Page 43 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation

14 Example of HOTS
HOTS Question

Given a curve y = 5x – 3x – 2. Find the coordinates
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of the point on the curve such that the gradient of the A research done on the movement of an object
found out that it follows the function f(x) = ax + bx
3
2
tangent at the point is 7. Hence, find the equation of + c such that a, b, and c are constants. Given that
the tangent. the curve of the function f(x) passes through points
Solution (–1, 0) and (0, 5) when the graph of the function f(x)
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2
y = 5x – 3x – 2 y = 5x – 3x –2 is drawn on the Cartesian plane. The tangents to the
2
curve f(x) at the points where x = 0 and x = 1 are
dy = 10x – 3 (x y ) parallel to the x-axis. Determine the function f(x) by
dx 1, 1 finding the values of a, b and c.
7 = 10x – 3 m = 7
10x = 10 Solution
x = 1 Given the point (–1, 0), 2
3

f(x) = ax + bx + c
Therefore, y = 5(1) – 3(1) – 2 0 = a(–1) + b(–1) + c
2
3
2
= 0 0 = –a + b + c …… 1
The coordinate of the point is (1, 0). For the point (0, 5),
3
2
Equation of tangent, 5 = a(0) + b(0) + c
c = 5
y – y = m (x – x ) Therefore, 0 = a + b + 5
1
1
1
y – 0 = 7(x – 1) a – b = 5 …… 2
y = 7x – 7
f(x) = ax + bx + c
3
2
SPM Tips f'(x) = 3ax + 2bx
2
For x = 1,
Make a rough sketch for easier understanding. 0 = 3a(1) + 2b(1)
2
3a + 2b = 0 …… 3
2 × 3, 3a – 3b = 15 …… 4
15 3 – 4, 5b = –15
b = –3
Given that the gradient of the curve y = hx + kx at the From 2, a – (–3) = 5
2
point (2, 8) is 6. Find the values of h and k. a + 3 = 5
a = 2
Solution \ f(x) = 2x – 3x + 5
2
3
Given the gradient = 6,
y = hx + kx
2
dy = 2hx + k REMEMBER!
dx
6 = 2h(2) + k The gradient of the tangent line parallel
4h + k = 6 …… 1 to the x-axis is 0.
Given the point (2, 8),
y = hx + kx Try this HOTS Question
2
Form 5
8 = h(2) + k(2)
2
8 = 4h + 2k …… 2 The motion of an object follows the equation
y = x – 6x + 5. The tangent to the curve at a point
2
2 – 1: k = 2 P on the curve is parallel to the straight line which
From 1, 4h + 2 = 6 joins the point A(1, 0) and B(7, 12). Given that the
4h = 4 normal line on the point P intersects the curve at
h = 1 two points. Find the equation of the normal at point
P and the coordinates of the points of intersection
SPM Tips between the normal and the curve.
1
The gradient of the curve refers to the gradient Answer: y = – x – 1; 1 3 , – 7 2
of the tangent. 2 2 4
Try Questions 2 – 5 in ‘Try This! 2.4’
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