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Higher Mathematics
Differentiation Past Papers Unit 1 Outcome 3
√
1. Differentiate 2 3 x with respect to x .
√
A. 6 x
√ 4
B. 23 3 x
C.
4
− √ 2
33x
D.
2
√
3 2
3 x
Key
D
Outcome
1.3
2
Grade
C
Facility
0.83
Disc.
0.38
Calculator
NC
Content
C2, C3
Source
HSN 091
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xy
2. Given f (x) = 3x 2 (2x − 1), find f 0 (−1).
[SQA]
3
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3. Find the coordinates of the point on the curve y = 2x 2 − 7x + 10 where the tangent
to the curve makes an angle of 45 ◦ with the positive direction of the x -axis.
[SQA]
Part
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•1
•2
•3
•4
Marks
4
sp:
pd:
ss:
pd:
Level
C
Calc.
NC
Content
G2, C4
know to diff., and differentiate
process gradient from angle
equate equivalent expressions
solve and complete
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Page 1
Answer
(2, 4)
U1 OC3
2002 P1 Q4
dy
•1 dx = 4x − 7
•2 mtang = tan 45◦ = 1
•3 4x − 7 = 1
•4 (2, 4)
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Higher Mathematics
4. If y = x2 − x , show that
[SQA]
2y
dy
= 1+
.
dx
x
3
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x−1
5. Find f 0 (4) where f (x) = √ .
x
[SQA]
5
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6. Find
[SQA]
√
dy
4
where y = 2 + x x .
dx
x
4
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Higher Mathematics
7. If f (x) = kx3 + 5x − 1 and f 0 (1) = 14, find the value of k.
[SQA]
3
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8. Find the x -coordinate of each of the points on the curve y = 2x 3 − 3x2 − 12x + 20
at which the tangent is parallel to the x -axis.
[SQA]
4
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9. Calculate, to the nearest degree, the angle between the x -axis and the tangent to
the curve with equation y = x3 − 4x − 5 at the point where x = 2.
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Higher Mathematics
10. The point P(−1, 7) lies on the curve with equation y = 5x 2 + 2. Find the equation
of the tangent to the curve at P.
[SQA]
4
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11. Find the equation of the tangent to the curve y = 4x 3 − 2 at the point where
x = −1.
[SQA]
4
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12. Find the equation of the tangent to the curve with equation y = 5x 3 − 6x2 at the
point where x = 1.
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Higher Mathematics
13. Find the equation of the tangent to the curve y = 3x 3 + 2 at the point where x = 1.
[SQA]
4
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14. The point P(−2, b) lies on the graph of the function f (x) = 3x 3 − x2 − 7x + 4.
[SQA]
(a) Find the value of b .
1
(b) Prove that this function is increasing at P.
3
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15. For what values of x is the function f (x) = 31 x3 − 2x2 − 5x − 4 increasing?
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dy
dy
16. Given that y = 2x 2 + x , find
and hence show that x 1 +
= 2y.
dx
dx
[SQA]
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17.
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Higher Mathematics
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18. The graph of a function f intersects the
x -axis at (−a, 0) and (e, 0) as shown.
[SQA]
y
There is a point of inflexion at (0, b) and a
maximum turning point at (c, d).
Sketch the graph of the derived function f 0 .
Part
Marks
3
•1 ic:
•2 ic:
•3 ic:
Level
C
Calc.
CN
Content
A3, C11
(c, d)
(0, b)
(−a, 0)
Answer
sketch
3
O
(e, 0) x
y = f (x)
U1 OC3
2002 P1 Q6
•1 roots at 0 and c (accept a statement to
this effect)
2
• min. at LH root, max. between roots
•3 both ‘tails’ correct
interpret stationary points
interpret main body of f
interpret tails of f
19.
[SQA]
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Higher Mathematics
20. A function f is defined by the formula f (x) = (x − 1) 2 (x + 2) where x ∈ R.
[SQA]
(a) Find the coordinates of the points where the curve with equation y = f (x)
crosses the x - and y-axes.
3
(b) Find the stationary points of this curve y = f (x) and determine their nature.
7
(c) Sketch the curve y = f (x).
2
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21.
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22.
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Higher Mathematics
23. A company spends x thousand
P
pounds a year on advertising
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and this results in a profit of P
thousand pounds. A mathematical
model , illustrated in the diagram,
y
suggests that P and x are related by
P = 12x3 − x4 for 0 ≤ x ≤ 12.
O
[SQA]
Find the value of x which gives the
maximum profit.
Part
•1
•2
•3
•4
•5
Marks
5
ss:
pd:
ss:
pd:
ic:
Level
C
Calc.
NC
Content
C11
(12, 0) x
5
Answer
x=9
U1 OC3
2001 P1 Q6
dP
2
3
•1 dP
dx = 36x . . . or dx = . . . − 4x
2
3
•2 dP
dx = 36x − 4x
•3 dP
dx = 0
•4 x = 0 and x = 9
•5 nature table about x = 0 and x = 9
start diff. process
process
set derivative to zero
process
interpret solutions
24.
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Higher Mathematics
25. A ball is thrown vertically upwards. The height h metres of the ball t seconds after
it is thrown, is given by the formula h = 20t − 5t2 .
[SQA]
(a) Find the speed of the ball when it is thrown (i.e. the rate of change of height
with respect to time of the ball when it is thrown).
3
(b) Find the speed of the ball after 2 seconds.
Explain your answer in terms of the movement of the ball.
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Higher Mathematics
26. A sketch of the graph of y = f (x) where f (x) = x 3 − 6x2 + 9x is shown below.
[SQA]
The graph has a maximum at A and a minimum at B(3, 0).
y
A
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y = f (x)
B(3, 0)
x
4
(a) Find the coordinates of the turning point at A.
(b) Hence sketch the graph of y = g(x) where g(x) = f (x + 2) + 4.
Indicate the coordinates of the turning points. There is no need to calculate
the coordinates of the points of intersection with the axes.
(c) Write down the range of values of k for which g(x) = k has 3 real roots.
Part
(a)
(b)
Marks
4
2
Level
C
C
Calc.
NC
NC
(c)
1
A/B
NC
•1
•2
•3
•4
ss:
pd:
ss:
pd:
Content
C8
A3
A2
interpret transformation
interpret transformation
•7 ic:
interpret sketch
U1 OC3
2000 P1 Q2
dy
know to differentiate
differentiate correctly
know gradient = 0
process
•5 ic:
•6 ic:
Answer
A(1, 4)
sketch (translate 4 up, 2
left)
4<k<8
•1 dx = . . .
dy
•2 dx = 3x2 − 12x + 9
3
• 3x2 − 12x + 9 = 0
•4 A = (1, 4)
translate f (x) 4 units up, 2 units left
•5 sketch with coord. of A 0 (−1, 8)
•6 sketch with coord. of B 0 (1, 4)
•7 4 < k < 8 (accept 4 ≤ k ≤ 8)
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1
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Higher Mathematics
27. A curve has equation y = x 4 − 4x3 + 3.
[SQA]
(a) Find algebraically the coordinates of the stationary points.
6
(b) Determine the nature of the stationary points.
2
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28. A curve has equation y = 2x 3 + 3x2 + 4x − 5.
[SQA]
5
Prove that this curve has no stationary points.
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Higher Mathematics
29. Find the values of x for which the function f (x) = 2x 3 − 3x2 − 36x is increasing.
[SQA]
4
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16
30. A curve has equation y = x − √ , x > 0.
x
[SQA]
6
Find the equation of the tangent at the point where x = 4.
Part
•1
•2
•3
•4
•5
•6
Marks
6
ic:
ss:
ss:
pd:
ss:
ic:
Level
C
Calc.
CN
Content
C4, C5
find corresponding y-coord.
express in standard form
start to differentiate
diff. fractional negative power
find gradient of tangent
write down equ. of tangent
Answer
y = 2x − 12
•1
•2
•3
•4
•5
•6
U1 OC3
2001 P2 Q2
(4, −4) stated or implied by •6
1
−16x − 2
dy
dx = 1 . . .
3
. . . + 8x − 2
m x=4 = 2
y − (−4) = 2(x − 4)
31. A ball is thrown vertically upwards.
[SQA]
After t seconds its height is h metres, where h = 1·2 + 19·6t − 4·9t 2 .
(a) Find the speed of the ball after 1 second.
3
(b) For how many seconds is the ball travelling upwards?
2
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32.
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33.
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34.
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35.
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36.
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37.
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Higher Mathematics
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38. A goldsmith has built up a solid which consists of a triangular
prism of fixed volume with a regular tetrahedron at each end.
x
The surface area, A, of the solid is given by
√ 3 3
16
2
A(x) =
x +
2
x
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where x is the length of each edge of the tetrahedron.
O
Find the value of x which the goldsmith should use to
y
minimise the amount of gold plating required to cover the
solid.
Part
•1
•2
•3
•4
•5
•6
Marks
6
ss:
pd:
ss:
pd:
pd:
ic:
Level
A/B
Calc.
CN
Content
C11
Answer
x=2
•1
•2
•3
•4
•5
•6
know to differentiate
process
know to set f 0 (x) = 0
deal with x −2
process
check for minimum
6
U1 OC3
2000 P2 Q6
A√0 (x) = . . .
√
√ −2
3 3
−2
2 (2x − 16x ) or 3 3x − 24 3x
A0 (x) = 0 √
− 16
or − 24x2 3
x2
x=2
x
2− 2 2+
0
A (x) −ve 0 +ve
so x = 2 is min.
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39.
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40.
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41.
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42.
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43. The shaded rectangle on this map
O
represents the planned extension to the
x
y
village hall. It is hoped to provide the
largest possible area for the extension.
The Vennel
[SQA]
Village hall
6m
8m
Manse Lane
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The coordinate diagram represents the
(0, 6)
right angled triangle of ground behind
the hall. The extension has length l
metres and breadth b metres, as shown.
One corner of the extension is at the point
(a, 0).
l
O
b
(a, 0)
(8, 0)
x
(a) (i) Show that l = 45 a.
(ii) Express b in terms of a and hence deduce that the area, A m 2 , of the
extension is given by A = 34 a(8 − a).
4
(b) Find the value of a which produces the largest area of the extension.
Part
(a)
(b)
Marks
3
4
Level
A/B
A/B
Calc.
CN
CN
•1 ss:
select strategy
through
•2 ss:
select strategy
through
•3 ic: complete proof
•4
•5
•6
•7
Content
0.1
C11
and
carry
and
carry
ss: know to set derivative to zero
pd: differentiate
pd: solve equation
ic: justify maximum, e.g. nature
table
Answer
proof
a=4
U1 OC3
2002 P2 Q10
•1 proof of l = 54 a
•2 b = 35 (8 − a)
•3 complete proof leading to A = . . .
•4 dA
da = . . . = 0
5
• 6 − 32 a
•6 a = 4
•7 e.g. nature table, comp. the square
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44.
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45.
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46.
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[END OF QUESTIONS]
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