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# MCQ in Differential Calculus (Limits and Derivatives) Part 2 | Math Board Exam

This is the Multiple Choice Questions Part 2 of the Series in Differential Calculus (Limits and Derivatives) topic in Engineering Mathematics. In Preparation for the ECE Board Exam make sure to expose yourself and familiarize in each and every questions compiled here taken from various sources including but not limited to past Board Examination Questions in Engineering Mathematics, Mathematics Books, Journals and other Mathematics References.

#### MCQ Topic Outline included in Mathematics Board Exam Syllabi

• MCQ in Derivatives | MCQ in Derivatives of Algebraic functions | MCQ in Derivatives of Exponential functions | MCQ in Derivatives of Logarithmic functions | MCQ in Derivatives of Trigonometric functions | MCQ in Derivatives of Inverse Trigonometric functions | MCQ in Derivatives of Hyperbolic functions

#### Continue Practice Exam Test Questions Part 2 of the Series

Choose the letter of the best answer in each questions.

51. Evaluate

A. 0

B. 1

C. 2

D. 3

Solution:

52. Simplify the expression:

A. 1

B. 8

C. 0

D. 16

Solution:

53. Evaluate the following limit:

A. 2/5

B. infinity

C. 0

D. 5/2

Solution:

54. Evaluate the limit ( x โ 4 ) / (x2 โ x โ 12) as x approaches 4.

A. 0

B. undefined

C. 1/7

D. infinity

Solution:

55. Evaluate the limit (1n x ) / x as x approaches positive infinity.

A. 1

B. 0

C. e

D. infinity

Solution:

56. Evaluate the following limit:

A. 1

B. Indefinite

C. 0

D. 2

Solution:

57. Evaluate:

A. 0

B. ยฝ

C. 2

D. -1/2

Solution:

58. Evaluate the following:

A. Infinity

B. eฯ

C. 0

D. e2/ฯ

Solution:

59. Find dy/dx if y = 52x – 1

A. 52x – 1 ln 5

B. 52x – 1 ln 25

C. 52x – 1 ln 10

D. 52x – 1 ln 2

Solution:

60. Find dy/dx if y = eโx

A. eโxย  / 2โx

B. eโx / โx

C. exย  / โx

D. eโx – 2โx

Solution:

61. Find dy/dx if y = x2 + 3x + 1 and x = t2 + 2.

A. 4t3 + 14t2

B. t3 + 4t

C. 4t3 + 14t

D. 4t3 + t

Solution:

62. Evaluate the first derivative of the implicit function: 4x2 + 2xy + y2 = 0

A. (4x + y) / (x + y)

B. โ[(4x + y) / (x + y)]

C. (4x โ y) / (x + y)

D. โ[(4x + y) / (x โy)]

Solution:

63. Find the derivative of (x + 5) / (x2 โ 1) with respect to x.

A. DF(x) = (-x2 โ 10x โ 1) / (x2 โ 1)2

B. DF(x) = (x2 + 10x โ 1) / (x2 โ 1)2

C. DF(x) = (x2 โ10x โ 1) / (x2 โ 1)2

D. DF(x) = (-x2 โ10x + 1) / (x2 โ 1)2

Solution:

64. If a simple constant, what is the derivative of y = xa?

A. a xa – 1

B. (a โ 1)x

C. xa – 1

D. ax

Solution:

65. Find the derivative of the function 2x2 + 8x + 9 with respect to x.

A. Df(x) = 4x โ 8

B. Df(x) = 2x + 9

C. Df(x) = 2x + 8

D. Df(x) = 4x + 8

Solution:

66. What is the first derivative dy/dx of the expression (xy)x = e?

A. โ y(1 + ln xy) / x

B. 0

C. โ y(1 โ ln xy) / x2

D. y/x

Solution:

67. find the derivative of (x + 1)3 / x

Solution:

68. Given the equation: y = (e ln x)2, find yโ.

A. ln x

B. 2 (ln x) / x

C. 2x

D. 2 e ln x

Solution:

69. Find the derivatives with respect to x of the function โ(2 โ 3x2)

A. -2x2 / โ(2 โ 3x2)

B. -3x / โ(2 โ 3x2)

C. -2x2 / โ(2 + 3x2)

D. -3x / โ(2 + 3x2)

Solution:

70. Differentiate ax2 + b to the ยฝ power.

A. -2ax

B. 2ax

C. 2ax + b

D. ax + 2b

Solution:

71. Find dy/dx if y = ln โx

A. โx / ln x

B. x / ln x

C. 1 / 2x

D. 2 / x

Solution:

72. Evaluate the differential of tan ัฒ.

A. ln sec ัฒ dัฒ

B. ln cos ัฒ dัฒ

C. sec ัฒ tan ัฒ dัฒ

D. sec2 ัฒ dัฒ

Solution:

73. If y = cos x, what is dy/dx?

A. sec x

B. โsec x

C. sin x

D. โsin x

Solution:

74. Find dy/dx: y = sin (ln x2).

A. 2 cos (ln x2)

B. 2 cos (ln x2) / x

C. 2x cos (ln x2)

D. 2 cos (ln x2) / x2

Solution:

75. The derivative of ln (cos x) is:

A. sec x

B. โsec x

C. โtan x

D. tan x

Solution:

76. Find the derivative of arc cos 4x with respect to x.

A. -4 / [1 โ (4x)^2]^2

B. -4 / [1 โ (4x)]^0.5

C. 4 / [1 โ (4x)^2]^0.5

D. -4 / [(4x)^2 – 1]^0.5

Solution:

77. What is the first derivative of y = arc sin 3x.

A. โ[3 / (1 + 9x2)]

B. 3 / (1 + 9x2)

C. โ[3 / โ(1 โ 9x2)]

D. 3 / โ(1 โ 9x2)

Solution:

78. If y = x (ln x), find d2y / dx2.

A. 1 / x2

B. -1 / x

C. 1 / x

D. -1 / x2

Solution:

79. Find the second derivative of y = x – 2 at x = 2.

A. 96

B. 0.375

C. -0.25

D. -0.875

Solution:

80. Given the function f(x) = x3 โ 5x + 2, find the value of the first derivative at x = 2, fโ (2).

A. 7

B. 3x2 โ 5

C. 2

D. 8

Solution:

81. Given the function f(x) = x to the 3rd power โ 6x + 2, find the value of the first derivative at x = 2, fโ(2)

A. 6

B. 3x2 โ 5

C. 7

D. 8

Solution:

82. Find the partial derivatives with respect to x of the function: xy2 โ 5y + 6.

A. y2 โ 5

B. xy โ 5y

C. y2

D. 2xy

Solution:

83. Find the point in the parabola y2 = 4x at which the rate of change of the ordinate and abscissa are equal.

A. (1, 2)

B. (2, 1)

C. (4, 4)

D. (-1, 4)

Solution:

84. Find the slope of the line tangent to the curve y = x3 โ 2x + 1 at x = 1.

A. 1

B. ยฝ

C. 1/3

D. ยผ

Solution:

85. Determine the slope of the curve x2 + y2 โ 6x โ 4y โ 21 = 0 at (0, 7).

A. 3/5

B. -2/5

C. -3/5

D. 2/5

Solution:

86. Find the slope of the tangent to a parabola y = x2 at a point on the curve where x = ยฝ.

A. 0

B. 1

C. ยผ

D. -1/2

Solution:

87. Find the slope of the ellipse x2 + 4y2 โ 10x + 16y + 5 = 0 at the point where y = -2 + 80.5 and x = 7.

A. -0.1654

B. -0.1538

C. -0.1768

D. -0.1463

Solution:

88. Find the slope of the tangent to the curve y = x4 โ 2x2 + 8 through point (2, 16).

A. 20

B. 1/24

C. 24

D. 1/20

Solution:

89. Find the slope of the tangent to the curve y2 = 3x2 + 4 through point (-2, 4)

A. -3/2

B. 3/2

C. 2/3

D. -2/3

Solution:

90. Find the slope of the line whose parametric equations are x = 4t + 6 and y = t โ 1.

A. -4

B. ยผ

C. 4

D. -1/4

Solution:

91. What is the slope of the curve x2 + y2 โ 6x + 10y + 5 = 0 at (1, 0).

A. 2/5

B. 5/2

C. -2/5

D. -5/2

Solution:

92. Find the slope of the curve y = 6(4 + x)ยฝ at (0, 12).

A. 0.67

B. 1.5

C. 1.33

D. 0.75

Solution:

93. Find the acute angle that the curve y = 1 โ 3x2 cut the x-axis.

A. 77ยฐ

B. 75ยฐ

C. 79ยฐ

D. 120ยฐ

Solution:

94. Find the angle that the line 2y โ 9x โ 18 = 0 makes with the x-axis.

A. 74.77ยฐ

B. 4.5ยฐ

C. 47.77ยฐ

D. 77.47ยฐ

Solution:

95. Find the equation of the tangent to the curve y = x + 2x1/3 through point (8, 12)

A. 7x โ 6y + 14 = 0

B. 8x + 5y + 21 = 0

C. 5x โ 6y โ 15 = 0

D. 3x โ 2y โ 1 = 0

Solution:

96. What is the radius of curvature at point (1, 2) of the curve 4x โ y2 = 0?

A. 6.21

B. 5.21

C. 5.66

D. 6.66

Solution:

97. Find the radius of curvature at any point of the curve y + ln (cos x) = 0.

A. cos x

B. 1.5707

C. sec x

D. 1

Solution:

98. Determine the radius of curvature at (4, 4) of the curve y2 โ 4x = 0.

A. 24.4

B. 25.4

C. 23.4

D. 22.4

Solution:

99. Find the radius of curvature of the curve x = y3 at (1, 1)

A. 4.72

B. 3.28

C. 4.67

D. 5.27

Solution:

100. The chords of the ellipse 64x2 + 25y2 = 1600 having equal slopes of 1/5 are bisected by its diameter. Determine the equation of the diameter of the ellipse.

A. 5x โ 64y = 0

B. 64x โ 5y = 0

C. 5x + 64y = 0

D. 64x + 5y = 0

Solution:

### Online Question and Answer in Differential Calculus (Limits and Derivatives) Series

Following is the list of multiple choice questions in this brand new series:

MCQ in Differential Calculus (Limits and Derivatives)
PART 1: MCQ from Number 1 โ 50                                Answer key: PART 1
PART 2: MCQ from Number 51 โ 100                           Answer key: PART 2

### Online Question and Answer in Differential Calculus (Maxima/Minima and Time Rates) Series

Following is the list of multiple choice questions in this brand new series:

MCQ in Differential Calculus (Maxima/Minima and Time Rates)
PART 1: MCQ from Number 1 โ 50                             Answer key: PART 1
PART 1: MCQ from Number 2 โ 100                           Answer key: PART 2

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