Category Archives: Mathematics

Includes Engineering Mathematics like Algebra, Geometry, Trigonometry, Calculus, Differential Equations, Probability, Statistics

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

4.5 The Logarithmic and Exponential Functions

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

4.9 The Hyperbolic Functions

Exercise 4.7

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Advertisements

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

Advertisements

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

Advertisements
Advertisements
Advertisements

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

Advertisements

PROBLEM:

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

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

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.

Advertisements
Advertisements
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

Advertisements

3.2 Angle Between Two Curves

Exercise 3.2

Problem 1

Problem 2

Problem 3

Problem 4

Problem 5

Problem 6

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements
Advertisements
Advertisements
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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements

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

Advertisements
Advertisements
Advertisements

Probability & Statistics for Engineers & Scientists Ninth Edition by Walpole Exercise 1.2

According to the journal Chemical Engineering, an important property of a fiber is its water absorbency. A random sample of 20 pieces of cotton fiber was taken and the absorbency on each piece was measured. The following are the absorbency values:

18.7121.4120.7221.8119.2922.4320.17
23.7119.4420.5018.9220.3323.0022.85
19.2521.7722.1119.7718.0421.12

(a) Calculate the sample mean and median for the above sample values.
(b) Compute the 10% trimmed mean.
(c) Do a dot plot of the absorbency data.
(d) Using only the values of the mean, median, and trimmed mean, do you have evidence of outliers in the data?

Solution:

Part A. The sample mean is computed as follows:

\begin{align*}
\bar x & = \frac{\Sigma x_{i}}{n} \\
\bar x & = \frac{18.71+21.41+20.72+\cdots +21.12}{20} \\
\bar x & = 20.77 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

We can solve for the sample median by arranging the data in increasing order first.

18.04, \
18.71, \
18.92, \
19.25, \
19.29, \
19.44, \
19.77, \
20.17, \
20.33, \
20.50 \\
20.72, \
21.12, \
21.41, \
21.77, \
21.81, \
22.11, \
22.43, \
22.85, \
23.00, \
23.71

Since there are 20 measurements (even), the middle measurements are the (20/2) 10th and the (20/2 + 1) 11th measurement. The 10th measurement is 20.50 and the 11th measurement is 20.72. The median is the average of these two measurements. 

\begin{align*}
\tilde x & = \frac{20.50+20.72}{2} \\
\tilde x & = 20.61 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part B. The 10% trimmed mean is calculated by removing 10% of the lowest data and 10% of the highest data. That is, removing the 2 lowest and 2 highest data. We are left with the following:

18.92, \
19.25, \
19.29, \
19.44, \
19.77, \
20.17, \
20.33, \
20.50 \\
20.72, \
21.12, \
21.41, \
21.77, \
21.81, \
22.11, \
22.43, \
22.85

The 10% trimmed mean, \bar x _{tr10} is

\begin{align*}
\bar x _{tr10} & = \frac{\Sigma x_{i}}{n} \\
\bar x _{tr10} & = \frac{18.92+19.25+\cdots+22.85}{16} \\
\bar x _{tr10} & = 20.74 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C. The dot plot is shown

dot plot for Probability & Statistics for Engineers & Scientists Ninth Edition by Walpole Exercise 1.2

Part D. Since the values of the mean, median, and trimmed mean are not actually far from each other, we can conclude that there are no outliers in the given measurements.

Advertisements
Advertisements

Probability & Statistics for Engineers & Scientists Ninth Edition by Walpole Exercise 1.1

The following measurements were recorded for the drying time, in hours, of a certain brand of latex paint.

3.42.54.82.93.6
2.83.35.63.72.8
4.44.05.23.04.8

Assume that the measurements are a simple random sample.
(a) What is the sample size for the above sample?
(b) Calculate the sample mean for these data.
(c) Calculate the sample median.
(d) Plot the data by way of a dot plot.
(e) Compute the 20% trimmed mean for the above data set.
(f) Is the sample mean for these data more or less descriptive as a center of location than the trimmed mean?

Solution:

Part A. Sample size, n is the total number of measurements.

n=15 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part B. The sample mean, \bar x is computed as follows:

\begin{align*}
\bar x & = \sum_{i=1}^{n}\frac{x_{i}}{n} \\
& = \frac{3.4+2.5+4.8+2.9+3.6+2.8+3.3+5.6+3.7+2.8+4.4+4.0+5.2+3.0+4.8}{15} \\
& = \frac{56.8}{15} \\
& = 3.79 \ \text{hours} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part C. The sample median, \tilde x is the number at the middle of the arranged measurements in increasing magnitude. There are 15 measurements, n=15. If we arranged the data in increasing magnitude, the median is the measurement in the middle.

2.5, \ 2.8, \ 2.8, \ 2.9, \ 3.0, \ 3.3, \ 3.4, \ \underset{\color{Blue} \text{middle number}}{3.6}, \ 3.7, \ 4.0, \ 4.4, \ 4.8, \ 4.8, \ 5.2, \ 5.6

The middle number is 3.6. That is

\tilde x= 3.6 \ \text{hours} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)

Part D. This dot plot has been created using the Statistical Software Rguroo

a dot plot for the data: 3.4, 2.5, 4.8, 2.9, 3.6, 2.8, 3.3, 5.6, 3.7, 2.8, 4.4, 4.0, 5.2, 3.0, 4.8. this dot plot was made possible through Rgurro at https://www.rguroo.com/

Part E. The 20% trimmed mean means the average of the measurements left after removing 20% highest and 20% lowest data. This means we remove the 3 highest and 3 lowest numbers. Therefore, the data becomes

\ 2.9, \ 3.0, \ 3.3, \ 3.4, \ 3.6,  \ 3.7, \ 4.0, \ 4.4, \ 4.8,

The 20% trimmed mean, \bar x _{tr20} is

\begin{align*}
\bar x _{tr20} & = \frac{2.9+3.0+ \cdots+4.8}{9} \\
\bar x _{tr20} & = \frac{33.1}{9} \\
\bar x _{tr20} & = 3.678 \ \text{hours} \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right)
\end{align*}

Part F. The sample mean for these data is \bar x = 3.79 \ \text{hours} while the 20% trimmed mean is \bar x _{tr20} = 3.678 \ \text{hours}. Seems like the two means are not really far from each other, but because of the elimination of the extreme values, we can treat the trimmed mean as a better descriptive mean.

Advertisements
Advertisements

Chapter 1: Introduction to Statistics and Data Analysis

Problem 1.1

Problem 1.2

Problem 1.3

Problem 1.4

Problem 1.5

Problem 1.6

Problem 1.7

Problem 1.8

Problem 1.9

Problem 1.10

Problem 1.11

Problem 1.12

Problem 1.13

Problem 1.14

Problem 1.15

Problem 1.16

Problem 1.17

Problem 1.18

Problem 1.19

Problem 1.20

Problem 1.21

Problem 1.22

Problem 1.23

Problem 1.24

Problem 1.25

Problem 1.26

Problem 1.27

Problem 1.28

Problem 1.29

Problem 1.30

Problem 1.31

Problem 1.32

Problem 1.33

Advertisements
Advertisements

Probability & Statistics for Scientists & Engineers Ninth Edition by Walpole, Myers, Myers, and Ye

Chapter 1: Introduction to Statistics and Data Analysis

Chapter 2: Probability

Chapter 3: Random Variables and Probability Distributions

Chapter 4: Mathematical Expectation

Chapter 5: Some Discrete Probability Distribution

Chapter 6: Some Continuous Probability Distribution

Chapter 7: Functions of Random Variables

Chapter 8: Fundamental Sampling Distributions and Data Descriptions

Chapter 9: One- and Two-Sample Estimation Problems

Chapter 10: One- and Two-Sample Tests of Hypotheses

Chapter 11: Simple Linear Regression and Correlation

Chapter 12: Multiple Linear Regression and Certain Nonlinear Regression Models

Chapter 13: One-Factor Experiments: General

Chapter 14: Factorial Experiments (Two or More Factors)

Chapter 15: 2k Factorial Experiments and Fractions

Chapter 16: Nonparametric Statistics

Chapter 17: Statistical Quality Control

Chapter 18: Bayesian Statistics

Textbook Solutions for Probability and Statistics

textbook cover page for probability and statistics for engineers and scientists ninth edition

Probability and Statistics for Engineers & Scientists, 9th Edition by Walpole, etc.

Advertisements
Advertisements

Solution Manuals

These are the available solution guides ready for purchase. Payments shall be done through PayPal, but you can still use your cards even without PayPal account. For inquiries, you can send an email to [email protected].

Mathematics Solution Manuals

Engineering Sciences Solution Manuals

Civil Engineering Solution Manuals