HOLIDAY TUITION August 2022 (The Calculus)
Write your Name and other particulars in the answer sheet.
ANSWER ALL THE QUESTIONS (Show your working)
𝑑𝑦
1 Find an expression of 𝑑𝑥 in each of the following functions
2
(a) y = 3x5 – 4x4 + 3x3 – 2x2 + x – 40,
4
1
(b) y = – 3x3 – 3 x3 – 𝑥 ,
[2]
[2]
Find the indefinite integrals of the following functions
(a) y = x2 + 4x - 1,
[2]
(b) y =
𝑥 3 + 2𝑥 2 + 4𝑥
𝑥
∫ (3
𝑥2
.
− 5𝑥 +
3
Determine
4
Find the gradient to the curve
at x = – 2 .
2
[2]
1
𝑥2
) 𝑑𝑥
[2]
2
1
y = 3 x3 – 2 x2 + 4,
[3]
HOLIDAY TUITION August 2022 (The Calculus)
Write your Name and other particulars in the answer sheet.
ANSWER ALL THE QUESTIONS (Show your working)
𝑑𝑦
1 Find an expression of 𝑑𝑥 in each of the following functions
2
(a) y = 3x5 – 4x4 + 3x3 – 2x2 + x – 40,
4
1
(b) y = – 3x3 – 3 x3 – 𝑥 ,
[2]
[2]
Find the indefinite integrals of the following functions
(a) y = x2 + 4x - 1,
[2]
(b) y =
𝑥 3 + 2𝑥 2 + 4𝑥
𝑥
∫ (3
𝑥2
.
− 5𝑥 +
3
Determine
4
Find the gradient to the curve
at x = – 2 .
2
[2]
1
𝑥2
) 𝑑𝑥
[2]
2
1
y = 3 x3 – 2 x2 + 4,
[3]
5
A curve is such that y = 2x3 – 4x + 5. Find the equation of
(a) the tangent to the curve at (–2, –3),
[3]
(b) the normal to the curve at x = –1.
[3]
6
The equation of a curve is
y = x – 6x – 15x + 3. Find
the coordinates of the stationary points.
[4]
7
Evaluate the following definite integrals
3
(a) ∫1 (𝑥 2 + 𝑥 + 14 )𝑑𝑥
3
2
[3]
0
(b) ∫−2(− 4𝑥 2 − 4𝑥 + 10 )𝑑𝑥
[3]
𝑑𝑦
8
Find the equation of the curve whose gradient is 𝑑𝑥 = 2x – 1
and it passes through the point (2, 6).
[4]
5
A curve is such that y = 2x3 – 4x + 5. Find the equation of
(a) the tangent to the curve at (–2, –3),
[3]
(b) the normal to the curve at x = –1.
[3]
6
The equation of a curve is
y = x3 – 6x2 – 15x + 3. Find
the coordinates of the stationary points.
[4]
7
Evaluate the following definite integrals
3
(a) ∫1 (𝑥 2 + 𝑥 + 14 )𝑑𝑥
[3]
0
(b) ∫−2(− 4𝑥 2 − 4𝑥 + 10 )𝑑𝑥
8
[3]
𝑑𝑦
Find the equation of the curve whose gradient is 𝑑𝑥 = 2x – 1
and it passes through the point (2, 6).
[4]
HOLIDAY TUITION August 2022 (The Calculus)
Write your Name and other particulars in the answer sheet.
ANSWER ALL THE QUESTIONS (Show your working)
𝑑𝑦
1 Find an expression of 𝑑𝑥 in each of the following functions
2
(a) y = 3x5 – 4x4 + 3x3 – 2x2 + x – 40,
4
1
(b) y = – 3x3 – 3 x3 – 𝑥 ,
[2]
[2]
Find the indefinite integrals of the following functions
(a) y = x2 + 4x - 1,
[2]
(b) y =
𝑥 3 + 2𝑥 2 + 4𝑥
𝑥
∫ (3
𝑥2
.
− 5𝑥 +
3
Determine
4
Find the gradient to the curve
at x = – 2 .
2
[2]
1
𝑥2
) 𝑑𝑥
[2]
2
1
y = 3 x3 – 2 x2 + 4,
[3]
HOLIDAY TUITION August 2022 (The Calculus)
Write your Name and other particulars in the answer sheet.
ANSWER ALL THE QUESTIONS (Show your working)
𝑑𝑦
1 Find an expression of 𝑑𝑥 in each of the following functions
2
(a) y = 3x5 – 4x4 + 3x3 – 2x2 + x – 40,
4
1
(b) y = – 3x3 – 3 x3 – 𝑥 ,
[2]
[2]
Find the indefinite integrals of the following functions
(a) y = x2 + 4x - 1,
[2]
(b) y =
𝑥 3 + 2𝑥 2 + 4𝑥
𝑥
∫ (3
𝑥2
.
− 5𝑥 +
3
Determine
4
Find the gradient to the curve
at x = – 2 .
2
[2]
1
𝑥2
) 𝑑𝑥
[2]
2
1
y = 3 x3 – 2 x2 + 4,
[3]
5
A curve is such that y = 2x3 – 4x + 5. Find the equation of
(a) the tangent to the curve at (–2, –3),
[3]
(b) the normal to the curve at x = –1.
[3]
6
The equation of a curve is
y = x – 6x – 15x + 3. Find
the coordinates of the stationary points.
[4]
7
Evaluate the following definite integrals
3
(a) ∫1 (𝑥 2 + 𝑥 + 14 )𝑑𝑥
3
2
[3]
0
(b) ∫−2(− 4𝑥 2 − 4𝑥 + 10 )𝑑𝑥
[3]
𝑑𝑦
8
Find the equation of the curve whose gradient is 𝑑𝑥 = 2x – 1
and it passes through the point (2, 6).
[4]
5
A curve is such that y = 2x3 – 4x + 5. Find the equation of
(a) the tangent to the curve at (–2, –3),
[3]
(b) the normal to the curve at x = –1.
[3]
6
The equation of a curve is
y = x3 – 6x2 – 15x + 3. Find
the coordinates of the stationary points.
[4]
7
Evaluate the following definite integrals
3
(a) ∫1 (𝑥 2 + 𝑥 + 14 )𝑑𝑥
[3]
0
(b) ∫−2(− 4𝑥 2 − 4𝑥 + 10 )𝑑𝑥
8
[3]
𝑑𝑦
Find the equation of the curve whose gradient is 𝑑𝑥 = 2x – 1
and it passes through the point (2, 6).
[4]