Page 51 - Focus SPM 2022 - Additional Mathematics

Additional Mathematics SPM Chapter 2 Differentiation
9. The sum of two positive numbers is 38. Determine
(b) dL = 3.4 – 3 = 0.4 the values of these two numbers such that the sum
When t = 1, x = 3 + 6(1) = 9 of the squares of these two numbers is minimum.
dL ≈ dL
dx dx 10. It is estimated that the cost RMK, to drive your
2 – 1 ≈ 0.4 own car from town A to town B follows the formula
3 dx 2 1
dx ≈ 1.2 K = 108V + V , such that V is the average speed
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in km h . Find the value of V if the cost is minimum.
–1
11. The diagram below shows a cuboid with length 3x
Try Questions 19 – 20 in ‘Try This! 2.4’ cm, width x cm and height h cm. Given that the sum
of the surface area of the cuboid is 96 cm .
2
Try This! 2.4
h cm
1. Find the gradient of the tangent to the curve
y = 2x – 5x at the point where the x-coordinate is 3x cm
2
3
–1. x cm
2. Find the equation of the tangent to the curve (a) Express the volume of the cuboid, V cm , in
3
1
y = x – 3 x + 2x at the point 1, 3 2 . Hence, find the terms of x.
2
3
2 2 (b) Hence, find the maximum volume of the cuboid.
coordinates of the other point such that the tangent to
the curve at the point is parallel to the tangent at the 12. Nizam plans to construct a closed cylinder with
1
point 1, 3 2 . radius r cm from a piece of metal with an area of
2
540 cm .
2
(a) Show that the height of the cylinder, h cm, is
3. Given that tangent to the curve y = 4x + px + q at 2
2
the point (–1, 10) is perpendicular to the straight line given by h = 270 – pr .
3y = x – 5. Find the values of p and q. pr
(b) Hence, find the values of r and h such that the
4. Find the equation of the normal to the curve 4y = x volume of the cylinder is maximum. Hence find
2
at the point (4, 4) and hence, find the coordinates of this maximum volume. [Use p = 3.142]
the point such that this normal intersects the curve
once again. 13. Given y = 8x – 5 x + 1 , find the rate of change
3
2
3 2
5
–1
5. M is a point on the curve y = such that the gradient of y when x increases with a rate of 3 unit s when
x
x = 1.
of the normal at point M is 5. The tangent and normal
at point M meet the y-axis at A and B respectively. If 14. Two positive variables h and k are related by
the x-coordinate of point M is positive, find k – 4
2
–1
(a) the coordinates of the midpoint of line AB, h = k . If k changes at a rate of 0.4 unit s , find
(b) the length of line AB. the rate of change of h when h = 3.
6. Find the turning points for y = x – 6x + 9x + 4 and
3
2
dy 15. A piece of ice cube with sides 3 cm melts at a rate of
determine the nature of each point by using and 0.03 cm s . Find the rate of change of its volume.
–1
d y . dx
2
Form 5
dx 2
16. A piece of metal in the shape of a sphere is heated
3 –1
7. Given that the graph of the function f(x) = x + h 2 has and expands at a rate of 0.6 cm s . Find
a turning point (3, q). x (a) the rate of change of its radius, and
(a) Find the values of h and q. (b) the rate of change of its surface area
3
(b) Determine the nature of this turning point. when its volume is 36p cm .
8. The curve y = x + px + qx + 1 has a turning point 17. Given y = 4 , find
2
3
(–1, 5). 2 – x
(a) Determine the values of p and q. (a) the small change in y when x increases from
(b) Find the other turning point and hence 1 to 1.05,
determine the nature of these two turning (b) the small change in x when y decreases from
2
points by using d y . 8 to 7.96.
dx 2
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