Category Archives: Engineering Mathematics Blog

Cover photo of Chapter 6 The Differential of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 6: The Differential

6.1 Differential: Definition and Interpretation

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6.2 Differential Formulas

Exercise 6.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

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6.3 Applications of the Differential

Exercise 6.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

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Cover photo of Chapter 5 The Indeterminate Forms of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 5: The Indeterminate Forms

5.1 Rolle’s Theorem

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5.2 Mean Value Theorem

Exercise 5.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

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5.3 L’Hospital’s Rule

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5.4 The Indeterminate Forms \displaystyle \frac{0}{0} and \displaystyle \frac{\infty }{\infty }

Exercise 5.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

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5.5 The Indeterminate Forms \displaystyle 0\left( \pm \infty \right) and \displaystyle \infty -\infty

Exercise 5.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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5.6 The Indeterminate Forms \displaystyle 0^0 , \displaystyle 1^\infty , and \displaystyle \infty ^0

Exercise 5.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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Cover photo of Chapter 4 Differentiation of Transcendental Functions of the book Differential and Integral Calculus by Feliciano and Uy

Chapter 4: Differentiation of Transcendental Functions

4.1 The Function \displaystyle \frac{\sin u}{u}

Exercise 4.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

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4.2 Differentiation of Trigonometric Functions

Exercise 4.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

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4.3 Differentiation of Inverse Trigonometric Functions

Exercise 4.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

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4.4 The functions \displaystyle \left( 1+u \right)^{\frac{1}{u}}

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4.5 The Logarithmic and Exponential Functions

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4.6 Differentiation of Logarithmic Functions

Exercise 4.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

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4.7 Logarithmic Differentiations

Exercise 4.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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4.8 Differentiation of Exponential Functions

Exercise 4.6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

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4.9 The Hyperbolic Functions

Exercise 4.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

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4.10 Differentiation of Hyperbolic Functions

Exercise 4.8

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

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4.11 Differentiation of Inverse Hyperbolic Functions

Exercise 4.9

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

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Finding the value/s of x for which a function is discontinuous – Problem 1.5.1

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PROBLEM:

Find the value or values of x for which the function is discontinuous.

\large \displaystyle f\left( x \right)=\frac{3x}{x-5}
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Solution:

A function \displaystyle f\left( x \right) is continuous at \displaystyle x=a if \displaystyle \lim_{x \to a} f\left( x \right)=f\left( a \right), which implies these three conditions:

  1. \displaystyle f\left( a \right) is defined.
  2. \displaystyle \lim_{x \to a} f\left( x \right)=L exists, and
  3. \displaystyle L=f\left( a \right)

We are given a rational function. A rational function is not defined when the denominator is equal to zero. If we equate the denominator to zero, we can compute the value/s of \displaystyle x where the function is discontinuous.

\begin{align*}
x-5 & = 0 \\
x & = 5 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

The function is not continuous at \displaystyle x=5.

The graph of the function \displaystyle f\left( x \right)=\frac{3x}{x-5} is drawn below. It can be seen that there is an infinite discontinuity at \displaystyle x=5.

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Cover photo of Chapter 3 Some Applications of the Derivatives of the textbook Differential and Integral Calculus by Feliciano and Uy

Chapter 3: Some Applications of the Derivative

3.1 Equations of Tangents and Normals

Exercise 3.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.2 Angle Between Two Curves

Exercise 3.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

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3.3 Increasing and Decreasing Functions

Exercise 3.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

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3.4 Maximum and Minimum Values of a Function

Exercise 3.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.5 Significance of the Second Derivative

Exercise 3.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

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3.6 Applications of the Maxima and Minima

Exercise 3.6

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

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3.7 Related Rates

Exercise 3.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

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3.8 Rectilinear Motion

Exercise 3.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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Cover photo for Chapter 2 Differentiation of Algebraic Functions of the textbook Differential and Integral Calculus by Feliciano and Uy

Chapter 2: Differentiation of Algebraic Functions

2.1 The Symbol Δ

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2.2 The Derivative of a Function

Exercise 2.1

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

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2.3 Geometric Significance of dy/dx

2.4 Rules for Differentiation

Exercise 2.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

Problem 23

Problem 24

Problem 25

Problem 26

Problem 27

Problem 28

Problem 29

Problem 30

Problem 31

Problem 32

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2.5 The Chain Rule

2.6 Differentiation of Inverse Functions

Exercise 2.3

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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2.7 Higher Derivatives

Exercise 2.4

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

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2.8 Implicit Differentiation

Exercise 2.5

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Problem 7

Problem 8

Problem 9

Problem 10

Problem 11

Problem 12

Problem 13

Problem 14

Problem 15

Problem 16

Problem 17

Problem 18

Problem 19

Problem 20

Problem 21

Problem 22

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Strength of Materials Problem 101 – Stress in each section of a composite bar

A composite bar consists of an aluminum section rigidly fastened between a bronze section and a steel section as shown in Fig. 1-8a. Axial loads are applied at the positions indicated. Determine the stress in each section.

Strength of Materials by Andrew Pytel and Ferdinand Singer Problem 101
Figure 1.8a

Solution:

We must first determine the axial load in each section to calculate the stresses. The free-body diagrams have been drawn by isolating the portion of the bar lying to the left of imaginary cutting planes. Identical results would be obtained if portions lying to the right of the cutting planes had been considered.

Solve for the internal axial load of the bronze

Free body diagram for the internal axial load of the bronze section for Problem 101 of Strength of Materials by Ferdinand Singer and Andrew Pytel
The free-body diagram of the bronze section
\begin{align*}
\sum_{}^{}F_x & = 0  \to  \\
-4000\ \text{lb}+P_{br} & = 0 \\
P_{br} & = 4000 \ \text{lb} \ \text{(tension)}
\end{align*}

Solve for the internal axial load of the aluminum

Free-body diagram of the aluminum section for problem 101 of Strength of materials by Andrew Pytel and Ferdinand Singer
The free-body diagram of the aluminum section
\begin{align*}
\sum_{}^{}F_x & = 0 \\
-4000 \ \text{lb} + 9000 \ \text{lb} - P_{al} & = 0 \\
P_{al} & = 5000 \ \text{lb} \ \text{(Compression)}
\end{align*}

Solve for the internal axial load of the aluminum

The free-body diagram of the steel section
\begin{align*}
\sum_{}^{}F_x & = 0 \\
-4000\ \text{lb} + 9000 \ \text{lb} + 2000\ \text{lb} - P_{st} & =0 \\
P_{st} & = 7000 \ \text{lb} \ \text{(Compression)}
\end{align*}

We can now solve the stresses in each section.

For the bronze

\begin{align*}
\sigma_{br} & = \frac{P_{br}}{A_{br}} \\
& = \frac{4000\ \text{lb}}{1.2 \ \text{in}^2} \\
& = 3330 \ \text{psi}\ \text{(Tension)} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

For the aluminum

\begin{align*}
\sigma_{al} & = \frac{P_{br}}{A_{al}} \\
& = \frac{5000\ \text{lb}}{1.8 \ \text{in}^2} \\
& = 2780 \ \text{psi}\ \text{(Compression)} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

For the steel

\begin{align*}
\sigma_{st} & = \frac{P_{st}}{A_{st}} \\
& = \frac{7000\ \text{lb}}{1.6 \ \text{in}^2} \\
& = 4380\ \text{psi}\ \text{(Compression)} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}
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Torsion Featured Image: Chapter 3 of the book of Andrew Pytel and Ferdinand Singer Strength of Materials 4th Edition

Chapter 3: Torsion

Problem 301

Problem 302

Problem 303

Problem 304

Problem 305

Problem 306

Problem 307

Problem 308

Problem 309

Problem 310

Problem 311

Problem 312

Problem 313

Problem 314

Problem 315

Problem 316

Problem 317

Problem 318

Problem 319

Problem 320

Problem 321

Problem 322

Problem 323

Problem 324

Problem 325

Problem 326

Problem 327

Problem 328

Problem 329

Problem 330

Problem 331

Problem 332

Problem 333

Problem 334

Problem 335

Problem 336

Problem 337

Problem 338

Problem 339

Problem 340

Problem 341

Problem 342

Problem 343

Problem 344

Problem 345

Problem 346

Problem 347

Problem 348

Problem 349

Problem 350

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Simple Strain Featured Image: Chapter 2 of the book of Andrew Pytel and Ferdinand Singer Strength of Materials 4th Edition

Chapter 2: Simple Strain

Problem 201

Problem 202

Problem 203

Problem 204

Problem 205

Problem 206

Problem 207

Problem 208

Problem 209

Problem 210

Problem 211

Problem 212

Problem 213

Problem 214

Problem 215

Problem 216

Problem 217

Problem 218

Problem 219

Problem 220

Problem 221

Problem 222

Problem 223

Problem 224

Problem 225

Problem 226

Problem 227

Problem 228

Problem 229

Problem 230

Problem 231

Problem 232

Problem 233

Problem 234

Problem 235

Problem 236

Problem 237

Problem 238

Problem 239

Problem 240

Problem 241

Problem 242

Problem 243

Problem 244

Problem 245

Problem 246

Problem 247

Problem 248

Problem 249

Problem 250

Problem 251

Problem 252

Problem 253

Problem 254

Problem 255

Problem 256

Problem 257

Problem 258

Problem 259

Problem 260

Problem 261

Problem 262

Problem 263

Problem 264

Problem 265

Problem 266

Problem 267

Problem 268

Problem 269

Problem 270

Problem 271

Problem 272

Problem 273

Problem 274

Problem 275

Problem 276

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Simple Stress Featured Image: Strength of Materials 4th Edition by Andrew Pytel and Ferdinand Singer

Chapter 1: Simple Stress

Problem 101

Problem 102

Problem 103

Problem 104

Problem 105

Problem 106

Problem 107

Problem 108

Problem 109

Problem 110

Problem 111

Problem 112

Problem 113

Problem 114

Problem 115

Problem 116

Problem 117

Problem 118

Problem 119

Problem 120

Problem 121

Problem 122

Problem 123

Problem 124

Problem 125

Problem 126

Problem 127

Problem 128

Problem 129

Problem 130

Problem 131

Problem 132

Problem 133

Problem 134

Problem 135

Problem 136

Problem 137

Problem 138

Problem 139

Problem 140

Problem 141

Problem 142

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