MT Chapter 08

LEE KIAN KEONG STPM MATHEMATICS (T) 8: Differentiation

8 Differentiation

1. [STPM ]Differentiate with respect to x and simplify your answer as far as possible:

(a)x2 x+ 1x2 + x 1 ,

(b) e2x[2 cos(3x) 3 sin(3x)].

[Answer : (a)2x(x 2)

(x2 + x 1)2 ; (b) 13e2x cos(3x)]

2. [STPM ]Differentiate with respect to x and simplify your answer as far as possible:

(a)cosx+ sinx

cosx sinx ,(b) xn loge x.

[Answer : (a)2

1 sin 2x ; (b) xn1(1 + n lnx)]

3. [STPM ]

(a) Finddy

dxwhere

i. y = (2x 1)3(3x+ 2)4 and express your result in the form of its factors,ii. y = e3x(2 cos 2x+ 3 sin 2x).

(b) If x = t 1t

and y = 2t+1

t, where t is a non-zero parameter, prove that

dy

dx= 2 3

t2 + 1.

Deduce that 1 < dydx

< 2.

[Answer : (a) (i) 42x(2x 1)2(3x+ 2)3 ; (ii) 13e2x sin 2x]

4. [STPM ]

Find the x-coordinate of the point on the curve y =lnx

x2(x > 0) such that

dy

dx= 0, and determine if it is a

maximum or minimum point. Sketch the curve for x > 0. You can assume that y 0 when x.

[Answer : x =e]

5. [STPM ]

(a) Differentiate x tanx+ 13

tan3 x with respect to x, and express your answer in terms of tanx.

(b) Given y = aemx cos px, prove thatd2y

dx2+ 2m

dy

dx+ (m2 + p2)y = 0.

(c) Given y = ln(1 + x) x+ 12x2, show that

dy

dx 0 for all values of x > 1.

[Answer : (a) tan4 x]

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6. [STPM ]

Given that y =cosx sinxcosx+ sinx

, show thatd2y

dx2+ 2y

dy

dx= 0.

7. [STPM ]A curve with the equation y = ax4 + bx3 + cx2 + dx + e, where a, b, c, d, e are constants, has the followingcharacteristics:

(a) It is symmetrical about the y-axis,

(b) It passes through the point (2,18) and has gradient zero at this point,(c) y = 0 when x = 1.

Show that b = d = 0 and find the values of a, c and e. Sketch the curve and give the coordinates of its turningpoints.

[Answer : a = 2, c = 16, e = 14]

8. [STPM ]

(a) Given that y = (x+ 2)2(3x 1)3, find dydx

as a product of its factors.

(b) If y =ex

1 + x2, show that (1 + x2)

dy

dx+ (1 + x)2y = 0.

[Answer : (a) (x+ 2)(15x+ 16)(3x 1)2]

9. [STPM ]The parametric equation of a curve are

x = a cos3 , y = a sin3

where a is a positive constant and 0 < 2pi.Find the equation of the tangent at the point with the parameter . This tangent meets the axes at L and M .Prove that the length of LM is independent of .

[Answer : y cos + x sin = a sin cos , LM = a]

10. [STPM ]A spherical balloon is being inflated at a constant rate of 500 m3 s1. Find the rate of increase in the totalsurface area of the balloon when its radius is 20 m.

[Answer : 50]

11. [STPM ]Given that y = e2x sin(x+ ), where and are constants, verify

d2y

dx2+ 4

dy

dx+ 5y = 0.

12. [STPM ]

(a) For the curve y = sinx cos3 x, where 0 x pi, find the x and y coordinates of the points where dydx

= 0.

Sketch this curve.

(b) For the curve y2 = sinx cos3 x, where 0 x 12pi, show that

(dy

dx

)2=

1

4cotx(cos2 x 3 sin2 x)2 on the

condition that x 6= 0. Sketch the curve.

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[Answer :

(pi

6,

3

3

16

),(pi

2, 0)

,

(5pi

6,3

3

16

)]

13. [STPM ]A curve is given by its parametric equations

x = t2, y = 1 1t, (t > 0).

The curve intersects the x-axis at P . Find the equation of the tangent to the curve at P .

[Answer : 2y x+ 1 = 0]

14. [STPM ]

The parametric equations of a curve are x = t2, y = t3. Expressdy

dxin terms of t. Find the equation of the

tangent to the curve at the point P (p2, p3).

[Answer :dy

dx=

3

2t ; 2y = 3px p3]

15. [STPM ]

If y = 3x+ sinx 8 sin(

1

2x

), find

dy

dxand express your answer in terms of cos

(1

2x

).

Deduce thatdy

dx 0 for all values of x.

[Answer : 2

(cos

1

2x 1

)2]

16. [STPM ]

(a) Find the equation of the asymptotes of the curve y =x 3

(x 2)(x+ 1) .

(b) Find the points where the curve intersects the axes, and find the stationary points on this curve.

(c) Sketch this curve.

(d) Find the values of k such that the equation (x 3) = k(x 2)(x+ 1) does not have real roots.

[Answer : (a) x = 2, x = 1, y = 0 ; (b) (3, 0), (0, 1.5), (1, 1), (5, 19

) ; (d)1

9< k < 1]

17. [STPM ]Find the coordinates of the stationary points of the curve y = x ln(1 + x) and sketch the curve.

[Answer : (0, 0)]

18. [STPM ]

(a) Find the points on the x-axis intersected by the curve y =1

2x3 x2 2x. Find also the maximum and

minimum points, as well as any points of inflection on this curve.

(b) Sketch this curve.

(c) Find the value of k if the equation1

2x3 x2 2x = k has a repeated root, and state this root.

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[Answer : (a) (0, 0), (1 +

5, 0), (1

5, 0) ;

Maximum point (23,

20

27) , Minimum point (2,4) , Point of inflection (2

3,44

27) ;

(c) k = 4, repeated root=2; k = 2027

, repeated root=23

]

19. [STPM ]If y = etan x, show that

d2y

dx2=dy

dx(1 + tanx)2.

20. [STPM ]

Show that the equation of the normal to the curve y = tan 2x at the point where the x-coordinate ispi

3is

3x+ 24y = pi 24

3.

21. [STPM ]The parametric equations of a curve are x = t2 2, y = t3 3. Find the equation of the normal to the curve atthe point where the parameter t = 2.

[Answer : x+ 3y = 17]

22. [STPM ]Differentiate with respect to x

(a) (x2 + 2x)ex2+2x,

(b)1 x21 + 2x

,

simplifying your answers.

[Answer : (a) 2(x+ 1)3ex2+2x ; (b) 3x

2 + 2x+ 1

(1 + 2x)32

]

23. [STPM ]Find the equations of the tangent and normal to the curve x2y + xy2 = 12 at the point (1,4).

[Answer : 8x 7y = 36 ; 7x+ 8y + 25 = 0]

24. [STPM ]A curve has the equation y2 = x2(x+ 3).

Show that the curve is symmetric about the x-axis.

Show that for all points on this curve, x 3.Find coordinates of the turning points of this curve.

Sketch the curve. Show clearly the shape of the curve near the origin.

Find the area bounded by the loop of this curve.

[Answer : (2, 2), (2,2) ; Area=245

3]

25. [STPM ]Two parallel sides of a rectangle respectively lengthen at a rate of 2 cm per second, while the other two parallelsides shorten such that the area of the rectangle is always 50 cm2. If, at the time t, the length of each lengtheningside is x, the length of each shortening side is y, and the perimeter of the rectangle is p, show that

dp

dt= 4

(1 y

x

).

Find the rate of change in the perimeter when

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(a) x=5 cm,

(b) y=5 cm.

Show that the perimeter of the rectangle is the least when x = y = 5

2 cm.

[Answer : (a) 4 ; (b) 2]

26. [STPM ]Differentiate with respect to x

(a) (2x3 + 1)ex2

,

(b) ln(x2ex),

[Answer : (a) 2xex2

(2x3 + 3x+ 1) ; (b)2

x 1]

27. [STPM ]

A curve has the equation y =x2

x2 4 .Write the equations of the asymptotes of this curve.

Find the coordinates of the turning point on this curve, and determine if this is a maximum or minimum point.Determine if there are any points of inflection on this curve.

Sketch this curve.

[Answer : x = 2, x = 2, y = 1 ; (0,0) is a local maximum point ; No points of inflexion]

28. [STPM ]If y = ln(sin px+ cos px), show that

d2y

dx2+

(dy

dx

)2+ p2 = 0.

29. [STPM ]Differentiate each of the following with respect to x.

(a) (x2 + 1)ex,

(b) cos2(x).

[Answer : (a) (x 1)2ex ; (b) sin 2x

2x

]

30. [STPM ]

Finddy

dxif ey =

x 13 x .

Determine the gradient of the curve y = ln

(x 13 x

)at the point where it intersects the x-axis.

[Answer :2

(x 1)(3 x) ; 2]

31. [STPM ]

A curve has the following equation y =1

x2 1x

.

Find the coordinates of the turning point of the curve, and determine if it is a maximum or minimum point.

Sketch this curve.

The tangent to the curve at the point A(1, 0) meets the curve once again at the point B. Find the coordinatesof B.

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[Answer : (2,14

) is minimum point, B = (1, 2)]

32. [STPM ]The equation of a curve is x2y + xy2 = 2. Find the equations of both tangents to the curve at the point x = 1.

[Answer : x+ y = 2 ; y = 2]

33. [STPM ]

If y =

sinx, show that

4y3d2y

dx2+ y4 + 1 = 0.

34. [STPM ]Differentiate each of the following with respect to x.

(a) ex lnx3,

(b)2x

1 + x4.

[Answer : (a)3ex

x(1 x lnx) ; (b) 2

x[(1 + x4) ln 2 4x3](1 + x4)2

]

35. [STPM ]

The variables x and y are connected by yxy x = 1. Find the values of y and dy

dxwhen x = 1.

[Answer : 4 , 43

]

36. [STPM ]The function f is defined by

f(x) = cosx+1

2cos 2x, 0 x 2pi.

(a) Find all values of x in the form of kpi, with k correct to one decimal place when f(x) = 0.

(b) Find all the pairs (x, f(x)) when f (x) = 0.

(c) Sketch the graph f .

(d) State the range of f .

[Answer : (a) 0.4pi, 1.6pi ; (b)

(0,

3

2

),

(pi,1

2

),

(2pi,

3

2

),

(2

3pi,3

4

),

(4

3pi,3

4

); (d) {y : 3

4 y 3

2}]

37. [STPM ]If y2 = ln(x2y) where x, y > 0,

(a) show thatdy

dx=

2y

x(2y2 1) ,

(b) finddy

dxwhen y = 1.

[Answer : (b)2e

]

38. [STPM ]

If x = sin3 2, y = cos3 2, finddy

dxin terms of . [6 marks]

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[Answer : cot 2]

39. [STPM ]

If y = (2ex 6x+ 5) 12 , show thatyd2y

dx2+

(dy

dx

)2= ex.

[4 marks]

40. [STPM ]

Figure above shows a composite solid which consists of a cuboid and a semicylindrical top with a common faceABCD.

The breadth and length of the cuboid is x cm and 2x cm respectively and its height is y cm. Given that thetotal surface area of this solid is 2400 cm2. Show that

y =1

24x[9600 (8 + 5pi)x2].

[3 marks]

If the volume of this solid is V cm3, express V in terms of x. Hence, show that V attains its maximum when

x =404 + pi

. [9 marks]

Find this maximum volume. [3 marks]

[Answer : V =1

12[9600x 2pix3 8x3] , 64000

3pi + 4

]

41. [STPM ]Find the gradient of the curve 2x2 + y2 + 2xy = 5 at the point (2,1). [3 marks]

[Answer : -3]

42. [STPM ]The parametric equations of a curve are

x = sec t tan t; y = cosec t cot t,

with 0 < t 0. [3 marks]

[Answer : Asymptotes are x = 1, x = 2, y = 1 ; 1 root]

44. [STPM ]

Finddy

dxin terms of x if x = e

t and y =

et. [4 marks]

[Answer :e(ln x)2

2

xlnx]

45. [STPM ]

The equation of a curve is y =e2kx 1e2kx + 1

where k is a positive constant.

(a) Show thatdy

dx> 0 for all values of x. [3 marks]

(b) Show thatdy

dx+ ky2 = k. Hence, show that

d2y

dx2 0 for x 0 and d

2y

dx2 0 for x 0. [8 marks]

(c) Sketch the curve. [4 marks]

46. [STPM ]

Given that y =sin kx

1 + cos kx, where k is a positive integer, show that

sin kxd2y

dx2= k2y2.

[6 marks]

47. [STPM ]The graphs of y = x3 + ax2 + bx+ c passes through (3,21) and has stationary points when x = 2 and x = 2.Find the values of a, b and c. [5 marks]

Find the coordinates of these stationary points and determine if they are local extremums. Find also the pointof inflexion of the curve. [7 marks]

Determine the set of x so thatdy

dx< 0. [3 marks]

[Answer : a = 0 , b = 12 , c = 12 ; (2,-28) is local minimum , (-2,4) is local maximum ;point of inflexion is (0,-12) ; {x : 2 < x < 2}]

48. [STPM ]A curve has parametric equations x = e2t 2t and y = et + t. Find the gradient of the curve at the point witht = ln 2. [5 marks]

[Answer :1

2]

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49. [STPM ]A curve with equation y = x3 + px2 + qx+ r cuts the y-axis at y = 34 and has stationary points at x = 3 andx = 5. Find the values of p, q, and r. [6 marks]

Show that the curve cuts the x-axis only at x = 1, and find the gradient of the curve at that point. [7 marks]

Sketch the curve. [2 marks]

[Answer : p = 12 , q = 45 , r = 34 ; 24]

50. [STPM ]Given a curve with parametric equation

x = a(t 3t3), y = 3at2,with a > 0 and t R.Determine the values of t when the curve cuts the y-axis and sketch the curve. [4 marks]

Show that

(dx

dt

)2+

(dy

dt

)2= a2(1 + 9t2)2. [3 marks]

[Answer : t = 0,

3

3,

3

3]

51. [STPM ]Find the equation of the normal to the curve x2y + xy2 = 12 at the point (3, 1). [6 marks]

[Answer : y =15

7x 38

7]

52. [STPM ]

Given that y = ex cosx, finddy

dxand

d2y

dx2when x = 0. [4 marks]

[Answer :dy

dx= 1, d

2y

dx2= 0]

53. [STPM ]Function f if defined by

f(x) =2x

(x+ 1)(x 2) .

Show that f (x) < 0 for all values of x in the domain of f . [5 marks]

Sketch the graph of y = f(x). Determine if f is a one to one function. Give reasons to your answer. [6 marks]

Sketch the graph of y = |f(x)|. Explain how the number of the roots of the equation |f(x)| = k(x 2) dependson k. [4 marks]

[Answer : f is not one to one function. If k 0, 1 root. If k < 0, 3 roots.]

54. [STPM ]

If y = lnxy, find the value of

dy

dxwhen y = 1. [5 marks]

[Answer :1

e2]

55. [STPM ]

A curve is defined by the parametric equations x = 1 2t, y = 2 + 2t. Find the equation of the normal to the

curve at the point A(3,4). [7 marks]The normal of the curve at the point A cuts the curve again at point B. Find the coordinates of B. [4 marks]

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[Answer : x+ y + 1 = 0 ; B(1, 0)]

56. [STPM ]

If y =cosx

x, where x 6= 0, show that xd

2y

dx2+ 2

dy

dx+ xy = 0. [4 marks]

57. [STPM ]

Find the coordinate of the stationary point on the curve y = x2 +1

xwhere x > 0; give the x-coordinate and

y-coordinate correct to three decimal places. Determine whether the stationary point is a minimum point or amaximum point. [5 marks]

[Answer : (0.794 , 1.890) , minimum]

58. [STPM ]If y = x ln(x+ 1), find an approximation for the increase in y when x increases by x.

Hence, estimate the value of ln 2.01 given that ln 2 = 0.6931. [6 marks]

[Answer : 0.698]

59. [STPM ]

The function f is defined by f(t) =4ekt 14ekt + 1

where k is a positive constant, t > 0.

(a) Find the value of f(0). [1 marks]

(b) Show that f (t) > 0. [5 marks]

(c) Show that k[1 f(t)2] = 2f (t) and hence show that f (t) < 0. [6 marks](d) Find lim

t f(t). [2 marks]

(e) Sketch the graph of f . [2 marks]

[Answer : (a)3

5; (d) 1]

60. [STPM ]

If y =x

1 + x2, show that x2

dy

dx= (1 x2)y2. [4 marks]

61. [STPM ]

Find the coordinates of the stationary points on the curve y =x3

x2 1 and determine their nature. [10 marks]Sketch the curve. [4 marks]

Determine the number of real roots of the equation x3 = k(x2 1), where k R, when k varies. [3 marks]

[Answer : (0, 0) is inflexion point , (

3,3

3

2) is local min. , (

3,3

3

2) is local max.

1 real root for 3

3

2< k

3

3

2]

62. [STPM ]

If y =sinx cosxsinx+ cosx

, show thatd2y

dx2= 2y

dy

dx. [6 marks]

63. [STPM ]

Show that the curve y =x

x2 1 is always decreasing. [3 marks]Determine the coordinates of the point of inflexion of the curve, and state the intervals for which the curve isconcave upwards. [5 marks]

Sketch the curve. [3 marks]

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[Answer : (0, 0) ; (1, 0) (1,)]

64. [STPM ]A curve is defined by x = cos (1 + cos ) , y = sin (1 + cos ).

(a) Show that (dx

d

)2+

(dy

d

)2= 2(1 + cos ).

[4 marks]

65. [STPM ]The line y+ x+ 3 = 0 is a tangent to the curve y = px2 + qx, where p 6= 0 at the point x = 1. Find the valuesof p and q. [6 marks]

[Answer : p = 3, q = 5]

66. [STPM ]A curve is defined by the parametric equations

x = t 2t

and y = 2t+1

t

where t 6= 0.

(a) Show thatdy

dx= 2 5

t2 + 2, and hence, deduce that 1

2 0 and y = f(u) is a differentiable function f . If

dy

du=

1u2 1 , show

thatdy

dx= 1. [5 marks]

68. [STPM ]The functions f and g are defined by

f : x x3 3x+ 2, x R.g : x x 1, x R.

(a) Find h(x) = (f g)(x), and determine the coordinates of the stationary points of h. [5 marks](b) Sketch the graph of y = h(x). [2 marks]

(c) On a separate diagram, sketch the graph of y =1

h(x). [3 marks]

Hence, determine the set of values of k such that the equation1

h(x)= k has

i. one root, [1 marks]

ii. two roots, [1 marks]

iii. three roots. [1 marks]

[Answer : (a) h(x) = x3 3x2 + 4 , (0,4) , (2,0) ; (c) (i) {k : k < 0, 0 < k < 14} ; (ii) {k : k = 1

4} ; (iii) {k : k > 1

4}]

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69. [STPM ]Given that y is differentiable and y

x = sinx, where x 6= 0. Using implicit differentiation, show that

x2d2y

dx2+ x

dy

dx+

(x2 1

4

)y = 0.

[6 marks]


f(x) =ln 2x

x2, where x > 0.

(a) State all asymptotes of f . [2 marks]

(b) Find the stationary point of f , and determine its nature. [6 marks]

(c) Obtain the intervals, where

i. f is concave upwards, and

ii. f is concave downwards.

Hence, determine the coordinates of the point of inflexion. [6 marks]

(d) Sketch the graph y = f(x). [2 marks]

[Answer : (a) x = 0, y = 0 ; (b)

(1

2e12 ,

2

e

)is a maximum point.

(c)(i) (1.15,) ; (ii) (0, 1.15) ; (1.15, 0.630)]

71. [STPM ]

Given that y = (2x)2x, finddy

dxin terms of x. [4 marks]

[Answer : (2x)2x(2 + 2 ln(2x))]


f(x) =ex1 + x2

, where x R,

(a) Show that

f (x) =ex(x2 + x+ 1)

(1 + x)32

.

[3 marks]

(b) Show that f is a decreasing function. [4 marks]

(c) Sketch the graph of f . [2 marks]

73. [STPM ]A curve is defined by the parametric equations x = ket cos t and y = ket sin t, where k is a constant.

(a) Show that

(dx

dt

)2+

(dy

dt

)2= 2k2e2t. [4 marks]

74. [STPM ]

Find the equation of the normal to the curve with parametric equations x = 1 2t and y = 2 + 2t

at the point

(3,4). [6 marks]

[Answer : y = x 1]

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75. [STPM ]A right circular cone of height a+ x, where a x a, is inscribed in a sphere of constant radius a, such thatthe vertex and all points on the circumference of the base lie on the surface of the sphere.

(a) Show that the volume V of the cone is given by V =1

3pi(a x)(a+ x)2. [3 marks]

(b) Determine the value of x for which V is maximum and find the maximum value of V . [6 marks]

(c) Sketch the graph of V against x. [2 marks]

(d) Determine the rate at which V changes when x =1

2a if x is increasing at a rate of

1

10a per minute.[4 marks]

[Answer : (b) x =a

3,

32

81pia3 ; (d) 1

40pia3]

76. [STPM ]Find the gradients of the curve y3 +y = x3 +x2 at the points where the curve meets the coordinate axes.[6 marks]

[Answer : 0 , 1]

77. [STPM ]The parametric equations of a curve are x = sin and y = 1 cos . Find the equation of the normal to thecurve at a point with parameter

1

2pi. [7 marks]

[Answer : y = x+ pi2

]

78. [STPM ]A curve is defined implicitly by the equation x2 + xy + y2 = 3.

(a) Show thatdy

dx+

2x+ y

x+ 2y= 0. [3 marks]

(b) Find the gradients of the curve at the points where the curve crosses the x-axis and y-axis. [5 marks]

(c) Show that the coordinates of the stationary points of the curve are (-1,2) and (1, -2). [5 marks]

(d) Sketch the curve. [2 marks]

[Answer : (b) 12

, -2]

79. [STPM ]A rectangle with a width 2x is inscribed in a circle of constant radius r.

(a) Express the area A of the rectangle in terms of x and r. [2 marks]

(b) Show that the rectangle is a square of side r

2 when A has a maximum value. [5 marks]

[Answer : (a) A = 4xr2 x2]

80. [STPM ]The graph of y = 2 cosx+ sin 2x for 0 x 2pi is shown below.

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The points A and C are local extremum points. The points B, D, E and F are points of inflexion.

(a) Determine the coordinates of

i. the points of local extremum. [5 marks]

ii. the points of inflexion. [5 marks]

(b) State the intervals where the graph is concave upward. [1 marks]

(c) Calculate the area of the region bounded by the curve and the x-axis. [4 marks]

[Answer : (a) (i) A =

(pi

6,

3

3

2

), C =

(5pi

6,3

3

2

);

(ii) B =(pi

2, 0)

, D = (3.39,1.45) , E =(

3pi

2, 0

), F = (6.03, 1.45) ; (b)

(pi2, 3.39

)(

3pi

2, 6.03

); (c) 4]

81. [STPM ]

The graph of y =3x 1

(x+ 1)3is shown below.

The graph has a local maximum at A and a point of inflexion at B.

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(a) Write the equations of the asymptotes of the graph. [1 marks]

(b) Determine the coordinates of the points A and B. [9 marks]

Hence, state

i. the set of values of x whendy

dx 0, [1 marks]

ii. the intervals where the graph is concave upward. [1 marks]

(c) Using the above graph of y =3x 1

(x+ 1)3, determine the set of values of k for which the equation 3x 1

k(x+ 1)3 = 0

i. has three distinct real roots, [2 marks]

ii. has only one positive root. [1 marks]

[Answer : (a) x = 1 , y = 0 ; (b) A(

1,1

4

), B

(5

3,

27

128

);

(i) {x : x < 1,1 < x 1} ; (ii) (,1) (

5

3,)

;

(c) (i) {k : 0 < k < 14} ; (ii) {k : 1 < k 0}]

82. [STPM ]The equation of a curve is y = x3e32x.

(a) Find the stationary points on the curve, and determine its nature. [7 marks]

(b) Sketch the curve. [3 marks]

[Answer : (a) (0,0)=point of inflexion,

(3

2,

27

8

)=local maximum]

83. [STPM ]

If y sin1 2x =

1 4x2, show that(1 4x2)dy

dx+ 4xy + 2y2 = 0.

[5 marks]

84. [STPM ]Differentiate with respect to t

(a) (t2 1)et21, [3 marks]

(b) ln

1 +

1

t. [3 marks]

[Answer : (a) 2t3et21 ; (b) 1

2(t2 + 1)]

85. [STPM ]For the graph of y = 3x4 + 16x3 + 24x2 6,

(a) determine the intervals on which the graph is concave upward and concave downward, [6 marks]

(b) find the points of inflexion, [3 marks]

(c) determine the extremum point and its nature. [3 marks]

Hence, sketch the graph of y = 3x4 + 16x3 + 24x2 6. [3 marks]

[Answer : (a) (,2) (2

3,)

,

(2,2

3

); (b)

(2

3,

14

27

)and (2, 10) ; (c) (0,6) minimum]

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86. [STPM ]A continuous function f is defined by

f(x) =

x3 1, 1 x < 2,1

2(x 3)2 + c, 2 x 8,

where c is a constant.

(a) Determine the value of c. [4 marks]

(b) Find the values of x such that f(x) = 0. [3 marks]

(c) Sketch the graph of f . [3 marks]

(d) Find the maximum and minimum values of f . [3 marks]

(e) State whether f is a one-to-one function or not. Give a reason for your answer. [2 marks]

[Answer : (a) c =15

2; (b) x = 1, 3 +

15 ; (d) max=

15

2, min=-5]

87. [STPM ]The equation of a curve is y = x(x 2)3.

(a) Find the set of values of x for which y 0. [3 marks](b) Determine the extremum point and the points of inflexion on the curve. [9 marks]

(c) Sketch the curve. [3 marks]

[Answer : (a) {x : x 0, x 2} ; (b) Extremum=(

1

2,27

16

), Inflexion=(1,1), (2, 0)]

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1 Functions2 Sequences and Series3 Matrices4 Complex Numbers5 Analytic Geometry6 Vectors7 Limits and Continuity8 Differentiation9 Integration10 Differential Equations11 Maclaurin Series12 Numerical Methods13 Data Description14 Probability15 Probability Distributions16 Sampling and Estimation17 Hypothesis Testing18 Chi-squared Tests

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MT Chapter 08