MCQs in Differential Calculus (Limits and Derivatives) Part II

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.

Multiple Choice Questions Topic Outline

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

Online Questions and Answers in Differential Calculus (Limits and Derivatives) Series

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

Differential Calculus (Limits and Derivatives) MCQs

PART 1: MCQs from Number 1 – 50                        Answer key: PART I

PART 2: MCQs from Number 51 – 100                   Answer key: PART II

Continue Practice Exam Test Questions Part II of the Series

Choose the letter of the best answer in each questions.

51. Evaluate

• A. 0
• B. 1
• C. 2
• D. 3

52. Simplify the expression:

• A. 1
• B. 8
• C. 0
• D. 16

53. Evaluate the following limit:

• A. 2/5
• B. infinity
• C. 0
• D. 5/2

54. Evaluate the limit ( x – 4 ) / (x² – x – 12) as x approaches 4.

• A. 0
• B. undefined
• C. 1/7
• D. infinity

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

• A. 1
• B. 0
• C. e
• D. infinity

56. Evaluate the following limit:

• A. 1
• B. Indefinite
• C. 0
• D. 2

57. Evaluate:

• A. 0
• B. ½
• C. 2
• D. -1/2

58. Evaluate the following:

• A. Infinity
• B. e^π
• C. 0
• D. e^2/π 

59. Find dy/dx if y = 5^2x-1

• A. 5^2x-1 ln 5
• B. 5^2x-1 ln 25
• C. 5^2x-1 ln 10
• D. 5^2x-1 ln 2

60. Find dy/dx if y = e^√x

• A. e^√x  / 2√x
• B. e^√x / √x
• C. e^x  / √x
• D. e^{√x - 2√x}

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

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

• A. (4x + y) / (x + y)
• B. –[(4x + y) / (x + y)]
• C. (4x – y) / (x + y)
• D. –[(4x + y) / (x –y)]

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

• A. DF(x) = (-x² – 10x – 1) / (x² – 1)²
• B. DF(x) = (x² + 10x – 1) / (x² – 1)²
• C. DF(x) = (x² –10x – 1) / (x² – 1)²
• D. DF(x) = (-x² –10x + 1) / (x² – 1)²

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

• A. a x^a-1
• B. (a – 1)x
• C. x^a-1
• D. ax

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

• A. Df(x) = 4x – 8
• B. Df(x) = 2x + 9
• C. Df(x) = 2x + 8
• D. Df(x) = 4x + 8

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

• A. – y(1 + ln xy) / x
• B. 0
• C. – y(1 – ln xy) / x²
• D. y/x

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

68. Given the equation: y = (e ^{ln x})², find y’.

• A. ln x
• B. 2 (ln x) / x
• C. 2x
• D. 2 e ^{ln x}

69. Find the derivatives with respect to x of the function √(2 – 3x²)

• A. -2x² / √(2 – 3x²)
• B. -3x / √(2 – 3x²)
• C. -2x² / √(2 + 3x²)
• D. -3x / √(2 + 3x²)

70. Differentiate ax² + b to the ½ power.

• A. -2ax
• B. 2ax
• C. 2ax + b
• D. ax + 2b

71. Find dy/dx if y = ln √x

• A. √x / ln x
• B. x / ln x
• C. 1 / 2x
• D. 2 / x

72. Evaluate the differential of tan Ѳ.

• A. ln sec Ѳ dѲ
• B. ln cos Ѳ dѲ
• C. sec Ѳ tan Ѳ dѲ
• D. sec² Ѳ dѲ

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

• A. sec x
• B. –sec x
• C. sin x
• D. –sin x

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

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

• A. sec x
• B. –sec x
• C. –tan x
• D. tan x

76. Find the derivative of arcos 4x with respect to x.

• A. -4 / [1 – (4x)^2]^2
• B. -4 / [1 – (4x)]^0.5
• C. 4 / [1 – (4x)^2]^0.5
• D. -4 / [(4x)^2 - 1]^0.5

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

• A. –[3 / (1 + 9x²)]
• B. 3 / (1 + 9x²)
• C. –[3 / √(1 – 9x²)]
• D. 3 / √(1 – 9x²)

78. If y = x (ln x), find d²y / dx².

• A. 1 / x²
• B. -1 / x
• C. 1 / x
• D. -1 / x²

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

• A. 96
• B. 0.375
• C. -0.25
• D. -0.875

80. Given the function f(x) = x³ – 5x + 2, find the value of the first derivative at x = 2, f’ (2).

• A. 7
• B. 3x² – 5
• C. 2
• D. 8

81. Given the function f(x) = x to the 3^rd power – 6x + 2, find the value of the first derivative at x = 2, f’(2)

• A. 6
• B. 3x² – 5
• C. 7
• D. 8

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

• A. y² – 5
• B. xy – 5y
• C. y²
• D. 2xy

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)

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

• A. 1
• B. ½
• C. 1/3
• D. ¼

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

• A. 3/5
• B. -2/5
• C. -3/5
• D. 2/5

86. Find the slope of the tangent to a parabola y = x² at a point on the curve where x = ½.

• A. 0
• B. 1
• C. ¼
• D. -1/2

87. Find the slope of the ellipse x² + 4y² – 10x + 16y + 5 = 0 at the point where y = -2 + 8^0.5 and x = 7.

• A. -0.1654
• B. -0.1538
• C. -0.1768
• D. -0.1463

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

• A. 20
• B. 1/24
• C. 24
• D. 1/20

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

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

• A. -4
• B. ¼
• C. 4
• D. -1/4

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

• A. 2/5
• B. 5/2
• C. -2/5
• D. -5/2

92. Find the slope of the curve y = 6(4 + x) ^½ at (0, 12).

• A. 0.67
• B. 1.5
• C. 1.33
• D. 0.75

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

• A. 77°
• B. 75°
• C. 79°
• D. 120°

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

• A. 74.77°
• B. 4.5°
• C. 47.77°
• D. 77.47°

95. Find the equation of the tangent to the curve y = x + 2x^1/3 through point (8, 12)

• A. 7x – 6y + 14 = 0
• B. 8x + 5y + 21 = 0
• C. 5x – 6y – 15 = 0
• D. 3x – 2y – 1 = 0

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

• A. 6.21
• B. 5.21
• C. 5.66
• D. 6.66

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

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

• A. 24.4
• B. 25.4
• C. 23.4
• D. 22.4

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

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 – 64y = 0
• B. 64x – 5y = 0
• C. 5x + 64y = 0
• D. 64x + 5y = 0

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