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
Problem 2: ECE Board April 1998
Evaluate the Limit:
A. undefined
B. 0
C. Infinity
D. 1/7
Answer: Option D
Solution: Evaluate: lim┬(x→4) {(x- 4)/(x^2-x-12)}
Problem 3: ME Board April 1998
Evaluate the Limit:
A. 0
B. 1
C. 8
D. 16
Answer: Option C
Solution: Evaluate: lim┬(x→4) {(x^2- 16)/(x-4)}
Problem 4: ECE Board April 1993
Evaluate the Limit:
A. 0
B. 2
C. 4
D. 6
Answer: Option C
Problem 5: EE Board April 1995
Evaluate the Limit:
A. 0
B. 1/2
C. 2
D. -1/2
Answer: Option B
Solution: Evaluate: lim┬(x→0) {(1-cosx)/x^2}
Problem 6: ME Board October 1997
Compute the following limit:
A. 1
B. 0
C. 2
D. Infinite
Answer: Option A
Solution: Compute the following limit: lim┬(x→∞) {(x+4)/(x-4)}
Problem 7: EE Board October 1994
Evaluate the Limit:
A. Undefined
B. 3/5
C. Infinity
D. Zero
Answer: Option C
Problem 8: ECE Board November 1991
Evaluate the Limit:
A. 24
B. 26
C. 28
D. 30
Answer: Option A
Solution: Evaluate: lim┬(x→4) (x^2+3x-4)
Problem 9: ECE Board November 1994
Evaluate the Limit:
A. e^2π
B. e^2/π
C. 0
D. α
Answer: Option B
Problem 10: EE Board October 1997
Differentiate y = e^x cos x^2
A. –e^x sin x^2
B. e^x (cos x^2 – 2xsin x^2)
C. e^x cos x^2 – 2xsin x^2
D. -2xe^x sin x
Answer: Option B
Solution: Differentiate y=e^x cosx^2
Problem 11: EE Board October 1997
Differentiate y = sec(x^2 + 2)
A. 2xcos (x^2 + 2)
B. –cos(x^2 + 2) cot(x^2 + 2)
C. 2xsec(x^2 + 2) tan(x^2 + 2)
D. cos(x^2 +2)
Answer: Option C
Solution: Differentiate y=sec(x^2+2)
Problem 12: CE Board October 1994
What is the derivative with respect to x of (x + 1)^3 – x^3?
A. 3x + 6
B. 3x – 3
C. 6x – 3
D. 6x + 3
Answer: Option D
Solution: What is the derivative with respect to x of (x+1)^3 – x^3?
Problem 13: EE Board October 1997
Differentiate y = log_10 (x^2 + 1)^2
A. 4x (x^2 + 1)
B. (4x log_10 e)/(x^2 + 1)
C. log e(x) (x^2 + 1)
D. 2x (x^2 + 1)
Answer: Option B
Solution: Differentiate y=log_10 (x^2+2)^2
Problem 14: EE Board October 1997
Differentiate (x^2 + 2)^1/2
A. ((x^2 + 2)^1/2)/2
B. x/(x^2 + 2)^1/2
C. (2x)/(x^2 + 2)^1/2
D. (x^2 + 2)^3/2
Answer: Option B
Solution: Differentiate (x^2+2)^(1/2)
Problem 15: EE Board October 1997
If y = (t^2 + 2)^2 and t = x^1/2, determine dy/dx
A. 3/2
B. (2x^2 + 2x)/3
C. 2(x + 2)
D. x^5/2 + x^1/2
Answer: Option C
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 + lnxy)/x)]
D. –y [(1 – lnxy)/x^2)]
Answer: Option C
Solution: What is the first derivative of the expression (xy)^x=e?
Problem 17: ME Board April 1998
Find the derivative with respect to x function √(2 – 3x^2)
A. (-2x^2)/√(2 – 3x^2)
B. (-3x)/√(2 – 3x^2)
C. (-3x^2)/√(2 – 3x^2)
D. (3x)/√(2 – 3x^2)
Answer: Option B
Solution: Find the derivative with respect to x function √(2-3x^2)
Problem 18: EE Board April 1995
Find y’ if y = arc sincosx
A. -1
B. -2
C. 1
D. 2
Answer: Option A
Solution: Find y’ if y=arcsincosx
Problem 19: CE Board May 1997
Find the derivative of arc cos4x.
A. -4/(1 – 16x^2)^0.5
B. 4/(1 – 16x^2)^0.5
C. -4/(1 – 4x^2)^0.5
D. 4/1 – 4x^2)^0.5
Answer: Option A
Solution: Find the derivative of arccos4x
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/x^3)
D. (3(x + 1)^2/x) – ((x + 1)^3/x^2)
Answer: Option D
Solution: Find the derivative of (x+1)^3/x
Problem 21: ECE Board November 1991
Differentiate the equation y = x^2/(x + 1)
A. (x^2 + 2x)/(x + 1)^2
B. x/(x + 1)
C. 2x
D. (2x^2)/(x + 1)
Answer: Option A
Solution: Differentiate the equation y=x^2/(x+1)
Problem 22: CE Board November 1995
The derivative with respect to x of 2cos^2 (x^2 + 2) is
A. 2sin(x^2 + 2) cos(x^2 + 2)
B. -2sin(x^2 + 2) cos(x^2 + 2)
C. 8x sin(x^2 + 2) cos(x^2 + 2)
D. -8x sin (x^2 + 2) cos(x^2 + 2)
Answer: Option C
Solution: What is the derivative with respect to x of 2cos^2(x^2+2)?
Problem 23: CE Board November 1993
Find the second derivative of y by implicit differentiation from the equation 4x^2 + 8y^2 = 36
A. 64x^2
B. (–9/4) y^3
C. 32xy
D. (-16/9) y^3
Answer: Option B
Solution: Find the second derivative of y by implicit differentiation
Problem 24: ME Board April 1998
Find the partial derivative with respect to x of the function xy^2 – 5y + 6.
A. y^2 – 5
B. y^2
C. xy – 5y
D. 2xy
Answer: Option B
Solution: Find the partial derivative with respect to x of the function xy^2-5y+6
Problem 25: ME Board October 1997
Find the second derivative of x^3 – 5x^2 + x = 0
A. 10x – 5
B. 6x – 10
C. 3x + 10
D. 3x^2 – 5x
Answer: Option B
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. 3x^2 – 5
D. 8
Answer: Option A
Solution: Given the function f(x) = x^3-6x+2. Find the first derivative at x=2
Problem 27: CE Board May 1996
Find the slope of the ellipse x^2 + 4y^2 – 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: Find the slope of the ellipse x^2+4y^2–10x–16y+5=0
Problem 28: EE Board October 1997
If y = 4cosx + sin2x, 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: If y=4cosx+sin2x, what is the slope of the curve when x=2 radians?
Problem 29: ECE Board November 1991
Find the slope of the line tangent to the curve y = x^3 – 2x + 1 at x = 1.
A. 1
B. 1/2
C. 1/3
D. 1/4
Answer: Option A
Solution: Find the slope of the line tangent to the curve y=x^3-2x+1 at x=1
Problem 30: ECE Board November 1991
Given the slope of the curve at the point (1, 1): y = (x^3/4) – 2x + 1
A. 1/4
B. -1/4
C. 1 1/4
D. -1 1/4
Answer: Option D
Solution: Give the slope of the curve at the point (1, 1): y = (x^3/4)-2x+1
Problem 31: ECE Board November 1998
Find the slope of x^2y = 8 at the point (2, 2)
A. 2
B. -1
C. -1/2
D. -2
Answer: Option D
Problem 32: CE Board May 1998
Find the slope of the curve x^2 + y^2 – 6x + 10y + 5 + 0 at point (1, 0).
A. 1/5
B. 2/5
C. 1/4
D. 2
Answer: Option B
Solution: Find the slope of the curve x^2+y^2–6x+10y+5+0 at point (1, 0)
Problem 33: CE Board May 1996
Find the slope of the tangent to the curve, y = 2x – x^2 + x^3 at (0, 2).
A. 1
B. 2
C. 3
D. 4
Answer: Option B
Solution: Find the slope of the tangent to the curve, y=2x–x^2+x^3 at (0, 2)
Problem 34: ECE Board April 1999
Find the coordinates of the vertex of the parabola y = x^2 – 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: Find the coordinates of the vertex of the parabola
Problem 35: ECE Board April 1999
Find the equation of the normal to x^2 + y^2 = 5 at the point (2, 1)
A. y = 2x
B. x = 2y
C. 2x + 3y = 3
D. x + y = 1
Answer: Option B
Solution: Find the equation of the normal to x^2+y^2=5 at the point (2, 1)
Problem 36: CE Board May 1995
What is the equation of the normal to the curve x^2 + y^2 = 25 at (4, 3)?
A. 5x + 3y = 0
B. 3x – 4y = 0
C. 3x + 4y = 0
D. 5x – 3y = 0
Answer: Option B
Solution: What is the equation of the normal to the curve x^2+y^2=25 at (4, 3)?
Problem 37: EE Board April 1997
Locate the points of inflection of the curve y = f(x) = x^2 e^x.
A. -2 ± √3
B. 2 ± √2
C. -2 ± √2
D. 2 ± √3
Answer: Option C
Solution: Locate the points of inflection of the curve y=f(x)=x^2 e^x
Problem 38: ECE Board November 1991
In the curve 2 + 12x – x^3, 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
Problem 39: CE Board November 1997
Find the radius of curvature of a parabola y^2 – 4x = 0 at point (4, 4).
A. 22.36 units
B. 25.78 units
C. 20.33 units
D. 15.42 units
Answer: Option A
Solution: Find the radius of curvature of a parabola y^2–4x=0 at point (4, 4)
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: Find the radius of curvature at any point in the curve y+lncos x=0
Online Question and Answer in Differential Calculus (Limits and Derivatives) Series
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Nice
Compute the valueor values of c that satisfy the mean value theorem for the function
f(x)=x+1/x on the interval [1/2,2]