Exercise unit 1 differential equation (hum3032)
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MARAThis document contains an exercise with 3 problems on determining the order and degree of differential equations, solving differential equations using separable variables and integrating factors. Problem 1 asks to determine the order and degree of 3 given differential equations. Problem 2 provides 3 differential equations to solve using separable variables. Problem 3 gives 4 differential equations to solve using the method of integrating factors, some with initial conditions.
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Exercise unit 1 hum3032
This document contains an exercise on solving differential equations. It includes determining the order and degree of given differential equations, solving equations using the method of separable variables, and solving equations using the method of integrating factors. The exercise contains multiple parts for each type of problem.
Tut 1
This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.
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This document contains 3 problems:
1) Showing that a sequence is increasing and less than 3.
2) Calculating the average value of a function over an interval.
3) Finding the solution to a partial differential equation of the form 2xyy' – y^2 + x^2 = 0. The solution is found to be y = √cx - x^2.
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This document contains the solutions to 4 differential equation problems using the method of separable variables.
1) The solution for dy/dx = sen(x+y) is given as y = 2arctan(-2-x-C/x+C) + 2πn - x.
2) The solution for dy/dx = 2 + √y - 2x + 3 is given as √y - 2x + 3 = x + C.
3) The solution for dy/dx = (x + y + 1)^2 is given as arctan(x + y + 1) = x + C.
4) The solution for dy/dx = 1
Tugas matematika menemukan konsep persamaan kuadrat
1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
Bresenham's line drawing algorithm
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
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This document discusses separable differential equations, which can be written in the form dx/dy = M(x)N(y), where M is a function of x only and N is a function of y only. Three examples are provided to illustrate solving separable equations: (1) solving a nonlinear separable equation, (2) solving an initial value problem for a separable equation, and (3) finding the implicit solution to an initial value problem for a separable equation. Graphs are shown to illustrate the solutions.
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Exercise unit 1 hum3032
This document contains an exercise on solving differential equations. It includes determining the order and degree of given differential equations, solving equations using the method of separable variables, and solving equations using the method of integrating factors. The exercise contains multiple parts for each type of problem.
Tut 1
This document contains 11 questions regarding differential equations. It asks the student to classify equations as order, linear or non-linear; formulate differential equations by eliminating constants; solve specific differential equations; prove properties of curves; find general solutions by substitution; and determine conditions for exactness and solve exact differential equations.
Tugas akhir matematika kelompok 1
This document contains 3 problems:
1) Showing that a sequence is increasing and less than 3.
2) Calculating the average value of a function over an interval.
3) Finding the solution to a partial differential equation of the form 2xyy' – y^2 + x^2 = 0. The solution is found to be y = √cx - x^2.
Leidy rivadeneira deber_1
This document contains the solutions to 4 differential equation problems using the method of separable variables.
1) The solution for dy/dx = sen(x+y) is given as y = 2arctan(-2-x-C/x+C) + 2πn - x.
2) The solution for dy/dx = 2 + √y - 2x + 3 is given as √y - 2x + 3 = x + C.
3) The solution for dy/dx = (x + y + 1)^2 is given as arctan(x + y + 1) = x + C.
4) The solution for dy/dx = 1
Tugas matematika menemukan konsep persamaan kuadrat
1. The document contains examples of solving quadratic equations by factorizing, completing the square, and using the quadratic formula.
2. The examples include finding the area of land by setting up and solving a quadratic equation, finding the volume of objects using quadratic equations, and determining the roots of various quadratic equations.
3. The key steps shown are factorizing, completing the square, setting the discriminant equal to zero, and using the quadratic formula to solve for the roots of the equations.
Bresenham's line drawing algorithm
The Bresenham's line drawing algorithm is an incremental scan conversion algorithm that uses only integer calculations. It works by calculating the difference between the ideal y value and the actual plotted pixel y value at each x value, and determining whether to plot the pixel above or below the line based on whether this difference is positive or negative. The algorithm takes the line endpoints as input, calculates the change in x and y per step, sets an initial decision parameter, and then iteratively plots pixels, updating the decision parameter to determine whether to plot above or below the line at each step.
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This document discusses separable differential equations, which can be written in the form dx/dy = M(x)N(y), where M is a function of x only and N is a function of y only. Three examples are provided to illustrate solving separable equations: (1) solving a nonlinear separable equation, (2) solving an initial value problem for a separable equation, and (3) finding the implicit solution to an initial value problem for a separable equation. Graphs are shown to illustrate the solutions.
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This document contains solutions to three equations involving fractions equal to values.
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The second equation, (x-1)/(x+1) = 3, is solved to find the value of x that makes the fraction equal to 3, which is x = -2.
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The document contains solutions to three differential equations:
1) y'' - 3y' = 8e^3x + 4sinx with solution Y = (C1 + C2e^3x) + (8/3)Xe^3x + (6/5)cosx - (2/5)sinx
2) y'' + y = xcosx - cosx with solution Y = C1cosx + C2sinx + (1/4)xcosx - (1/2)xsinx + (1/4)x^2sinx
3) y'' + 4y = 4cosx + 3sinx - 8 with solution Y
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Assignment 2 HUM 3032 ENGINEERING MATHEMATICS 3
This document contains instructions for a group assignment on engineering mathematics. It provides 6 questions involving data presentation using bar charts, component bar charts, multiple bar charts, pie charts, and calculating percentages. Students are asked to graphically represent various sets of data involving monthly sales, employee demographics, factory absenteeism rates, accident causes in factories, company costs, and times when cars were stolen. The assignment is due as a hardcopy within one week's time.
Assignment 2 HUM 3032 ENGINEERING MATHEMATICS 3
This document contains instructions for a group assignment involving using different types of graphs to represent various sets of data. It includes 6 questions involving creating bar charts, component bar charts, multiple bar charts, pie charts to visualize monthly sales revenue, employee demographics by age over time, factory absenteeism rates by year, causes of factory accidents by percentage, company costs by category, and times when cars were stolen in a town. Students are instructed to complete all working and answers using a computer and submit a hard copy of the assignment by the specified due date.
Exercise unit 1 hum3032
This document contains an exercise on solving differential equations. It includes determining the order and degree of given differential equations, solving equations using the method of separable variables, and solving equations using the method of integrating factors. The exercise contains multiple parts for each type of problem.
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Bresenham's line algorithm is an efficient method for drawing lines on a discrete grid such as a display. It works by calculating the next pixel to plot along the line based on the difference between the true and discrete slopes. It starts at the initial point and calculates a decision variable di to determine whether to plot the next point horizontally or diagonally. This variable is updated using the slope to continually track the closest path to the true line.
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The document summarizes Bresenham's line drawing algorithm. It derives the equations for calculating the next pixel position when drawing a line on a digital display. It considers cases where the slope is less than or equal to 1 and greater than 1. For each case, it calculates the distance from intersection points to pixel positions and derives the decision parameter equation to determine the next pixel.
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This document contains solutions to three equations involving fractions equal to values.
The first equation, x^2 + x - 2 = 0, is solved to find the values of x that satisfy the equation, which are x = 1 and x = -2.
The second equation, (x-1)/(x+1) = 3, is solved to find the value of x that makes the fraction equal to 3, which is x = -2.
The third equation, x/(x-1) - 1/(x+2) - 3/(x-1)(x+2) = 0, is solved but does not have any real number solutions for x.
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Circle algorithm
The document describes the midpoint circle algorithm for drawing circles on a pixel screen. It explains how the algorithm determines the midpoint between the next two possible consecutive pixels and checks if the midpoint is inside or outside the circle to determine which pixel to illuminate. It provides the mathematical equations and steps used to iteratively calculate the x and y coordinates of each pixel on the circle. The algorithm is implemented in a C++ program to draw a circle on a graphics screen.
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The document provides information on algorithms for drawing basic geometric shapes like lines, circles, and ellipses digitally. It discusses:
1. The line drawing algorithm DDA which samples a line at integer intervals by determining corresponding y-values based on the slope.
2. Bressenham's line algorithm which determines which pixel to plot by comparing the distance of potential pixels to the true line using a decision parameter that is updated iteratively.
3. The midpoint circle algorithm which uses a decision variable to iteratively choose pixels on the circumference of a circle based on whether their midpoint is inside or outside the circle.
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The document contains solutions to three differential equations:
1) y'' - 3y' = 8e^3x + 4sinx with solution Y = (C1 + C2e^3x) + (8/3)Xe^3x + (6/5)cosx - (2/5)sinx
2) y'' + y = xcosx - cosx with solution Y = C1cosx + C2sinx + (1/4)xcosx - (1/2)xsinx + (1/4)x^2sinx
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The document derives Bresenham's line algorithm for drawing lines on a discrete grid. It starts with the line equation and defines variables for the slope and intercept. It then calculates the distance d1 and d2 from the line to two possible pixel locations and expresses their difference in terms of the slope and intercept. By multiplying this difference by the change in x, it removes the floating point slope value, resulting in an integer comparison expression. This is defined recursively to draw each subsequent pixel, using pre-computed constants. The initial p0 value is also derived from the line endpoint coordinates.
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1) This document discusses solving quadratic equations through factorization, using formulas, and rearranging into general form.
2) Methods covered include factorization to solve 1 + 2x^2 = 5x + 4, using the quadratic formula to solve equations like 3 - x^2 = -3(4x - 3) and 05xx3c6x2x8bxx920xa2=+−−=−=+−, and rearranging equations like (2x + 5) = x/7 and 7x - 1 = x^2/4 into the general form ax^2 + bx + c.
3) The goal is to be able to solve a variety of quadratic equations
2.1 solutions of algebraic equations
1) Solve the equations 97x^2 = 10x + 15 and 2x + 1x = 1.
2) Solve the equations 6/7 - 5/2 = -j and 94 - 6 = -k.
3) Solve the equations 98 = -x, 58y - = +, and m - 6 = 10 - 3m.
4) A coffee maker originally priced at $x was discounted $20. The customer paid $50, so the original price was $x = $50 + $20 = $70.
5) If half a number + 5 is 11, then half the number is 6, so the number is 12.
1.3 transposition of formulae
1) The document provides 10 examples of transposing formulas to make different variables the subject of each equation. It involves rearranging formulas that relate variables like velocity (v), time (t), acceleration (a), distance (s), weight (w), height (h), radius (r), and more.
2) Learners are asked to change the subject of each formula by rearranging the terms, and in some cases calculate the value of the variable if other values in the equation are given.
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2) Factorize expressions like 18x + 27, 4x2 - 8x, 9x4 - 18x3 - 27x2, and others.
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Syllabus
Chapter-1
Introduction to objective, scope and outcome the subject
Chapter 2
Introduction: Scope and Specialization of Civil Engineering, Role of civil Engineer in Society, Impact of infrastructural development on economy of country.
Chapter 3
Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
Linear Measurements: Instruments used. Linear Measurement by Tape, Ranging out Survey Lines and overcoming Obstructions; Measurements on sloping ground; Tape corrections, conventional symbols. Angular Measurements: Instruments used; Introduction to Compass Surveying, Bearings and Longitude & Latitude of a Line, Introduction to total station.
Levelling: Instrument used Object of levelling, Methods of levelling in brief, and Contour maps.
Chapter 4
Buildings: Selection of site for Buildings, Layout of Building Plan, Types of buildings, Plinth area, carpet area, floor space index, Introduction to building byelaws, concept of sun light & ventilation. Components of Buildings & their functions, Basic concept of R.C.C., Introduction to types of foundation
Chapter 5
Transportation: Introduction to Transportation Engineering; Traffic and Road Safety: Types and Characteristics of Various Modes of Transportation; Various Road Traffic Signs, Causes of Accidents and Road Safety Measures.
Chapter 6
Environmental Engineering: Environmental Pollution, Environmental Acts and Regulations, Functional Concepts of Ecology, Basics of Species, Biodiversity, Ecosystem, Hydrological Cycle; Chemical Cycles: Carbon, Nitrogen & Phosphorus; Energy Flow in Ecosystems.
Water Pollution: Water Quality standards, Introduction to Treatment & Disposal of Waste Water. Reuse and Saving of Water, Rain Water Harvesting. Solid Waste Management: Classification of Solid Waste, Collection, Transportation and Disposal of Solid. Recycling of Solid Waste: Energy Recovery, Sanitary Landfill, On-Site Sanitation. Air & Noise Pollution: Primary and Secondary air pollutants, Harmful effects of Air Pollution, Control of Air Pollution. . Noise Pollution Harmful Effects of noise pollution, control of noise pollution, Global warming & Climate Change, Ozone depletion, Greenhouse effect
Text Books:
1. Palancharmy, Basic Civil Engineering, McGraw Hill publishers.
2. Satheesh Gopi, Basic Civil Engineering, Pearson Publishers.
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