The document provides information about trigonometric functions and solving algebraic equations related to engineering problems. It includes:
1) An example of expressing the displacement of an oscillating structure as a trigonometric function and sketching the displacement over time.
2) Solving a trigonometric equation using trigonometric identities.
3) Evaluating the horizontal range of a stone thrown at an angle on a sloped hill using kinematics equations.
4) Setting up algebraic equations to represent how much of different yogurt blends can be made based on available ingredients.
5) Proving an identity involving fractions and solving simultaneous equations using matrices.
Dokumen tersebut membahas tentang teorema faktor dan akar-akar rasional dari persamaan sukubanyak. Teorema faktor menyatakan hubungan antara faktor sukubanyak dengan nilai nol persamaannya, sedangkan akar-akar rasional adalah faktor-faktor bulat dari koefisien paling besar. Dokumen ini juga menjelaskan cara menentukan jumlah dan hasil kali akar-akar suatu persamaan.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
Solutions manual for calculus an applied approach brief international metric ...Larson612
Solutions Manual for Calculus An Applied Approach Brief International Metric Edition 10th Edition by Larson IBSN 9781337290579
Full download: https://goo.gl/RtxZKH
This document provides examples and exercises on summation notation. Example C demonstrates how to evaluate summations such as Σ fk, Σ ai, and Σ xjyj. Example D shows properties of summation notation including the formula for the sum of natural numbers from 1 to n. Example E evaluates the summation Σ (2k - 5) from k=1 to 45. The exercises provide practice writing out terms of sequences defined by fn and finding formulas for sequences defined by patterns in their terms.
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.
Dokumen tersebut membahas tentang teorema faktor dan akar-akar rasional dari persamaan sukubanyak. Teorema faktor menyatakan hubungan antara faktor sukubanyak dengan nilai nol persamaannya, sedangkan akar-akar rasional adalah faktor-faktor bulat dari koefisien paling besar. Dokumen ini juga menjelaskan cara menentukan jumlah dan hasil kali akar-akar suatu persamaan.
This document discusses Newton's forward and backward difference interpolation formulas for equally spaced data points. It provides the formulations for calculating the forward and backward differences up to the kth order. For equally spaced points, the forward difference formula approximates a function f(x) using its kth forward difference at the initial point x0. Similarly, the backward difference formula approximates f(x) using its kth backward difference at x0. The document includes an example problem of using these formulas to estimate the Bessel function and exercises involving interpolation of the gamma function and exponential function.
Solutions manual for calculus an applied approach brief international metric ...Larson612
Solutions Manual for Calculus An Applied Approach Brief International Metric Edition 10th Edition by Larson IBSN 9781337290579
Full download: https://goo.gl/RtxZKH
This document provides examples and exercises on summation notation. Example C demonstrates how to evaluate summations such as Σ fk, Σ ai, and Σ xjyj. Example D shows properties of summation notation including the formula for the sum of natural numbers from 1 to n. Example E evaluates the summation Σ (2k - 5) from k=1 to 45. The exercises provide practice writing out terms of sequences defined by fn and finding formulas for sequences defined by patterns in their terms.
This document discusses binomial expansion, which is a method for expanding binomials like (x + a)^n without lengthy multiplication. It introduces key concepts like Pascal's triangle for finding coefficients and the binomial theorem for determining the general pattern of terms in the expansion. Examples are worked through to demonstrate expanding specific binomials like (2x - 3y)^6 according to this method.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
1. Bab II membahas kegiatan pembelajaran tentang turunan fungsi aljabar. Definisi turunan fungsi dijelaskan dengan contoh penentuan turunan dari f(x) = 4x - 3 dan f(x) = 3x^2.
2. Teorema-teorema turunan fungsi aljabar dijelaskan, seperti turunan fungsi konstan, turunan fungsi aljabar, dan turunan hasil perkalian/pembagian fungsi aljabar. Contoh soal diberikan
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
Series expansion of exponential and logarithmic functionsindu psthakur
The document summarizes series expansions of exponential and logarithmic functions. It provides the series expansions of ex, e-x, ex+c, and logarithmic functions like log(1+x). It discusses the properties of the natural logarithm and exponential functions. Examples are given of finding sums of various exponential and logarithmic series. Questions with hints are also provided about proving properties related to logarithmic and exponential series.
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
- A matrix is singular if its determinant is zero, meaning its rows and columns are linearly dependent. The rank of a matrix is the size of the largest non-singular sub-matrix. A full rank matrix has
The document discusses directional derivatives and the gradient. It defines the directional slope of a vector v as the ratio of the y-component to the x-component of v. It then gives an example of calculating the directional slopes of several vectors. It explains that the directional slope indicates the steepness and direction of v. For a differentiable function f(x), the directional derivative in the direction of the unit vector <1,0> is the partial derivative df/dx, while the opposite direction has derivative -df/dx.
This document provides formulas for integrals of common functions. It includes integrals of polynomials, rational functions, radicals, logarithms, exponentials, trigonometric functions, and their combinations. There are a total of 94 formulas organized into sections based on function type. The integrals cover basic forms, rational functions, radicals, logarithms, exponentials, and trigonometric functions.
Dokumen tersebut memberikan penjelasan tentang definisi matriks, jenis-jenis matriks, operasi yang dapat dilakukan pada matriks seperti penjumlahan, pengurangan, perkalian skalar dan perkalian matriks, serta teorema-teorema yang berkaitan dengan operasi pada matriks.
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
The document provides steps to calculate the area between two curves by finding the intersection points using "y=y" and then integrating between those points. It works through multiple examples of finding the area between various curve combinations, including quadratic, cubic, and linear functions. The key steps are finding the intersection points using "y=y", setting up the integral as the top curve minus the bottom curve, and integrating between the bounds found from the intersection points.
This document summarizes key concepts in Boolean algebra including:
1) Defining a Boolean algebra using a set of elements and binary operations that follow specific axioms.
2) The two-valued Boolean algebra uses the elements 0 and 1 with AND, OR, and NOT operations.
3) Boolean functions consist of variables, constants, and logic operations and can be represented using truth tables.
4) Logic circuits can be designed to implement Boolean functions using gates like AND, OR, NOT, NAND, and NOR.
This document discusses different types of geometrical transformations including translations, reflections, rotations, and dilations. It provides examples and notations for transformations and notes that dilation is not an isometric transformation. Reflections are further categorized by the line they are reflected across such as the x-axis, y-axis, lines of the form y=x, y=-x, y=k, x=k, and y=mx+c. Two examples of finding the line of reflection are given when points are mapped before and after a reflection.
The document discusses regular singular points of differential equations. A point x0 is a regular singular point if the functions P, Q, and R are analytic at x0 or their limits exist and are finite as x approaches x0. Several examples are provided to illustrate regular singular points, including the Bessel and Legendre equations. An irregular singular point is defined as one where these limits do not exist. The behavior of solutions near regular singular points can be analyzed using power series methods.
1) The document discusses logarithms and how they were developed to simplify calculations of large numbers before calculators. Logarithms allow expressing a number as an exponent by finding what power of a base number equals the target number.
2) It provides examples of converting between exponential and logarithmic forms, using logarithmic properties to solve equations, and using change of base formulas to solve when the calculator uses a different base than the problem.
3) Key steps are shown to solve the equation 5x = 15,000 by rewriting in logarithmic form, using the change of base formula to solve for x, and checking the exponential form of the answer.
This document discusses continuity of functions. It provides examples of functions and examines whether they are continuous at various points. The key points are:
1) A function f(x) is continuous at a point a if the left-hand and right-hand limits of f(x) as x approaches a are equal to f(a).
2) An example function is examined and found to be continuous at x=2 because the left and right limits match the value of f(x) at x=2.
3) Another function is discontinuous at x=0 because the left and right limits do not match at that point.
1. This document provides a review of concepts and sample problems related to multiple integrals. It covers topics such as iterated integrals, changing the order of integration, and evaluating double integrals over various regions.
2. The problem set contains 14 problems evaluating double integrals over different regions using techniques like iterated integration and changing the order of integration.
3. Multiple integrals are used to find volumes, masses, moments, and other physical quantities over regions in 2D and 3D space. The document demonstrates how to set up and evaluate multiple integrals to solve applied problems.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
This document discusses the inverse of matrices. It defines the cofactor method for finding the inverse of a matrix, which involves calculating the matrix of cofactors and then taking its transpose divided by the determinant of the original matrix. Several examples are worked through, including calculating the inverse of a 3x3 matrix. The document also discusses using matrices to represent and solve systems of simultaneous linear equations, developing the general matrix solution of x = A^-1b where A is the coefficient matrix, x is the vector of unknowns, and b is the vector of constants.
1. Bab II membahas kegiatan pembelajaran tentang turunan fungsi aljabar. Definisi turunan fungsi dijelaskan dengan contoh penentuan turunan dari f(x) = 4x - 3 dan f(x) = 3x^2.
2. Teorema-teorema turunan fungsi aljabar dijelaskan, seperti turunan fungsi konstan, turunan fungsi aljabar, dan turunan hasil perkalian/pembagian fungsi aljabar. Contoh soal diberikan
This document provides the answers to a mathematics exam for 10th grade students. It includes multiple choice questions with explanations and word problems with step-by-step solutions. The topics covered include algebra, logarithms, quadratic equations, and inequalities.
Series expansion of exponential and logarithmic functionsindu psthakur
The document summarizes series expansions of exponential and logarithmic functions. It provides the series expansions of ex, e-x, ex+c, and logarithmic functions like log(1+x). It discusses the properties of the natural logarithm and exponential functions. Examples are given of finding sums of various exponential and logarithmic series. Questions with hints are also provided about proving properties related to logarithmic and exponential series.
This document discusses matrix algebra concepts such as determinants, inverses, eigenvalues, and rank. It provides the following key points:
- The determinant of a square matrix is a number that characterizes properties like singularity. It is defined as the sum of products of the matrix elements.
- Cramer's rule provides a formula for solving systems of linear equations using determinants, but it is only practical for small matrices up to 3x3 or 4x4 due to computational complexity.
- A matrix is singular if its determinant is zero, meaning its rows and columns are linearly dependent. The rank of a matrix is the size of the largest non-singular sub-matrix. A full rank matrix has
The document discusses directional derivatives and the gradient. It defines the directional slope of a vector v as the ratio of the y-component to the x-component of v. It then gives an example of calculating the directional slopes of several vectors. It explains that the directional slope indicates the steepness and direction of v. For a differentiable function f(x), the directional derivative in the direction of the unit vector <1,0> is the partial derivative df/dx, while the opposite direction has derivative -df/dx.
This document provides formulas for integrals of common functions. It includes integrals of polynomials, rational functions, radicals, logarithms, exponentials, trigonometric functions, and their combinations. There are a total of 94 formulas organized into sections based on function type. The integrals cover basic forms, rational functions, radicals, logarithms, exponentials, and trigonometric functions.
Dokumen tersebut memberikan penjelasan tentang definisi matriks, jenis-jenis matriks, operasi yang dapat dilakukan pada matriks seperti penjumlahan, pengurangan, perkalian skalar dan perkalian matriks, serta teorema-teorema yang berkaitan dengan operasi pada matriks.
This document provides an overview of matrix algebra concepts including:
- Matrix addition is defined as adding corresponding elements and is commutative and associative.
- Matrix multiplication is defined as taking the dot product of rows and columns. It is associative but not commutative.
- The transpose of a matrix is obtained by flipping rows and columns.
- Properties of matrix operations like addition, multiplication, and transposition are discussed.
The document discusses higher order differential equations. It defines nth order differential equations and describes their general forms. For homogeneous equations, the general solution method involves making an operator form, constructing an auxiliary equation, solving for roots, and finding the complementary solution. For non-homogeneous equations, the method of undetermined coefficients is used to find a particular solution and the general solution is the sum of the complementary and particular solutions. Examples are provided to illustrate the solution methods.
The document provides steps to calculate the area between two curves by finding the intersection points using "y=y" and then integrating between those points. It works through multiple examples of finding the area between various curve combinations, including quadratic, cubic, and linear functions. The key steps are finding the intersection points using "y=y", setting up the integral as the top curve minus the bottom curve, and integrating between the bounds found from the intersection points.
This document summarizes key concepts in Boolean algebra including:
1) Defining a Boolean algebra using a set of elements and binary operations that follow specific axioms.
2) The two-valued Boolean algebra uses the elements 0 and 1 with AND, OR, and NOT operations.
3) Boolean functions consist of variables, constants, and logic operations and can be represented using truth tables.
4) Logic circuits can be designed to implement Boolean functions using gates like AND, OR, NOT, NAND, and NOR.
This document discusses different types of geometrical transformations including translations, reflections, rotations, and dilations. It provides examples and notations for transformations and notes that dilation is not an isometric transformation. Reflections are further categorized by the line they are reflected across such as the x-axis, y-axis, lines of the form y=x, y=-x, y=k, x=k, and y=mx+c. Two examples of finding the line of reflection are given when points are mapped before and after a reflection.
The document discusses regular singular points of differential equations. A point x0 is a regular singular point if the functions P, Q, and R are analytic at x0 or their limits exist and are finite as x approaches x0. Several examples are provided to illustrate regular singular points, including the Bessel and Legendre equations. An irregular singular point is defined as one where these limits do not exist. The behavior of solutions near regular singular points can be analyzed using power series methods.
1) The document discusses logarithms and how they were developed to simplify calculations of large numbers before calculators. Logarithms allow expressing a number as an exponent by finding what power of a base number equals the target number.
2) It provides examples of converting between exponential and logarithmic forms, using logarithmic properties to solve equations, and using change of base formulas to solve when the calculator uses a different base than the problem.
3) Key steps are shown to solve the equation 5x = 15,000 by rewriting in logarithmic form, using the change of base formula to solve for x, and checking the exponential form of the answer.
This document discusses continuity of functions. It provides examples of functions and examines whether they are continuous at various points. The key points are:
1) A function f(x) is continuous at a point a if the left-hand and right-hand limits of f(x) as x approaches a are equal to f(a).
2) An example function is examined and found to be continuous at x=2 because the left and right limits match the value of f(x) at x=2.
3) Another function is discontinuous at x=0 because the left and right limits do not match at that point.
1. This document provides a review of concepts and sample problems related to multiple integrals. It covers topics such as iterated integrals, changing the order of integration, and evaluating double integrals over various regions.
2. The problem set contains 14 problems evaluating double integrals over different regions using techniques like iterated integration and changing the order of integration.
3. Multiple integrals are used to find volumes, masses, moments, and other physical quantities over regions in 2D and 3D space. The document demonstrates how to set up and evaluate multiple integrals to solve applied problems.
Howard, anton calculo i- um novo horizonte - exercicio resolvidos v1cideni
This document contains exercises related to functions and graphs. Exercise set 1.1 contains word problems involving various functional relationships and graphs. Exercise set 1.2 involves evaluating and sketching functions, determining domains and ranges, and identifying piecewise functions. Exercise set 1.3 involves selecting appropriate axis ranges and scales to graph functions over specified domains.
This document provides information about Section I, Part A of the Calculus AB exam. It includes 30 multiple choice questions covering topics like limits, derivatives, integrals, and other calculus concepts. A calculator is not allowed for this section. The questions cover skills like evaluating limits, finding derivatives and integrals, solving related rate and optimization problems, and interpreting graphs.
(1) The document discusses various integration techniques including: review of integral formulas, integration by parts, trigonometric integrals involving products of sines and cosines, trigonometric substitutions, and integration of rational functions using partial fractions.
(2) Examples are provided to demonstrate each technique, such as using integration by parts to evaluate integrals of the form ∫udv, using trigonometric identities to reduce powers of trigonometric functions, and using partial fractions to break down rational functions into simpler fractions.
(3) The key techniques discussed are integration by parts, trigonometric substitutions to transform integrals involving quadratic expressions into simpler forms, and partial fractions to decompose rational functions for integration. Various examples illustrate the
The document discusses numerical integration methods such as Newton-Cotes formulas, the trapezoidal rule, and Simpson's rules. The trapezoidal rule approximates the integral of a function f(x) between bounds a and b by taking the average of f(a) and f(b) and multiplying by the width b-a. Simpson's rules use higher order polynomials to connect function values for a more accurate approximation of the integral. Gauss quadrature implements strategic positioning of points to define straight lines that balance positive and negative errors, improving the integral estimate.
The document provides calculations to estimate the cost of constructing a masonry water platform based on given drawings and specifications. It first estimates quantities of various construction materials needed based on dimensions. It then lists rates for different construction items and uses the estimated quantities to calculate the total cost of each item and the overall project cost, including contingencies and work charges. The total estimated cost is 36,365.60 Sri Lankan Rupees. The document also contains calculations for other construction-related tasks involving geometry, trigonometry, and statistics.
20230411 Discussion of Integration by Substitution.docxSharon Liu
1) The document discusses integration by substitution and evaluates some simple integrals using this method, such as ∫b(ax) dx and ∫(xa)b dx.
2) It then looks at calculating the integral of sin(3x) between 0 and 180 degrees by numerically evaluating the area under the curve in an Excel spreadsheet.
3) Some issues with using the standard table of integrals to evaluate ∫sin(x) dx from 0 to 360 degrees are noted, as it does not give the correct numeric result of 0.
The document discusses solving quadratic equations by factorizing. It provides examples of factorizing quadratic expressions and equations to find their roots. In one example, the quadratic equation x^2 - 2x + 2 = 4 is factored into (x - 2)(x + 2) = 0, showing it has only one real root of x = 2. Another example factors a quadratic expression f(x) = x^2 - x - 1 to find its two roots of 1 and -1. The document demonstrates how to factorize quadratic expressions and equations in order to solve for their real roots.
x
y
2.5 3.0 3.5
-1.0 6 7 8
1.0 0 1 2
3.0 -6 -5 -4
MATH 223
FINAL EXAM REVIEW PACKET ANSWERS
(Fall 2012)
1. (a) increasing (b) decreasing
2. (a) 2 2( 3) 25y z− + = This is a cylinder parallel to the x-axis with radius 5.
(b) 3x = , 3x = − . These are vertical planes parallel to the yz-plane.
(c) 2 2 2z x y= + . This is a cone (one opening up and one opening down) centered on the z-axis.
3. There are many possible answers.
(a) 0x = produces the curve 23y z= − .
(b) 1y = produces the curves 23 cosz x= − and 23 cosz x= − − .
(c)
2
x
π
= produces the curves 3z = and 3z = − .
4. (a) (b) (i) 1 (ii) Increase (iii) Decrease
5. (a) Paraboloids centered on the x-axis, opening up in the positive x direction. 2 2x y z c= + +
(b) Spheres centered at the origin with radius 1 ln c− for 0 c e< ≤ . 2 2 2 1 lnx y z c+ + = −
6. (a) 6 am 11:30 am
(b) Temperature as a function of time at a depth of 20 cm.
(c) Temperature as a function of depth at noon.
7. ( , ) 2 3 2z f x y x y= = − −
8. (a) II, III, IV, VI (b) I (c) I, III, VI (d) VI (e) I, V
9. (a)
12
4 12
5
z x y= − + (b) There are many possible answers.
12
4
5
i j k+ −
(c)
3 569
2
10. (a) iii, vii (b) iv (c) viii (d) ii (e) v, vi (f) i, ix
11. There are many possible answers.
(a) ( )5 4 3
26
i j k− +
or ( )5 4 3
26
i j k− − +
(b) 2 3i j− +
(c)
4
cos
442
θ = , 1.38θ ≈ radians (d) ( )4 4 3
26
i j k− +
(e) 4 11 17i j k− − −
12. (a)
3
5
a = − (b)
1
3
a = (c) 2( 1) ( 2) 3( 3) 0x y z− − + + − = (d)
1 2 , 2 , 3 3x t y t z t= + = − − = +
13. 6 39i
or 6 39i−
14. (a)
( )
2
23 2 2
3 2
3 1
z x y x
x x y x y
∂
= −
∂ + + +
(b)
( )4
10 4 3
5
H
H T
f
H
+ +
=
−
(c)
2
2 2
1 1z
x y y x
∂
= − −
∂ ∂
15. (a) 2 2 24 ( 1) 3 ( 2) 2z e x e y e= − + − + (b) 4( 3) 8( 3) 6( 6) 0x y z− + − + − =
16. (a)
2sin(2 ) cos(2 )
5 5
v v
ds dv d
α α
α= +
(b) The distance s decreases if the angle α increases and the initial speed v remains constant.
(c) 0.0886α∆ ≈ − . The angle decreases by about 0.089 radians.
17. (a) The water is getting shallower.
4
( 1, 2)
17
uh − = −
(b) There are many possible answers. 3i j+
(c) 72 ft/min
18. (a)
( )
2 2 2
22 2 22
2 2
1 1 11
yz xyz z yz
grad i j k
x x xx
= − + + + + + +
(b) ( ) ( ) ( )( )2 2 2 2curl x y z i y z j xz k i zj yk+ + − + + = + −
(c) ( ) ( ) ( )( )2 3 3cos sec 2 cos sin sec tan 3z zdiv x i x y j e k x x x y y e+ + = − + +
(d)
37
3
(e) ( , , ) sin zg x y z xy e c= + +
19. ( , ) 4 3vG a b = −
20. (a) positive (b) negative (c) negative (d) negative (e) positive (f) zero
21. (a)
(.
Calculus Early Transcendentals 10th Edition Anton Solutions Manualnodyligomi
This document contains a table of contents for a calculus textbook, outlining 15 chapters that cover topics from limits and continuity to vector calculus and partial derivatives. It also includes 3 appendix sections on graphing functions, trigonometry review, and solving polynomial equations.
IIT Jam math 2016 solutions BY TrajectoryeducationDev Singh
The document contains a mathematics exam question paper with 10 single mark questions (Q1-Q10) and 20 two mark questions (Q11-Q30). The questions cover topics like sequences, linear transformations, integrals, permutations, differential equations etc. Some key questions asked about the nature of a sequence involving sines, order of a permutation, evaluating a limit, checking if a differential equation is exact etc. and provided solutions to them.
This document contains solutions to chapter 14 problems from the 7th edition of Engineering Circuit Analysis. It includes multiple pages with solutions to circuit analysis problems involving complex frequencies. The solutions provide the complex frequency (s) associated with different circuit variables like voltage and current. Key information includes identifying the complex frequency of signals, calculating impedances of circuit elements, and determining voltages and currents using complex frequency analysis.
This document provides information about integration in higher mathematics. It begins with an overview of integration as the opposite of differentiation. It then discusses using antidifferentiation to find integrals by reversing the power rule for differentiation. Several examples are provided to illustrate integrating polynomials. The document also discusses using integrals to find the area under a curve or between two curves. It provides examples of calculating areas bounded by graphs and the x-axis. Finally, it presents some exam-style integration questions for practice.
This is the entrance exam paper for ISI MSQE Entrance Exam for the year 2008. Much more information on the ISI MSQE Entrance Exam and ISI MSQE Entrance preparation help available on http://crackdse.com
The document provides an introduction to the binomial theorem. It defines binomial coefficients through the Pascal triangle and gives an explicit formula for computing them using factorials. The binomial theorem is then derived and stated, providing a formula for expanding expressions of the form (a + b)^n in terms of binomial coefficients. Several examples are worked out to demonstrate expanding expressions and finding coefficients using the binomial theorem. Applications to estimating interest calculations are also briefly discussed.
1. The document discusses equations, inequalities, and systems of equations for modeling real-world situations. It covers topics such as linear equations, quadratic equations, simultaneous equations, and inequalities.
2. Examples are provided to demonstrate modeling problems using different equation and inequality types. Variables are identified and equations or inequalities are set up to represent the relationships in the word problems.
3. The solutions show solving the equations or systems of equations to find the values of the variables that satisfy the constraints, thus addressing the questions asked.
This document provides a review of exercises for a Math 112 final exam. It contains 31 multi-part exercises covering topics like graphing, logarithms, trigonometry, and word problems. The review is intended to help students practice problems similar to what may appear on the exam. The exam will have two parts, one allowing a calculator and one not.
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This document outlines the responsibilities and roles of different team members in a construction company regarding health and safety. It defines the roles of the Managing Director, who is responsible for overall health and safety arrangements, and the Director Responsible for Health and Safety, who is accountable to the Managing Director. It also outlines some of the key responsibilities of each role, such as ensuring policies and legal requirements are met and communicated. The document emphasizes the importance of effective communication and an overall company policy to make all employees aware of their individual health and safety responsibilities.
- An inclined U-tube manometer allows more accurate measurement of small pressure changes compared to a standard U-tube manometer due to greater deflection of the liquid level for the same pressure change.
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Measuring tendering and estimating for CBESahl Buhary
The client plays a major role in the tender process, including preparing tender documents, reviewing contractor submissions, and selecting the winning contractor. There are various constraints that must be considered in the tender process, including the client's objectives, financial constraints, physical constraints of the construction site, legal restrictions, and design requirements. The contractual documentation required to support the tender process includes the notice of tender, conditions of contract, drawings, bill of quantities, form of tender, specifications, and instructions for bidders. Estimating unit costs requires considering factors such as labor rates based on wages and output, material costs including purchasing, transport, and wastage, plant costs such as rental fees and maintenance, and overhead costs associated with both the office and
This document discusses employability skills for a civil engineering career. It covers developing responsibilities and performance objectives, evaluating performance goals, creating a personal development plan, reviewing motivation techniques, communication styles, time management strategies, and team dynamics. For developing responsibilities, it provides examples of personal responsibilities and performance objectives for a civil engineer. It also includes templates for evaluating goals, a personal development plan, and a motivation analysis. When discussing communication, it outlines different communication styles and problem solving approaches. It proposes time management strategies like creating to-do lists and schedules. Finally, it describes the roles and dynamics of effective teamwork, including allocating resources, communicating, fulfilling requirements, and providing training.
This document discusses construction methods for tunnels and hydraulic structures. For tunnels, it lists various construction methods including cut-and-cover, boring machines, drill and blast, and others. It then discusses the New Austrian Tunneling Method and pipe jacking/microtunneling in more detail. For hydraulic structures, it outlines classifications based on function and then explains the construction methods for earth dams, aqueducts, and sluice gates in detail. Key steps for earth dam construction include site preparation, spillway design, and compacting soil layers to increase stability.
Rainfall intensity duration frequency curve statistical analysis and modeling...bijceesjournal
Using data from 41 years in Patna’ India’ the study’s goal is to analyze the trends of how often it rains on a weekly, seasonal, and annual basis (1981−2020). First, utilizing the intensity-duration-frequency (IDF) curve and the relationship by statistically analyzing rainfall’ the historical rainfall data set for Patna’ India’ during a 41 year period (1981−2020), was evaluated for its quality. Changes in the hydrologic cycle as a result of increased greenhouse gas emissions are expected to induce variations in the intensity, length, and frequency of precipitation events. One strategy to lessen vulnerability is to quantify probable changes and adapt to them. Techniques such as log-normal, normal, and Gumbel are used (EV-I). Distributions were created with durations of 1, 2, 3, 6, and 24 h and return times of 2, 5, 10, 25, and 100 years. There were also mathematical correlations discovered between rainfall and recurrence interval.
Findings: Based on findings, the Gumbel approach produced the highest intensity values, whereas the other approaches produced values that were close to each other. The data indicates that 461.9 mm of rain fell during the monsoon season’s 301st week. However, it was found that the 29th week had the greatest average rainfall, 92.6 mm. With 952.6 mm on average, the monsoon season saw the highest rainfall. Calculations revealed that the yearly rainfall averaged 1171.1 mm. Using Weibull’s method, the study was subsequently expanded to examine rainfall distribution at different recurrence intervals of 2, 5, 10, and 25 years. Rainfall and recurrence interval mathematical correlations were also developed. Further regression analysis revealed that short wave irrigation, wind direction, wind speed, pressure, relative humidity, and temperature all had a substantial influence on rainfall.
Originality and value: The results of the rainfall IDF curves can provide useful information to policymakers in making appropriate decisions in managing and minimizing floods in the study area.
Introduction- e - waste – definition - sources of e-waste– hazardous substances in e-waste - effects of e-waste on environment and human health- need for e-waste management– e-waste handling rules - waste minimization techniques for managing e-waste – recycling of e-waste - disposal treatment methods of e- waste – mechanism of extraction of precious metal from leaching solution-global Scenario of E-waste – E-waste in India- case studies.
VARIABLE FREQUENCY DRIVE. VFDs are widely used in industrial applications for...PIMR BHOPAL
Variable frequency drive .A Variable Frequency Drive (VFD) is an electronic device used to control the speed and torque of an electric motor by varying the frequency and voltage of its power supply. VFDs are widely used in industrial applications for motor control, providing significant energy savings and precise motor operation.
AI for Legal Research with applications, toolsmahaffeycheryld
AI applications in legal research include rapid document analysis, case law review, and statute interpretation. AI-powered tools can sift through vast legal databases to find relevant precedents and citations, enhancing research accuracy and speed. They assist in legal writing by drafting and proofreading documents. Predictive analytics help foresee case outcomes based on historical data, aiding in strategic decision-making. AI also automates routine tasks like contract review and due diligence, freeing up lawyers to focus on complex legal issues. These applications make legal research more efficient, cost-effective, and accessible.
Use PyCharm for remote debugging of WSL on a Windo cf5c162d672e4e58b4dde5d797...shadow0702a
This document serves as a comprehensive step-by-step guide on how to effectively use PyCharm for remote debugging of the Windows Subsystem for Linux (WSL) on a local Windows machine. It meticulously outlines several critical steps in the process, starting with the crucial task of enabling permissions, followed by the installation and configuration of WSL.
The guide then proceeds to explain how to set up the SSH service within the WSL environment, an integral part of the process. Alongside this, it also provides detailed instructions on how to modify the inbound rules of the Windows firewall to facilitate the process, ensuring that there are no connectivity issues that could potentially hinder the debugging process.
The document further emphasizes on the importance of checking the connection between the Windows and WSL environments, providing instructions on how to ensure that the connection is optimal and ready for remote debugging.
It also offers an in-depth guide on how to configure the WSL interpreter and files within the PyCharm environment. This is essential for ensuring that the debugging process is set up correctly and that the program can be run effectively within the WSL terminal.
Additionally, the document provides guidance on how to set up breakpoints for debugging, a fundamental aspect of the debugging process which allows the developer to stop the execution of their code at certain points and inspect their program at those stages.
Finally, the document concludes by providing a link to a reference blog. This blog offers additional information and guidance on configuring the remote Python interpreter in PyCharm, providing the reader with a well-rounded understanding of the process.
Redefining brain tumor segmentation: a cutting-edge convolutional neural netw...IJECEIAES
Medical image analysis has witnessed significant advancements with deep learning techniques. In the domain of brain tumor segmentation, the ability to
precisely delineate tumor boundaries from magnetic resonance imaging (MRI)
scans holds profound implications for diagnosis. This study presents an ensemble convolutional neural network (CNN) with transfer learning, integrating
the state-of-the-art Deeplabv3+ architecture with the ResNet18 backbone. The
model is rigorously trained and evaluated, exhibiting remarkable performance
metrics, including an impressive global accuracy of 99.286%, a high-class accuracy of 82.191%, a mean intersection over union (IoU) of 79.900%, a weighted
IoU of 98.620%, and a Boundary F1 (BF) score of 83.303%. Notably, a detailed comparative analysis with existing methods showcases the superiority of
our proposed model. These findings underscore the model’s competence in precise brain tumor localization, underscoring its potential to revolutionize medical
image analysis and enhance healthcare outcomes. This research paves the way
for future exploration and optimization of advanced CNN models in medical
imaging, emphasizing addressing false positives and resource efficiency.
Electric vehicle and photovoltaic advanced roles in enhancing the financial p...IJECEIAES
Climate change's impact on the planet forced the United Nations and governments to promote green energies and electric transportation. The deployments of photovoltaic (PV) and electric vehicle (EV) systems gained stronger momentum due to their numerous advantages over fossil fuel types. The advantages go beyond sustainability to reach financial support and stability. The work in this paper introduces the hybrid system between PV and EV to support industrial and commercial plants. This paper covers the theoretical framework of the proposed hybrid system including the required equation to complete the cost analysis when PV and EV are present. In addition, the proposed design diagram which sets the priorities and requirements of the system is presented. The proposed approach allows setup to advance their power stability, especially during power outages. The presented information supports researchers and plant owners to complete the necessary analysis while promoting the deployment of clean energy. The result of a case study that represents a dairy milk farmer supports the theoretical works and highlights its advanced benefits to existing plants. The short return on investment of the proposed approach supports the paper's novelty approach for the sustainable electrical system. In addition, the proposed system allows for an isolated power setup without the need for a transmission line which enhances the safety of the electrical network
Null Bangalore | Pentesters Approach to AWS IAMDivyanshu
#Abstract:
- Learn more about the real-world methods for auditing AWS IAM (Identity and Access Management) as a pentester. So let us proceed with a brief discussion of IAM as well as some typical misconfigurations and their potential exploits in order to reinforce the understanding of IAM security best practices.
- Gain actionable insights into AWS IAM policies and roles, using hands on approach.
#Prerequisites:
- Basic understanding of AWS services and architecture
- Familiarity with cloud security concepts
- Experience using the AWS Management Console or AWS CLI.
- For hands on lab create account on [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
# Scenario Covered:
- Basics of IAM in AWS
- Implementing IAM Policies with Least Privilege to Manage S3 Bucket
- Objective: Create an S3 bucket with least privilege IAM policy and validate access.
- Steps:
- Create S3 bucket.
- Attach least privilege policy to IAM user.
- Validate access.
- Exploiting IAM PassRole Misconfiguration
-Allows a user to pass a specific IAM role to an AWS service (ec2), typically used for service access delegation. Then exploit PassRole Misconfiguration granting unauthorized access to sensitive resources.
- Objective: Demonstrate how a PassRole misconfiguration can grant unauthorized access.
- Steps:
- Allow user to pass IAM role to EC2.
- Exploit misconfiguration for unauthorized access.
- Access sensitive resources.
- Exploiting IAM AssumeRole Misconfiguration with Overly Permissive Role
- An overly permissive IAM role configuration can lead to privilege escalation by creating a role with administrative privileges and allow a user to assume this role.
- Objective: Show how overly permissive IAM roles can lead to privilege escalation.
- Steps:
- Create role with administrative privileges.
- Allow user to assume the role.
- Perform administrative actions.
- Differentiation between PassRole vs AssumeRole
Try at [killercoda.com](https://killercoda.com/cloudsecurity-scenario/)
Generative AI Use cases applications solutions and implementation.pdfmahaffeycheryld
Generative AI solutions encompass a range of capabilities from content creation to complex problem-solving across industries. Implementing generative AI involves identifying specific business needs, developing tailored AI models using techniques like GANs and VAEs, and integrating these models into existing workflows. Data quality and continuous model refinement are crucial for effective implementation. Businesses must also consider ethical implications and ensure transparency in AI decision-making. Generative AI's implementation aims to enhance efficiency, creativity, and innovation by leveraging autonomous generation and sophisticated learning algorithms to meet diverse business challenges.
https://www.leewayhertz.com/generative-ai-use-cases-and-applications/
Embedded machine learning-based road conditions and driving behavior monitoringIJECEIAES
Car accident rates have increased in recent years, resulting in losses in human lives, properties, and other financial costs. An embedded machine learning-based system is developed to address this critical issue. The system can monitor road conditions, detect driving patterns, and identify aggressive driving behaviors. The system is based on neural networks trained on a comprehensive dataset of driving events, driving styles, and road conditions. The system effectively detects potential risks and helps mitigate the frequency and impact of accidents. The primary goal is to ensure the safety of drivers and vehicles. Collecting data involved gathering information on three key road events: normal street and normal drive, speed bumps, circular yellow speed bumps, and three aggressive driving actions: sudden start, sudden stop, and sudden entry. The gathered data is processed and analyzed using a machine learning system designed for limited power and memory devices. The developed system resulted in 91.9% accuracy, 93.6% precision, and 92% recall. The achieved inference time on an Arduino Nano 33 BLE Sense with a 32-bit CPU running at 64 MHz is 34 ms and requires 2.6 kB peak RAM and 139.9 kB program flash memory, making it suitable for resource-constrained embedded systems.
Comparative analysis between traditional aquaponics and reconstructed aquapon...bijceesjournal
The aquaponic system of planting is a method that does not require soil usage. It is a method that only needs water, fish, lava rocks (a substitute for soil), and plants. Aquaponic systems are sustainable and environmentally friendly. Its use not only helps to plant in small spaces but also helps reduce artificial chemical use and minimizes excess water use, as aquaponics consumes 90% less water than soil-based gardening. The study applied a descriptive and experimental design to assess and compare conventional and reconstructed aquaponic methods for reproducing tomatoes. The researchers created an observation checklist to determine the significant factors of the study. The study aims to determine the significant difference between traditional aquaponics and reconstructed aquaponics systems propagating tomatoes in terms of height, weight, girth, and number of fruits. The reconstructed aquaponics system’s higher growth yield results in a much more nourished crop than the traditional aquaponics system. It is superior in its number of fruits, height, weight, and girth measurement. Moreover, the reconstructed aquaponics system is proven to eliminate all the hindrances present in the traditional aquaponics system, which are overcrowding of fish, algae growth, pest problems, contaminated water, and dead fish.
Comparative analysis between traditional aquaponics and reconstructed aquapon...
Applied mathematics for complex engineering
1. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 1
LO1 Understand the use of trigonometric functions.
1.1.During a model study of structure by sending vibration force to the structure, its
deformation was measured which was oscillating with a maximum displacement of
32 cm and a frequency of 50 Hz. At time = 0 the displacement is 150 mm.
Express the displacement in the general form sin( ± ). Sketch the
displacement of structure for first 0.02 sec.
Answers
sin( ± )
Amplitude = maximum displacement = 0.32m.
Angular velocity, = 2 = 2 (50) = 100 rad/s.
Hence, = . ( + ) .
When = 0, displacement = 0.15 m, t = 0
hence, 0.15 = 0.32sin(100 × 0 + )
0.15 = 0.32sin(0 + )
sin =
0.15
0.32
= 0.4687
= sin 0.4687
= 27.95° = 0.487rad.
Thus, = . ( + . ) .
1.2.Solve the trigonometric equation√2cosx + sinx = 1,for values of x in the interval of
(−π < x < π).
Answers
√2 sin + cos = 1
√2 sin = 1 − cos
Square both sides
(√2 sin ) = (1 − cos )
2 sin = 1 − 2 cos + cos
2(1 − cos ) = 1 − 2 cos + cos
2 − 2 cos ) = 1 − 2 cos + cos
3 cos − 2 cos − 1 = 0
3 cos − 3 cos + cos − 1 = 0
3 cos (cos − 1) + 1(cos − 1) = 0
(3 cos + 1)(cos − 1) = 0
∴ (3 cos + 1) = 0 (cos − 1) = 0
Hence, cos = −
1
3
1
2. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 2
= cos −
1
3
cos 1
= 109.47˚ 0˚ (−π < x < π).
1.3 A person is standing on top of a hill that slopes downwards uniformly at an angle ∅
with respect to the horizontal. The person throws a stone at an initial angle from
the horizontal with an initial speed of . You may neglect air resistance. Evaluate
the problem and show that the horizontal range of the stone when the stone strikes
the ground as (0.5 sin 2 + tan ∅)?
Answers
=
ℎ
⟶ = +
1
2
= . ∗ − − − − − 1 =
.
↓ ℎ = (− . ) ∗ +
1
2
ℎ = − ∗ +
1
2
. =
− . ∗
.
+
1
2
∗
²
² ∗ ²
θ = − +
1
2
∗
² ²
= − +
1
2 ² ∗ ²
2( + ) ² + ²
=
3. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 3
=
2 ²
∗
∗ ∗
+ ∗
=
²
+ ² ∗
LO2 Be able to solve algebraic equations representing engineering problems
2.1.A Yogurt Company makes three yogurt blends: Lime Orange, using 2 quarts of lime
yogurt and 2 quarts of orange yogurt per gallon; Lime Lemon, using 3 quarts of lime
yogurt and 1 quart of lemon yogurt per gallon; and Orange Lemon, using 3 quarts of
orange yogurt and 1 quart of lemon yogurt per gallon. Each day the company has
800quarts of lime yogurt, 650 quarts of orange yogurt, and 350 quarts of lemon
yogurt available. How many gallons of each blend should it make each day if it
wants to use up all of the supplies? (Construct algebraic equations).
Answers
x = the number of gallons of Lime Lemon
y = the number of gallons of Lime Orange
z = the number of gallons of Orange Lemon.
If Softflow makes x quarts of LimeOrange, y quarts of LimeLemon, and z quarts of
OrangeLemon, it will need a total of
2x + 3y
quarts of lime yoghurt. Since Softflow has a total of 800 quarts of lime yogurt on hand, and it
wants nothing left over, we must have; Amount used = Amount Available
2x + 3y = 800
Similarly, we get two more equations for orange and lemon yogurt:
2x + 3z = 650
Y + Z = 350
We can organize the given information in a table. To set up the table, do the following:
Place the categories corresponding to the unknowns along the top.
Add an extra column for the "Total Available"
Place the "ingredients" down the side.
6. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 6
= + + + + + − −
= + + +
= 1 +
1
+
1
+
1
(b)
1 2 3
1 3 5
2 5 1
6
9
8
∆ =
1 2 3
1 2 5
2 5 1
= 1(−22) − 2(−9) + 3(−1) = −
=
6 2 3
9 2 5
8 5 1
∆
=
6(−22) − 2(−31) + 3(21)
−7
=
−7
−7
= 1
=
1 6 3
1 9 5
2 8 1
∆
=
1(−31) − 6(−9) + 3(−10)
−7
=
−7
−7
= 1
=
1 2 6
1 2 9
2 5 8
∆
=
1(−21) − 2(−10) + 6(−1)
−7
=
−7
−7
= 1
2.3 (a) Find a root of the function f(x)= 5 + 11 − 17 = 0 using
bisection method (Pre-specified relative error tolerance εs = 6%)
(b) You are working for ‘DOWN THE TOILET COMPANY’ that
makes floats for ABC commodes. The floating ball has a specific gravity
of 0.6 and has a radiusof 5.5 cm. You are asked to find the depth to
which the ball is submerged when floating in water.
7. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 7
The equation that gives the depth x in meters to which the ball is submerged under
water is given by
x − 0.165x + 3.993 × 10 = 0
Use the Newton-Raphson method of finding roots of equations to find
i) The depth x to which the ball is submerged under water. Conduct three iterations to
estimate the root of the above equation.
ii) The absolute relative approximate error at the end of each iteration.
(Hint: assume the initial guess of the root of f (x) = 0 is x= 0.05 m)
Answers
From the physics of the problem, the ball would be submerged between x = 0 and x = 2R,
where R = radius of the ball, that is
11.00
055.020
20
x
x
Rx
8. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 8
Let us assume
11.0
00.0
ux
xl
Check if the function changes sign between xl and xu
4423
4423
10662.210993.311.0165.011.011.0
10993.310993.30165.000
fxf
fxf
u
l
Hence,
010662.210993.311.00 44
ffxfxf ul
So there is at least on root between xl and xu, that is between 0 and 0.11
Iteration 1
The estimate of the root is
055.0
2
11.00
2
u
m
xx
x l
010655.610993.3055.00
10655.610993.3055.0165.0055.0055.0
54
5423
ffxfxf
fxf
ml
m
9. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 9
Hence the root is bracketed between xm and xu, that is, between 0.055 and 0.11. So, the lower
and upper limits of the new bracket are
11.0,055.0 ul xx
At this point, the absolute relative approximate error cannot be calculated as we do not
have a previous approximation.
The absolute relative approximate error at the end of Iteration 3 is
%20
100
06875.0
0825.006875.0
100
new
m
old
m
new
m
a
x
xx
Still none of the significant digits are at least correct in the estimated root of the equation as the
absolute relative approximate error is greater than 5%.
Seven more iterations were conducted and these iterations are shown in Table 1.
Root of f(x)=0 as function of number of iterations for bisection method.
a
a
10. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 10
Iteration x xu xm
a % f(xm)
1
2
3
4
5
6
7
8
9
10
0.00000
0.055
0.055
0.055
0.06188
0.06188
0.06188
0.06188
0.0623
0.0623
0.11
0.11
0.0825
0.06875
0.06875
0.06531
0.06359
0.06273
0.06273
0.06252
0.055
0.0825
0.06875
0.06188
0.06531
0.06359
0.06273
0.0623
0.06252
0.06241
----------
33.33
20.00
11.11
5.263
2.702
1.370
0.6897
0.3436
0.1721
6.655×10−5
−1.622×10−4
−5.563×10−5
4.484×10−6
−2.593×10−5
−1.0804×10−5
−3.176×10−6
6.497×10−7
−1.265×10−6
−3.0768×10−7
Hence the number of significant digits at least correct is given by the largest value or m for
which
463.23442.0log2
23442.0log
103442.0
105.01721.0
105.0
2
2
2
m
m
m
m
m
a
So
2m
The number of significant digits at least correct in the estimated root of 0.06241 at the end of the
10th
iteration is 2.
11. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 11
LO3 Be able to apply calculus to engineering problems
3.1. The Wave Equation: We generally visit beach and if we stand on an ocean shore
and take a snapshot of the waves, the picture shows a regular pattern of peaks and
valleys in an instant of time. We see periodic vertical motion in space, with respect to
distance. If we stand in the water, we can feel the rise and fall of the water as the waves
go by. We see periodic vertical motion in time. This beautiful symmetry is expressed by
the one-dimensional wave equation
=
Where w is the wave height, x is the distance variable, t is the time variable, and c is the
velocity with which the waves are propagated (see the accompanying figure).
Show that the following functions are all solutions of the wave equation by determining
relevant partial derivatives. (P3.1)
a. = sin( + )
b. = cos(2 + 2 )
c. = sin( + ) + cos(2 + 2 )
d. = ln(2 + 2 )
e. = tan(2 − 2 )
f. = 5 cos(3 + 3 ) +
Answers
=
15. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 15
3.2 A manufacturing company is to be made in the form of a rectangular room as a
store room. They decided to cover the surfaces of the room with canvas covering on the
top, back and sides. They have to minimize the material cost for canvas covering with
effective usage of floor area. Determine the minimum surface area of canvas necessary
if the volume of the box is to be 250 m3 by classify stationary points of function of the
surface area of room. (P3.2 & M1.2)
Answers
− ∗ ∗ ℎ
=
⇒ = ∗ ℎ
250 = ∗ ℎ
=
250
ℎ
= ( ) ∗ 2 + ( ∗ ℎ) ∗ 2 ∗ (ℎ ∗ )
= 2 + 2 ℎ + ℎ
= 2 ∗
250
ℎ
+
2 ∗ 250
+
250
500
ℎ
+
500
+
250
500
ℎ
+
500 ∗ ℎ
250
+
250
= 0 + 2 ∗ ℎ −
250
²
= 0 , 2ℎ = 250
² =
17. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 17
= −3, = −3
( , )(− , − ) ( , )
3.3.Evaluate
Determine the integral of followings: (P3.3)
1. 2
=
1
2
sin 2 2
=
1
2
sin 2
=− cos 2 × +
= −
1
4
cos 2 +
Here C is an arbitary constant
2. √4 + 1
=
1
4
4 + 1 4
=
1
4
√ + 1
=
1
4
( + 1) /
=
1
4
2
3
( + 1) /
=
1
6
( + 1) /
=
1
6
(626 + 1)
=
1
6
× 15 661.5
= 2610.25
3. 2
= − − 2
= −
= − +
= − +
=
= 2
= 2
=
= 4
= 4
= −
= −2
= −2
18. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 18
4. sin cos .
=
=
2
+
=
2
+
5. sin ( |cos |)
= − |cos | cos − (− cos ).
1
cos
(− sin ) .
= − |cos | cos − sin .
= − |cos | cos . + cos +
6. cos
= sin − sin
= 2 − 0 + cos
= 2 + 0 − 1
= 2 − 1
= 0.5708
7. sin
= (− cos ) − (− ) 2
= ( − 0) + 2 cos
= + 2 sin − 2 sin
= + 0 + 2 cos
= + 2 −1 − 1
= − 4
= sin
= cos
= cos
= . − . .
20. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 20
3.4
The diagram shows the curve Y=√(3x+1) and the points P(0,1) and Q(1,2) on the curve.
(i) Find the area enclosed between the curve, the Y axis, the X axis and the line
X=1
(ii) Find the volume of the solid generated when the above area is rotated about
the X axis by 3600
Answers
I. = √3 + 1
=
2
3
(3 + 1) /
3
=
2
9
(8 − 1) =
14
9
= 1.556
II. =
= (3 + 1)
=
3
2
+
=
3
2
+ 1 =
5
2
= 7.854.
21. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 21
LO5 Be able to apply statistical techniques to engineering problems.
4.1 In an experiment to determine the relationship between force and momentum, a force, X, is applied to
a mass, by placing the mass on an inclined plane, and the time, Y, for the velocity to change from u m/s to
v m/s is measured. The results obtained are as follows:
Force
(N)
11.4 18.7 11.7 12.3 14.7 18.8 19.6
Time
(s)
0.56 0.35 0.55 0.52 0.43 0.34 0.31
By determine the lines of best fit find
(a) The time corresponding to a force of 16 N, and
(b) The force at a time of 0.25 s, assuming the relationship is linear outside of the range of values given.
Answers
Find XY, X2
, Y2
X
value
Y
Value
X*X X*Y Y*Y
11.4 0.56 11.4*11.4=129.96 11.4*0.56=6.384 0.56*0.56=0.3136
18.7 0.35 18.7*18.7=349.69 18.7*0.35=6.545 0.35*0.35=0.1225
11.7 0.55 11.7*11.7=136.89 11.7*0.55=6.435 0.55*0.55=0.3025
12.3 0.52 12.3*12.3=151.29 12.3*0.52=6.396 0.52*0.52=0.2704
14.7 0.43 14.7*14.7=216.09 14.7*0.43=6.321 0.43*0.43=0.1849
18.8 0.34 18.8*18.8=353.44 18.8*0.34=6.392 0.34*0.34=0.1156
19.6 0.31 19.6*19.6=384.16 19.6*0.31=6.076 0.31*0.31=0.0961
Table 3.21
Find ΣX, ΣY, ΣXY, ΣX2
, ΣY2
.
ΣX = 11.4+18.7+11.7+12.3+14.7+18.8+19.6 = 107.2
ΣY = 0.56+0.35+0.55+0.52+0.43+0.34+0.31 = 3.06
ΣXY = 6.384+6.545+6.435+6.396+6.321+6.392+6.076=44.549
22. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 22
ΣX2
= 129.96+349.69+136.89+151.29+216.09+353.44+384.16=1721.52
ΣY2
= 0.3136+0.1225+0.3025+0.2704+0.1849+0.1156+0.0961= 1.4056
The equation of the regression line of force on times is of the form Y=a0 + a1X and the constants
a0 and a1 are determined from the normal equations:
ΣY = a0N + a1ΣX
ΣXY = a0ΣX + a1ΣX2
Thus 3.06 = 7a0 + 107.2a1
And 44.549 = 107.2a0 + 1721.52a1
Solving for a1
3.06 = 7a0 + 107.2a1
3.06 - 107.2a1 = 7a0
(3.06 - 107.2a1)/7= a0
44.549 = 107.2a0 + 1721.52a1
44.549 = 107.2/7(3.06-107.2a1) + 1721.52a1
44.549 = 46.86171429-1641.691429a1 + 1721.52a1
44.549 = 46.8617429 +79.828571a1
-2.3127429 = 79.828571a1
a1 = -0.02897136790786346407228058736013
a1 = -0.029 (correct to 3 significant figures)
Solving for a0
3.06 = 7a0 + 107.2a1
3.06 = 7a0 + 107.2*-0.029
3.06 = 7a0-3.1088
a0 = 0.88125714285714285714285714285714
23. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 23
a0 = 0.881(correct to 3 significant figures)
Solving these simultaneous equations gives a0 = 4.873 and a1 = -0.029. Thus the equation of
the regression line of force on times is :
Y = a0 + a1
Y = 0.881- 0.029X
The equation of the regression line of force on times is of the form X=b0 + b1Y and the constants
b0 and b1 are determined from the normal equations:
ΣX = b0N + b1ΣY
ΣXY = b0ΣY + b1ΣY2
Thus 107.2 = 7b0 + 3.06b1
And 44.549 = 3.06b0 + 1.4056b1
Solving for b0
107.2 = 7b0 + 3.06b1
107.2 – 7b0 = 3.06b1
b1= (107.2 – 7b0)/ 3.06
44.549 = 3.06b0 + 1.4056b1
44.549 = 3.06b0 + 1.4056/3.06(107.2 – 7b0)
44.549 = 3.06b0 + 49.2419346 - 3.2154248b0
0.1554248b0= 4.6929346
b0 = 30.19424571
b0 = 30.194 (correct to 3 significant figures)
Solving for b1
24. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 24
107.2 = 7b0 + 3.06b1
107.2 = 7*30.194 + 3.06b1
107.2 = 211.358 + 3.06b1
3.06b1 = -104.158
b1 = -34.0385620915
b1 = -34.039 (correct to 3 significant figures)
Solving these simultaneous equations gives b0 = 30.194 and b1 = -34.039 Thus the equation of
the regression line of force on times is :
X = b0 + b1
X = 30.194 -34.039Y
For X = 30.194 -34.039Y
x y
0 0.887041
5 0.740151
10 0.593261
15 0.44637
20 0.29948
25 0.15259
30 0.005699
Table 3.2.2
For Y = 0.881 - 0.29X
X Y
0 0.881
5 0.736
10 0.591
15 0.446
20 0.301
25 0.156
30 0.011
Table 3.2.3
25. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 25
Diagram 3.2
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
TimesS
Force in N
X=30.194-34.039Y
Y=0.881-0.29X
26. HND in Construction & Built Environment (Civil Engineering) BCAS DOHA QATAR
Unit 36: Applied Mathematics for Complex Engineering Problems Page 26
Diagram 3.2.1
See the diagram 3.21, when the force is 16N, the times should be 0.41S, when the time is 0.25s,
the force is 22.5N.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 5 10 15 20 25 30
TimesS
Force in N
X=30.194-34.039Y
Y=0.881-0.29X