SUEC 高中 Adv Maths (Domain, Codomain, Range)
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SUEC 高中 Adv Maths (Interval)
Sources:
Visual - various maths sites (credits to original creator)
Questions - Dong Zong's Textbook
suitable for SPM (Add Maths) too
10 fluid pressures x
The document discusses calculating the force exerted on plates submerged in fluids. It explains that the force is calculated by dividing the plate into thin strips, determining the cross-sectional area and depth of each strip, and taking the limit of the sum as the widths approach zero. This gives an integral representing the total force as the density of the fluid multiplied by the cross-sectional length times the depth integrated over the length of the plate. Examples are presented to demonstrate calculating the force on specific shapes, such as a square plate, triangular plate, and semi-circular plate.
Surface area cylinder (1)
This document describes how to calculate the surface area of a cylinder. It provides the formulas to calculate the area of the circular top and bottom (Area = πr^2) as well as the rectangular label (Area = Height x Width). The areas of the top, bottom, and label are added to find the total surface area. As an example, it calculates the surface area of a can with a radius of 3.75 cm and height of 11 cm to be 347.3625 square cm.
Maxima and minima
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This is a ______ graph
This document discusses different integration methods including the shell method, disc method, and washer method. The shell method is best used when rotating around the y-axis, while the disc method and washer method are also integration techniques discussed in the document.
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This document discusses the area formulas for different types of quadrilaterals. It provides the formulas for calculating the area of rectangles, squares, parallelograms, rhombi, trapezoids, and general quadrilaterals. Examples are given to demonstrate how to use the formulas to calculate areas. The key defines the variables used in the formulas such as base, height, side, and diagonal lengths.
Calculas IMPROPER INTEGRALS AND APPLICATION OF INTEGRATION ppt
The document provides an overview of improper integrals and applications of integration. It discusses three types of improper integrals: integrals with infinite limits, integrals with discontinuous integrands, and integrals that are a combination of the first two types. It also discusses how to calculate the volume of solids of revolution using disk, washer, and cylindrical shell methods, and provides examples of each. The document is a study guide for a group project on this topic presented by 11 students.
IMPROPER INTEGRALS AND APPLICATION OF INTEGRATION
This ppt includes all the Topics of this chapter.
IMPROPER INTEGRALS AND
APPLICATION OF INTEGRATION ppt of Calculus Sem 1.
Intergration Sector!
Recommended
SUEC 高中 Adv Maths (Interval)
Sources:
Visual - various maths sites (credits to original creator)
Questions - Dong Zong's Textbook
suitable for SPM (Add Maths) too
10 fluid pressures x
The document discusses calculating the force exerted on plates submerged in fluids. It explains that the force is calculated by dividing the plate into thin strips, determining the cross-sectional area and depth of each strip, and taking the limit of the sum as the widths approach zero. This gives an integral representing the total force as the density of the fluid multiplied by the cross-sectional length times the depth integrated over the length of the plate. Examples are presented to demonstrate calculating the force on specific shapes, such as a square plate, triangular plate, and semi-circular plate.
Surface area cylinder (1)
This document describes how to calculate the surface area of a cylinder. It provides the formulas to calculate the area of the circular top and bottom (Area = πr^2) as well as the rectangular label (Area = Height x Width). The areas of the top, bottom, and label are added to find the total surface area. As an example, it calculates the surface area of a can with a radius of 3.75 cm and height of 11 cm to be 347.3625 square cm.
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This document discusses maxima and minima of functions of two independent variables. It provides the definitions of a relative minimum/maximum point as one where the function value at that point is less than or equal to/greater than or equal to the value at neighboring points. It describes stationary points as those where the function is maximum or minimum, and extreme values as the function values at stationary points. A working rule is given to determine maxima and minima, involving taking partial derivatives and checking signs of second-order derivatives. An example finds the maximum of the function f(x,y)=x^2+y^2+6x+12 at the stationary point (-3,0), with maximum value of 3.
This is a ______ graph
This document discusses different integration methods including the shell method, disc method, and washer method. The shell method is best used when rotating around the y-axis, while the disc method and washer method are also integration techniques discussed in the document.
Area of quadrilaterals
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The Trapezoidal rule approximates the area under a curve by dividing it into trapezoids and calculating their individual areas. It works by taking the ordinates at evenly spaced intervals along the x-axis and using the formula: Area = (1/2) * (first ordinate + last ordinate + 2 * sum of middle ordinates) * width. This provides an estimate of the definite integral. The more trapezoids used, the more accurate the estimate. The estimate will be an overestimate if the gradient is increasing and an underestimate if decreasing over the interval.
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This document discusses methods for calculating the volume of solids of revolution generated by rotating plane regions about lines. It covers the disk and washer methods for rotation about axes in the plane of the region. Formulas are provided for calculating volumes of revolution in Cartesian, parametric, and polar coordinate systems by integrating the area of thin cross-sectional disks/washers. Several examples demonstrate applying these formulas to find the volumes of solids generated by rotating curves like ellipses, lines, and cardioids.
The disk method
1) The document provides instructions for calculating the volume of a cylinder given its height and total volume. It defines a cylinder as a solid of revolution formed by rotating a rectangle about an axis.
2) The volume of a cylinder (solid of revolution formed by a disk) is calculated using the formula: Volume = πR2w, where R is the radius and w is the width (height) of the disk.
3) Examples are provided to demonstrate calculating the volume of solids of revolution using the disk method, which treats the solid as a series of thin circular disks and sums their individual volumes.
Application of integration
This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
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The document discusses using the shell method to find the volume of solids of revolution. The shell method uses cylindrical shells to calculate volume, whereas the disk method uses disks. It is best to use the shell method when the representative rectangle forming each shell layer is parallel to the axis of rotation, as this allows identifying the radius using the correct variable in the integral. The disk method is best when the rectangle is perpendicular to the axis of rotation. Examples are provided to illustrate setting up integrals for both methods.
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2. The slicing method is used when the cross sections are all the same regular shape. The disk and washer methods are used when revolving an area about an axis, with disk for solids without hollow parts and washer for solids with hollow parts.
3. Examples show how to set up the integrals to calculate volume using the appropriate method, finding radii of cross sections and limits of integration.
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The document discusses the trapezoidal rule for approximating definite integrals. It introduces the trapezoidal rule, which works by approximating the area under a function's graph as a trapezoid. It then provides the formula for calculating the area of a trapezoid. Finally, it discusses how the trapezoidal rule can be applied to both uniform and non-uniform grids to approximate definite integrals, and notes applications of the rule in fields like engineering.
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Hoho
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This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.
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1. This document discusses deriving and using the formula to find the area of a triangle given two side lengths and the measure of the included angle.
2. Examples are provided to demonstrate finding the area of various triangles using this formula, including regular hexagons and triangles with specified side lengths and angles.
3. Readers are prompted to try applying the formula to find the area of shapes with given characteristics, such as a parallelogram or triangles with specified side lengths and total area.
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This document discusses methods for calculating the volume of solids of revolution generated by rotating plane regions about lines. It covers the disk and washer methods for rotation about axes in the plane of the region. Formulas are provided for calculating volumes of revolution in Cartesian, parametric, and polar coordinate systems by integrating the area of thin cross-sectional disks/washers. Several examples demonstrate applying these formulas to find the volumes of solids generated by rotating curves like ellipses, lines, and cardioids.
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2) The volume of a cylinder (solid of revolution formed by a disk) is calculated using the formula: Volume = πR2w, where R is the radius and w is the width (height) of the disk.
3) Examples are provided to demonstrate calculating the volume of solids of revolution using the disk method, which treats the solid as a series of thin circular disks and sums their individual volumes.
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This document discusses using the method of cylindrical shells to calculate volumes of solids of revolution. It provides an example calculating the volume of the solid obtained by rotating the region between the curves y=2x^2 - x^3 and y=0 about the y-axis. The method involves imagining the solid as being composed of cylindrical shells and using the formula V=2π∫_{a}^{b} x*f(x) dx to calculate the volume, where f(x) is the height of each shell.
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Explains about the concept ,formula,and solving problems on area of a square and parallelogram.The development of a formula is also explained with the help of examples for both triangle and parallelogram.The power point is made for VIIth standard s.s.c board text book.
Intro To Formula Project
The document provides examples of working with literal equations, which are formulas written as equations. It shows how to transform a formula by solving it for a different variable. For example, rewriting the volume formula V=lwh to solve for width by isolating w on one side. Students are assigned a project to research two formulas, write them, describe the variables, show a real-life application by substituting values, and present their findings.
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The document discusses using the shell method to find the volume of solids of revolution. The shell method uses cylindrical shells to calculate volume, whereas the disk method uses disks. It is best to use the shell method when the representative rectangle forming each shell layer is parallel to the axis of rotation, as this allows identifying the radius using the correct variable in the integral. The disk method is best when the rectangle is perpendicular to the axis of rotation. Examples are provided to illustrate setting up integrals for both methods.
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1. The document discusses several methods for calculating the volume of solids of revolution, including slicing, disk, washer, and specific examples of their use.
2. The slicing method is used when the cross sections are all the same regular shape. The disk and washer methods are used when revolving an area about an axis, with disk for solids without hollow parts and washer for solids with hollow parts.
3. Examples show how to set up the integrals to calculate volume using the appropriate method, finding radii of cross sections and limits of integration.
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Hoho
The document provides information on mathematical formulas for calculating the area of squares, rectangles, and triangles as well as the perimeter of polygons. It states that the area of a square can be calculated as s2, where s is the length of one side. The area of a rectangle can be found using the diagonals method, which is half the product of the two equal diagonal lengths. It also provides the formula for calculating the perimeter of a regular polygon as ns, where n is the number of sides and s is the length of one side. Finally, it shares the formula for calculating the area of a triangle using the x and y coordinates of its three vertices.
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Hasse Diagram
The document defines and provides examples of Hasse diagrams. It can be summarized as:
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This document discusses different methods for calculating the volumes of solids of revolution: the disk method and washer method. It provides step-by-step explanations of how to set up and evaluate the definite integrals needed to calculate these volumes, whether the region is revolved about an axis that forms a border or not. Examples are given to illustrate each method. The key steps are to divide the solid into slices, approximate the volume of each slice, add the slice volumes using a limit of a Riemann sum, and evaluate the resulting definite integral.
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2. Examples are provided to demonstrate finding the area of various triangles using this formula, including regular hexagons and triangles with specified side lengths and angles.
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Gaussian Process Regression
References:
"Gaussian Process", Lectured by Professor Il-Chul Moon
"Gaussian Processes", Cornell CS4780 , Lectured by Professor
Kilian Weinberger
Bayesian Deep Learning by Sungjoon Choi
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Surveying: Object Principles & Types of Surveying; Site Plans, Plans & Maps; Scales & Unit of different Measurements.
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Chapter 6
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- 1. 对应域 定义域 值域 𝑫𝒇 = 𝑨 𝑹𝒇 = 𝒃 𝑩
- 2. 𝟏𝟗 𝒚𝟐 − 𝟒𝒚𝒛 + 𝟒𝒛𝟐 + 𝟑 𝒕𝟐 + 𝟑 𝟏𝟐 𝒙𝟐 −𝟒𝒙 + 𝟕 𝟐𝟎 𝒂𝟒 − 𝟑𝒂𝟐 + 𝟐 𝒙𝟒 − 𝟔𝒙𝟑 + 𝟏𝟎𝒙𝟐 − 𝟑𝒙 − 𝟑𝟎 𝟓𝒙𝟐 −𝟏𝟓𝒙 + 𝟐𝟐 𝟐𝒙 + 𝒉 − 𝟑 𝒑𝒈 𝟏𝟓 𝒅𝒐𝒎𝒂𝒊𝒏 定义域 𝑫𝒇 = 𝑨 𝟕 𝒂𝟐 − 𝟐𝒂 + 𝟔 𝟏𝟒 𝟔 𝟐𝟓
- 3. 𝒑𝒈 𝟏𝟔 𝒅𝒐𝒎𝒂𝒊𝒏 定义域 𝑫𝒇 = 𝑨 𝟏𝟓 − 𝟏 − 𝟒 − 𝟎. 𝟓𝟖 − 𝟒 − 𝟏 −𝟐 − 𝟓 𝟏𝟒 𝟐 − 𝟏 𝒇 𝟏 = 𝟏 𝒇 𝟎 = 𝟎 𝒇 − 𝟏 𝟐 = − 𝟏 𝒇 𝟏 = 𝟏 𝒇 𝟎 = 𝟏 𝒇 − 𝟏 𝟐 = 𝟎
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- 5. 𝒅𝒐𝒎𝒂𝒊𝒏 𝒑𝒈 𝟏𝟕 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 值域 : 𝑹𝒇 = 𝒃 = 𝟏, − 𝟐, 𝟏𝟎 值域 : 𝑹𝒇 = 𝒃 = 𝟏 𝟑 , 𝟏 𝟐 , 𝟏 定义域 : 𝑫𝒇 = 𝑨 = −𝟏, 𝟎, 𝟏, 𝟐 定义域 : 𝑫𝒇 = 𝑨 = −𝟏, 𝟎, 𝟏
- 7. 𝒑𝒈 𝟏𝟑 𝒅𝒐𝒎𝒂𝒊𝒏 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 定义域 : 𝑫𝒇 = 𝑹 对应域 : 𝑹 值域 : 𝑹𝒇 = 𝑹 定义域 : 𝑫𝒈 = 𝑹 对应域 : 𝑹 值域 : 𝑹𝒈 = 𝑹+ ∪ 𝟎 定义域 : 𝑫𝒉 = 𝑵 对应域 : 𝑵 值域 : 𝑹𝒉 = 𝒚 | 𝒚 ∈ 𝑵 且 𝒚 ≠ 𝟏 𝒏 = 𝒏𝒂𝒕𝒖𝒓𝒂𝒍 𝒏𝒖𝒎𝒃𝒆𝒓𝒔 = 𝟏, 𝟐, 𝟑, 𝟒 … “ y, such that y "
- 8. 𝒅𝒐𝒎𝒂𝒊𝒏 𝒑𝒈 𝟏𝟑, 𝟏𝟒 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 定义域 : 𝑫𝒇 = 𝒙 | 𝒙 ∈ R, 𝒙 ≥ 𝟏 值域 : 𝑹𝒇 = 𝒚 | 𝒚 ∈ R, 𝒚 ≥ 𝟎 “ x, such that x " 定义域 : 𝑫𝒈 = 𝒙 | 𝒙 ∈ R , 𝒙 ≠ − 𝟏 𝟐 值域 : 𝑹𝒈 = 𝒚 | 𝒚 ∈ R, 𝒚 ≠ 𝟎 定义域 : 𝑫𝒌 = 𝒙 | 𝒙 ∈ R , 𝒙 > 𝟎 值域 : 𝑹𝒌 = 𝑹
- 9. 𝒅𝒐𝒎𝒂𝒊𝒏 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 𝒑𝒈 𝟏𝟑, 𝟏𝟒 定义域 : 𝑫𝒉 = 𝑹 值域 : 𝑹𝒉 = 𝒚 | 𝒚 ∈ R, 𝒚 ≤ 𝟒
- 10. 𝒅𝒐𝒎𝒂𝒊𝒏 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 𝒑𝒈 𝟏𝟖 定义域 : 𝑫𝒇 = 𝒙 | 𝒙 ∈ R, 𝒙 ≥ 𝟎 定义域 : 𝑫𝒇 = 𝒙 | 𝒙 ∈ R, 𝒙 ≠ 𝟎 定义域 : 𝑫𝒇 = 𝒙 ∈ 𝑹 定义域 : 𝑫𝒇 = 𝒙 ∈ 𝑹
- 11. 𝒅𝒐𝒎𝒂𝒊𝒏 𝒄𝒐 − 𝒅𝒐𝒎𝒂𝒊𝒏 𝒓𝒂𝒏𝒈𝒆 定义域 𝑫𝒇 = 𝑨 对应域 𝑩 值域 𝑹𝒇 = 𝒃 𝒑𝒈 𝟏𝟖 定义域不同 定义域相同 值域不同 值域不同 定义域相同 值域相同
Editor's Notes
- Source: Online Math Learning