coordinates of the point P, where C crosses the positive x-axis, and the point Q,
where C crosses the y-axis.
b Find an equation of the tangent to C at P.
c Find the coordinates of the point where the tangent to C at P meets the tangent to C at Q.
16
The straight line l is a tangent to the curve y = x
2
− 5x + 3 at the point A on the curve.
a find the coordinates of the point A,
Given that l is parallel to the line 3x + y = 0,
b find the equation of the line l in the form y = mx + c.
17
18
The line with equation y = 2x + k is a normal to the curve with equation y =
Find the value of the constant k.
16
x
2
.
A ball is thrown vertically downwards from the top of a cliff. The distance, s metres, of the ball
from the top of the cliff after t seconds is given by s = 3t + 5t
2
.
Find the rate at which the distance the ball has travelled is increasing when
a t = 0.6,
b s = 54.
19 Water is poured into a vase such that the depth, h cm, of the water in the vase after t seconds is
given by h = kt
3
, where k is a constant. Given that when t = 1, the depth of the water in the vase
is increasing at the rate of 3 cm per second,
1
a find the value of k,
b find the rate at which h is increasing when t = 8.