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. e^{2π}

B. e^{2/π}

C. 0

D. α

Answer: **Option B**

Solution: Review Solution for Number 9

**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} – 2x sin x^{2})

C. e^{x} cos x^{2} – 2x sin x^{2}

D. -2xe^{x} sin x

Answer: **Option B**

Solution: Review Solution for Number 10

**Problem 11: EE Board October 1997**

Differentiate y = sec (x^{2} + 2)

A. 2x cos (x^{2} + 2)

B. –cos (x^{2} + 2) cot (x^{2} + 2)

C. 2x sec (x^{2} + 2) tan (x^{2} + 2)

D. cos (x^{2} +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} – x^{3}?

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 = 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: Review Solution for Number 13

**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: Review Solution for Number 14

**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**

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) / x^{2})]

Answer: **Option C**

Solution: Review Solution for Number 16

**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: 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 – 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: 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} / x^{3})

D. (3(x + 1)^{2} / x) – ((x + 1)^{3} / x^{2})

Answer: **Option D**

Solution: Review Solution for Number 20

**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: Review Solution for Number 21

**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: Review Solution for Number 22

**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: Review Solution for Number 23

**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: Review Solution for Number 24

**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**

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. 3x^{2} – 5

D. 8

Answer: **Option A**

Solution: Review Solution for Number 26

**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: 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 = x^{3} – 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 = (x^{3}/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 x^{2}y = 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 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: Review Solution for Number 32

**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: Review Solution for Number 33

**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: Review Solution for Number 34

**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: Review Solution for Number 35

**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: Review Solution for Number 36

**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: Review Solution for Number 37

**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**

Solution: Review Solution for Number 38

**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: 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

### 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)**

**MCQ from Number 1 – 50**Answer key:

**PART 1**

**MCQ from Number 51 – 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)**

**MCQ from Number 1 – 50**Answer key:

**PART 1**

**MCQ from Number 2 – 100**Answer key:

**PART 2**

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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]