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This is the additional review for the first part of chapter 4. (The second of the 3 reviews). AP CALCULUS AB REVIEW SHEET LIMITS sin LIMITS LAWS lim 𝑥→ ( )=𝐿 exists if an only is lim 𝑥→ − ( )=lim 𝑥→ + ( )=𝐿 lim 𝑥→0 𝑥 𝑥 =1, lim 𝑥→0 cos𝑥−1 𝑥 =0 lim 𝑥→∞ sin𝑥 𝑥 =0 lim 𝑥→0 sin2𝑥 𝑥 =0 CONTINUITY A function is continuous at a if lim 𝑥→ − ( )=lim. Product 3.6 Connections Between The Dot Product And Cross Product 4.0 BASIC CALCULUS 4.1 Functions 4.1.1 Domain of a function 4.1.2 Combination of functions 4.2 Linear Equations 4.2.1 Slope of Line 4.2.2 Equation of Line 4.2.3 Parallel and Perpendicular line 4.2.4 Intercepts and Graph 4.2.5 Distance between two points 1,2,3 2 0.5 0.5 3 6 4.0. AP CALCULUS AB REVIEW SHEET LIMITS sin LIMITS LAWS lim 𝑥→ ( )=𝐿 exists if an only is lim 𝑥→ − ( )=lim 𝑥→ + ( )=𝐿 lim 𝑥→0 𝑥 𝑥 =1, lim 𝑥→0 cos𝑥−1 𝑥 =0 lim 𝑥→∞ sin𝑥 𝑥 =0 lim 𝑥→0 sin2𝑥 𝑥 =0 CONTINUITY A function is continuous at a if lim 𝑥→ − ( )=lim.

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 x²

A. –e^x sin x²

B. e^x (cos x² – 2x sin x²)

C. e^x cos x² – 2x sin x²

D. -2xe^x 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 = x^1/2, determine dy/dx

A. 3/2

B. (2x² + 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) / 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

4.4 Analytical Connections F F' F' (part 1)ap Calculus Solver

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).

4.4 Analytical Connections F F' F' (part 1)ap Calculus Textbook

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 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 – x³, find the critical points.

A. (2, 18) and (-2, -14)

B. (2, 18) and (2, -14)

4.4 Analytical Connections F F' F' (part 1)ap Calculus 4th Edition

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

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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