E
T
x
P
i Q es i n
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P3
Pas P
Differentiation
r
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3.
The point P lies on the curve with equation y = ln ⎛⎜ 1 x ⎞⎟ . The x-coordinate of P is 3.
⎝3 ⎠
Find an equation of the normal to the curve at the point P in the form y = ax + b, where
a and b are constants.
(5)
1
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4.
(a) Differentiate with respect to x
(i) x2e3x + 2,
(4)
3
(ii) cos(2 x ) .
3x
(b) Given that x = 4 sin(2y + 6), find
(4)
dy
in terms of x.
dx
(5)
2
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2.
Differentiate, with respect to x,
(a) e3x + ln 2x,
(3)
3
2
(b) (5 + x ) 2 .
(3)
(Total 6 marks)
3
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3.
The curve C has equation
x = 2 sin y.
(a) Show that the point P
(b) Show that
,
π
4
lies on C.
(1)
dy
1
at P.
=
dx
(4)
(c) Find an equation of the normal to C at P. Give your answer in the form y = mx + c,
where m and c are exact constants.
(4)
4
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4.
(i) The curve C has equation
y=
x
.
9 + x2
Use calculus to find the coordinates of the turning points of C.
(6)
(ii) Given that
3
y = (1 + e 2 x ) 2 ,
find the value of
dy
1
at x = ln 3 .
dx
2
(5)
5
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3.
A curve C has equation
y = x2ex.
(a) Find
dy
, using the product rule for differentiation.
dx
(3)
(b) Hence find the coordinates of the turning points of C.
(3)
(c) Find
d2 y
.
dx 2
(2)
(d) Determine the nature of each turning point of the curve C.
(2)
6
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2.
A curve C has equation
y = e 2 x tan x,
x ≠ (2n + 1)
π
.
2
(a) Show that the turning points on C occur where tan x = −1 .
(6)
(b) Find an equation of the tangent to C at the point where x = 0 .
(2)
7
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7.
A curve C has equation
y = 3 sin 2 x + 4 cos 2 x, - π
x
π.
The point A(0, 4) lies on C.
(a) Find an equation of the normal to the curve C at A.
(5)
(b) Express y in the form
Give the value of
R sin(2 x + ), where R > 0 and 0
π
.
2
to 3 significant figures.
(4)
(c) Find the coordinates of the points of intersection of the curve C with the x-axis.
Give your answers to 2 decimal places.
(4)
8
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1.
The point P lies on the curve with equation
y = 4e 2 x +1.
The y-coordinate of P is 8.
(a) Find, in terms of ln 2, the x-coordinate of P.
(2)
(b) Find the equation of the tangent to the curve at the point P in the form y = ax + b,
where a and b are exact constants to be found.
(4)
9
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6.
(a) Differentiate with respect to x,
(i) e3 x (sin x + 2 cos x) ,
(3)
(ii)
(3)
Given that y =
3x 2 + 6 x − 7
,
( x + 1) 2
(b) show that
dy
20
=
.
dx ( x + 1)3
x ≠ −1 ,
(5)
2
(c) Hence find
2
d y
d y
15
and the real values of x for which 2 = − .
2
dx
dx
4
(3)
10
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1.
(a) Find the value of
dy
at the point where x = 2 on the curve with equation
dx
y = x2 (5x – 1).
(6)
(b) Differentiate
sin 2 x
with respect to x.
x2
(4)
11
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4.
Find the equation of the tangent to the curve x = cos(2 y + π ) at 0,
π
.
4
Give your answer in the form y = ax + b, where a and b are constants to be found.
(6)
12
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2.
A curve C has equation
y=
3
,
(5 − 3 x) 2
x≠
5
3
The point P on C has x-coordinate 2. Find an equation of the normal to C at P in the form
ax + by + c = 0, where a, b and c are integers.
(7)
13
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7.
The curve C has equation
y=
3 + sin 2 x
2 + cos 2 x
(a) Show that
dy 6sin 2 x + 4 cos 2 x + 2
=
2
dx
( 2 + cos 2 x )
(4)
π
.
2
Write your answer in the form y = ax + b, where a and b are exact constants.
(b) Find an equation of the tangent to C at the point on C where x =
(4)
14
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8.
(a) Given that
d
( cos x ) = − sin x
dx
show that
d
( sec x ) = sec x tan x .
dx
(3)
Given that
x = sec 2 y
(b) find
dx
in terms of y.
dy
(2)
(c) Hence find
dy
in terms of x.
dx
(4)
15
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1.
Differentiate with respect to x
(
(a) ln x 2 + 3 x + 5
(b)
)
(2)
cos x
x2
(3)
16
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7.
f ( x) =
4x − 5
− 22 x ,
(2 x + 1)( x − 3) x − 9
f ( x) =
5
(2 x + 1)( x + 3)
x ≠ ±3, x ≠ − 1
2
(a) Show that
(5)
5⎞
⎛
The curve C has equation y = f (x). The point P ⎜ −1, − ⎟ lies on C.
⎝
2⎠
(b) Find an equation of the normal to C at P.
(8)
17
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1.
Differentiate with respect to x, giving your answer in its simplest form,
(a) x 2 ln ( 3 x )
(4)
(b)
sin 4 x
x3
(5)
18
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4.
π
π⎞
⎛
The point P is the point on the curve x = 2 tan ⎜ y + ⎟ with y-coordinate .
⎝
12 ⎠
4
Find an equation of the normal to the curve at P.
(7)
19
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3.
y
P
C
x
O
Figure 1
Figure 1 shows a sketch of the curve C which has equation
y = e x˲3 sin 3 x ,
− π ! x! π
3
3
(a) Find the x coordinate of the turning point P on C, for which x ! 0
Give your answer as a multiple of ".
(6)
(b) Find an equation of the normal to C at the point where x = 0
(3)
20
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7.
(a) Differentiate with respect to x,
1
(i)
x 2 ln(3 x)
(ii)
1 − 10 x
, giving your answer in its simplest form.
(2 x − 1)5
(b) Given that x = 3 tan 2 y find
dy
in terms of x.
dx
21
(6)
(5)
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1.
The curve C has equation
y = (2 x − 3)5
The point P lies on C and has coordinates (w, – 32).
Find
(a) the value of w,
(2)
(b) the equation of the tangent to C at the point P in the form y = mx + c , where m and
c are constants.
(5)
22
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5.
(i) Differentiate with respect to x
(a) y = x3 ln 2 x
(b) y = ( x + sin 2 x)3
(6)
Given that x = cot y ,
(ii) show that
dy
= −1
dx 1 + x 2
(5)
23
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5.
Given that
x = sec2 3y,
(a) find
0<y<
π
6
dx
in terms of y.
dy
(2)
(b) Hence show that
dy
1
=
1
dx 6 x( x − 1) 2
(4)
2
(c) Find an expression for
d y
in terms of x. Give your answer in its simplest form.
dx 2
(4)
24