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.

### 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**

**MCQs from Number 1 – 50**Answer key:

**PART I**

**MCQs from Number 51 – 100**Answer key:

**PART II**

### Start Practice Exam Test Questions Part I 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

**Problem 2: ECE Board April 1998**

Evaluate the Limit:

- A. undefined
- B. 0
- C. Infinity
- D. 1/7

**Problem 3: ME Board April 1998**

Evaluate the Limit:

- A. 0
- B. 1
- C. 8
- D. 16

**Problem 4: ECE Board April 1993**

Evaluate the Limit:

- A. 0
- B. 2
- C. 4
- D. 6

**Problem 5: EE Board April 1995**

Evaluate the Limit:

- A. 0
- B. 1/2
- C. 2
- D. -1/2

**Problem 6: ME Board October 1997**

Compute the following limit:

- A. 1
- B. 0
- C. 2
- D. Infinite

**Problem 7: EE Board October 1994**

Evaluate the Limit:

- A. Undefined
- B. 3/5
- C. Infinity
- D. Zero

**Problem 8: ECE Board November 1991**

Evaluate the Limit:

- A. 24
- B. 26
- C. 28
- D. 30

**Problem 9: ECE Board November 1994**

Evaluate the Limit:

- A. e2π
- B. e2/π
- C. 0
- D. α

**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

**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)

**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

**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)

**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

**Problem 15: EE Board October 1997**

If y = (t2 + 2)2 and t = x1/2, determine dy/dx

- A. 3/2
- B. (2×2 + 2x) / 3
- C. 2(x + 2)
- D. x5/2 + x1/2

**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)]

**Problem 17: ME Board April 1998**

Find the derivative with respect to x function √(2 – 3×2)

- A. (-2×2) / √(2 – 3×2)
- B. (-3x) / √(2 – 3×2)
- C. (-3×2) / √(2 – 3×2)
- D. (3x) / √(2 – 3×2)

**Problem 18: EE Board April 1995**

Find y’ if y = arcsin cos x

- A. -1
- B. -2
- C. 1
- D. 2

**Problem 19: CE Board May 1997**

Find the derivative of arccos 4x.

- A. -4 / (1 – 16×2)0.5
- B. 4 / (1 – 16×2)0.5
- C. -4 / (1 – 4×2)0.5
- D. 4 / (1 – 4×2)0.5

**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)

**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. (2×2) / (x + 1)

**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)

**Problem 23: CE Board November 1993**

Find the second derivative of y by implicit differentiation from the equation 4×2 + 8y2 = 36

- A. 64×2
- B. (– 9/4) y3
- C. 32xy
- D. (- 16/9) y3

**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

**Problem 25: ME Board October 1997**

Find the second derivative of x3 – 5×2 + x = 0

- A. 10x – 5
- B. 6x – 10
- C. 3x + 10
- D. 3×2 – 5x

**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. 3×2 – 5
- D. 8

**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

**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

**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

**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

**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

**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

**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

**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)

**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

**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

**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

**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)

**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
- B. 25.78 units
- C. 20.33 units
- D. 15.42 units

**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