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

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6.3 Applications of the Differential

Exercise 6.2

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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

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Problem 6

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5.3 L’Hospital’s Rule

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5.4 The Indeterminate Forms 00\displaystyle \frac{0}{0} and ∞∞\displaystyle \frac{\infty }{\infty }

Exercise 5.2

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

Exercise 5.3

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5.6 The Indeterminate Forms 00\displaystyle 0^0 , 1∞\displaystyle 1^\infty , and ∞0\displaystyle \infty ^0

Exercise 5.4

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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 sin⁡uu\displaystyle \frac{\sin u}{u}

Exercise 4.1

Problem 1

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4.2 Differentiation of Trigonometric Functions

Exercise 4.2

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4.3 Differentiation of Inverse Trigonometric Functions

Exercise 4.3

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4.4 The functions (1+u)1u\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

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4.7 Logarithmic Differentiations

Exercise 4.5

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4.8 Differentiation of Exponential Functions

Exercise 4.6

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4.9 The Hyperbolic Functions

Exercise 4.7

Problem 1

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4.10 Differentiation of Hyperbolic Functions

Exercise 4.8

Problem 1

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4.11 Differentiation of Inverse Hyperbolic Functions

Exercise 4.9

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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.

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

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

  1. f(a)\displaystyle f\left( a \right) is defined.
  2. lim⁡x→af(x)=L\displaystyle \lim_{x \to a} f\left( x \right)=L exists, and
  3. L=f(a)\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 x\displaystyle x where the function is discontinuous.

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

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

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

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Problem 12

Problem 13

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3.2 Angle Between Two Curves

Exercise 3.2

Problem 1

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3.3 Increasing and Decreasing Functions

Exercise 3.3

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3.4 Maximum and Minimum Values of a Function

Exercise 3.4

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3.5 Significance of the Second Derivative

Exercise 3.5

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3.6 Applications of the Maxima and Minima

Exercise 3.6

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3.7 Related Rates

Exercise 3.7

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3.8 Rectilinear Motion

Exercise 3.7

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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

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Problem 5

Problem 6

Problem 7

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Problem 9

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Problem 15

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2.3 Geometric Significance of dy/dx

2.4 Rules for Differentiation

Exercise 2.2

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Problem 31

Problem 32

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2.5 The Chain Rule

2.6 Differentiation of Inverse Functions

Exercise 2.3

Problem 1

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2.7 Higher Derivatives

Exercise 2.4

Problem 1

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2.8 Implicit Differentiation

Exercise 2.5

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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
∑Fx=0→−4000 lb+Pbr=0Pbr=4000 lb (tension)\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
∑Fx=0−4000 lb+9000 lb−Pal=0Pal=5000 lb (Compression)\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
∑Fx=0−4000 lb+9000 lb+2000 lb−Pst=0Pst=7000 lb (Compression)\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

σbr=PbrAbr=4000 lb1.2 in2=3330 psi (Tension)  (Answer)\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

σal=PbrAal=5000 lb1.8 in2=2780 psi (Compression)  (Answer)\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

σst=PstAst=7000 lb1.6 in2=4380 psi (Compression)  (Answer)\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

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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

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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

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