5.2 MVT & Rolle's Theorem Video Notes Review Average Rate of Change and Instantaneous Rate of Change (Day 1) Nov 24; Video Notes Rolle's Theorem (Day 1. 4.1-4.3 Quiz Left, Right, Midpoint, or Trapezoid Area Approximation, Indefinite Integrals(Including Trig), Particular Solutions, Basic Definite Integrals, Problems 73 from Section 4.4, Given v(t) and an Initial Value, then Find s(2) and a(6). The Area A of the region bounded by the curves y = f(x), y = g(x) and the lines x = a, x = b, where f and g are continuous f(x) ≥ g(x) for all x in a, b is. The following diagrams illustrate area under a curve and area between two curves. Scroll down the page for examples and solutions.
(Last Updated On: January 6, 2021)Part A: 30 questions; 60 minutes (calculator not permitted). Part B: 15 questions; 45 minutes (graphing calculator required). Questions include algebraic, exponential, logarithmic, trigonometric, and general types of functions. Questions include analytical, graphical, tabular, and verbal types of representations. Section 2: Free Response.
This is the Multiple Choice Questions Part 1 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
Start Practice Exam Test Questions Part 1 of the Series
Choose the letter of the best answer in each questions.
Problem 1: CE Board November 1997
Evaluate the Limit:
A. 1/5
B. 2/5
C. 3/5
D. 4/5
Answer: Option B
Solution: Review Solution for Number 1
Problem 2: ECE Board April 1998
Evaluate the Limit:
A. undefined
B. 0
C. Infinity
D. 1/7
Answer: Option D
Solution: Review Solution for Number 2
Problem 3: ME Board April 1998
Evaluate the Limit:
A. 0
B. 1
C. 8
D. 16
Answer: Option C
Solution: Review Solution for Number 3
Problem 4: ECE Board April 1993
Evaluate the Limit:
A. 0
B. 2
C. 4
D. 6
Answer: Option C
Solution: Review Solution for Number 4
Problem 5: EE Board April 1995
Evaluate the Limit:
A. 0
B. 1/2
C. 2
D. -1/2
Answer: Option B
Solution: Review Solution for Number 5
Problem 6: ME Board October 1997
Compute the following limit:
A. 1
B. 0
C. 2
D. Infinite
Answer: Option A
Solution: Review Solution for Number 6
Problem 7: EE Board October 1994
Evaluate the Limit:
A. Undefined
B. 3/5
C. Infinity
D. Zero
Answer: Option C
Solution: Review Solution for Number 7
Problem 8: ECE Board November 1991
Evaluate the Limit:
A. 24
B. 26
C. 28
D. 30
Answer: Option A
Solution: Review Solution for Number 8
Problem 9: ECE Board November 1994
Evaluate the Limit:
A. e2π
B. e2/π
C. 0
D. α
Answer: Option B
Solution: Review Solution for Number 9
Problem 10: EE Board October 1997
Differentiate y = ex cos x2
A. –ex sin x2
B. ex (cos x2 – 2x sin x2)
C. ex cos x2 – 2x sin x2
D. -2xex sin x
Answer: Option B
Solution: Review Solution for Number 10
Problem 11: EE Board October 1997
Differentiate y = sec (x2 + 2)
A. 2x cos (x2 + 2)
B. –cos (x2 + 2) cot (x2 + 2)
C. 2x sec (x2 + 2) tan (x2 + 2)
D. cos (x2 +2)
Answer: Option C
Solution: Review Solution for Number 11
Problem 12: CE Board October 1994
What is the derivative with respect to x of (x + 1)3 – x3?
A. 3x + 6
B. 3x – 3
C. 6x – 3
D. 6x + 3
Answer: Option D
Solution: Review Solution for Number 12
Problem 13: EE Board October 1997
Differentiate y = log10 (x2 + 1)2
A. 4x (x2 + 1)
B. (4x log10 e) / (x2 + 1)
C. log e(x) (x2 + 1)
D. 2x (x2 + 1)
Answer: Option B
Solution: Review Solution for Number 13
Problem 14: EE Board October 1997
Differentiate (x2 + 2)1/2
A. ((x2 + 2)1/2) / 2
B. x / (x2 + 2)1/2
C. (2x) / (x2 + 2)1/2
D. (x2 + 2)3/2
Answer: Option B
Solution: Review Solution for Number 14
Problem 15: EE Board October 1997
If y = (t2 + 2)2 and t = x1/2, determine dy/dx
A. 3/2
B. (2x2 + 2x) / 3
C. 2(x + 2)
D. x5/2 + x1/2
Answer: Option C
Solution: Review Solution for Number 15
Problem 16: ME Board April 1997
What is the first derivative of the expression (xy)x = e?
A. 0
B. x/y
C. –y [(1 + ln xy) / x)]
D. –y [(1 – ln xy) / x2)]
Answer: Option C
Solution: Review Solution for Number 16
Problem 17: ME Board April 1998
Find the derivative with respect to x function √(2 – 3x2)
A. (-2x2) / √(2 – 3x2)
B. (-3x) / √(2 – 3x2)
C. (-3x2) / √(2 – 3x2)
D. (3x) / √(2 – 3x2)
Answer: Option B
Solution: Review Solution for Number 17
Problem 18: EE Board April 1995
Find y’ if y = arc sin cos x
A. -1
B. -2
C. 1
D. 2
Answer: Option A
Solution: Review Solution for Number 18
Problem 19: CE Board May 1997
Find the derivative of arc cos 4x.
A. -4 / (1 – 16x2)0.5
B. 4 / (1 – 16x2)0.5
C. -4 / (1 – 4x2)0.5
D. 4 / (1 – 4x2)0.5
Answer: Option A
Solution: Review Solution for Number 19
Problem 20: CE Board November 1996
Find the derivative of (x + 1)3 / x
A. ((x + 1)2 / x) – ((x + 1)3 / x)
B. (4(x + 1)2 / x) – (2(x + 1)3 / x)
C. (2(x + 1)3 / x) – ((x + 1)3 / x3)
D. (3(x + 1)2 / x) – ((x + 1)3 / x2)
Answer: Option D
Solution: Review Solution for Number 20
Problem 21: ECE Board November 1991
Differentiate the equation y = x2 / (x + 1)
A. (x2 + 2x) / (x + 1)2
B. x / (x + 1)
C. 2x
D. (2x2) / (x + 1)
Answer: Option A
Solution: Review Solution for Number 21
Problem 22: CE Board November 1995
The derivative with respect to x of 2cos2 (x2 + 2) is
A. 2sin (x2 + 2) cos (x2 + 2)
B. -2sin (x2 + 2) cos (x2 + 2)
C. 8x sin (x2 + 2) cos (x2 + 2)
D. -8x sin (x2 + 2) cos (x2 + 2)
Answer: Option C
Solution: Review Solution for Number 22
Problem 23: CE Board November 1993
Find the second derivative of y by implicit differentiation from the equation 4x2 + 8y2 = 36
A. 64x2
B. (– 9/4) y3
C. 32xy
D. (- 16/9) y3
Answer: Option B
Solution: Review Solution for Number 23
Problem 24: ME Board April 1998
Find the partial derivative with respect to x of the function xy2 – 5y + 6.
A. y2 – 5
B. y2
C. xy – 5y
D. 2xy
Answer: Option B
Solution: Review Solution for Number 24
Problem 25: ME Board October 1997
Find the second derivative of x3 – 5x2 + x = 0
A. 10x – 5
B. 6x – 10
C. 3x + 10
D. 3x2 – 5x
Answer: Option B
Solution: Review Solution for Number 25
Problem 26: ME Board April 1998
Given the function f(x) = x to the 3rd power – 6x + 2. Find the first derivative at x = 2.
A. 6
B. 7
C. 3x2 – 5
D. 8
Answer: Option A
Solution: Review Solution for Number 26
Problem 27: CE Board May 1996
Find the slope of the ellipse x2 + 4y2 – 10x – 16y + 5 = 0 at the point where y = 2 + 80.5 and x = 7.
A. -0.1463
B. -0.1538
C. -0.1654
D. -0.1768
Answer: Option D
Solution: Review Solution for Number 27
Problem 28: EE Board October 1997
If y = 4cos x + sin 2x, what is the slope of the curve when x = 2 radians?
A. -2.21
B. -4.94
C. -3.95
D. 2.21
Answer: Option B
Solution: Review Solution for Number 28
Problem 29: ECE Board November 1991
Find the slope of the line tangent to the curve y = x3 – 2x + 1 at x = 1.
A. 1
B. 1/2
C. 1/3
D. 1/4
Answer: Option A
Solution: Review Solution for Number 29
Problem 30: ECE Board November 1991
Given the slope of the curve at the point (1, 1): y = (x3/4) – 2x + 1
A. 1/4
B. -1/4
C. 1 1/4
D. -1 1/4
Answer: Option D
Solution: Review Solution for Number 30
Problem 31: ECE Board November 1998
Find the slope of x2y = 8 at the point (2, 2)
A. 2
B. -1
C. -1/2
D. -2
Answer: Option D
Solution: Review Solution for Number 31
Problem 32: CE Board May 1998
Find the slope of the curve x2 + y2 – 6x + 10y + 5 + 0 at point (1, 0).
A. 1/5
B. 2/5
C. 1/4
D. 2
Answer: Option B
Solution: Review Solution for Number 32
Problem 33: CE Board May 1996
Find the slope of the tangent to the curve, y = 2x – x2 + x3 at (0, 2).
A. 1
B. 2
C. 3
D. 4
Answer: Option B
Solution: Review Solution for Number 33
Problem 34: ECE Board April 1999
Find the coordinates of the vertex of the parabola y = x2 – 4x + 1 by making use of the fact that at the vertex, the slope of the tangent is zero.
A. (2, -3)
B. (3, -2)
C. (-1, -3)
D. (-2, -3)
Answer: Option A
Solution: Review Solution for Number 34
Problem 35: ECE Board April 1999
Find the equation of the normal to x2 + y2 = 5 at the point (2, 1)
A. y = 2x
B. x = 2y
C. 2x + 3y = 3
D. x + y = 1
Answer: Option B
Solution: Review Solution for Number 35
Problem 36: CE Board May 1995
What is the equation of the normal to the curve x2 + y2 = 25 at (4, 3)?
A. 5x + 3y = 0
B. 3x – 4y = 0
C. 3x + 4y = 0
D. 5x – 3y = 0
Answer: Option B
Solution: Review Solution for Number 36
Problem 37: EE Board April 1997
Locate the points of inflection of the curve y = f(x) = x2 ex.
A. -2 ± √3
B. 2 ± √2
C. -2 ± √2
D. 2 ± √3
Answer: Option C
Solution: Review Solution for Number 37
Problem 38: ECE Board November 1991
In the curve 2 + 12x – x3, find the critical points.
A. (2, 18) and (-2, -14)
B. (2, 18) and (2, -14)
C. (-2, 18) and (2, -14)
D. (-2, 18) and (-2, 14)
Answer: Option A
Solution: Review Solution for Number 38
Problem 39: CE Board November 1997
Find the radius of curvature of a parabola y2 – 4x = 0 at point (4, 4).
A. 22.36 units
4.5 Analytical Connections F F' F' (part 2)ap Calculus 2nd Edition
B. 25.78 units
C. 20.33 units
D. 15.42 units
Answer: Option A
Solution: Review Solution for Number 39
Problem 40: ECE Board November 1996
Find the radius of curvature at any point in the curve y + ln cos x = 0.
A. cos x
B. 1.5707
C. sec x
D. 1
Answer: Option C
Solution: Review Solution for Number 40
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