Ejercicios victor
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This document contains several math exercises: 1) Solving equations involving functions f(x)=x^2-4 and g(x)=x-4. The solutions are x=0, 1. 2) Finding where the curves y=x^3 and y=4x intersect. The solutions are x=0, -2, 2. 3) Calculating several integrals, including finding the area under curves such as y=cos^2(x) and y=e^2 between limits.
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![0 2
A = (X3 4X)dx + ( 4X - X3)dx
-2 0
2
A = 2 ( 4X - X3)dx
0
2
A = 2 [ 4X族 - X4 ]
2 4 0
2
A = 2 [ 2X族 - X4 ]
4 0
2
A = 2 [ (2)族 - (2)4 ]
4 0
A = 2 [ 8 - 4 ]
c) X = 12 , X = 0, Y = 1, Y = 亠2
Y
GRAFICO
A = 8 堂2](https://image.slidesharecdn.com/ejerciciosvictor-140904230518-phpapp02/85/Ejercicios-victor-4-320.jpg)
![亠2
A = 12 dy
1 Y
亠2
A = 12 dy
1 Y
亠2
12 Ln Y
1
12 [Ln 亠2 - Ln 1]
12 [ 2 Ln 亠]
A = 24 堂2](https://image.slidesharecdn.com/ejerciciosvictor-140904230518-phpapp02/85/Ejercicios-victor-5-320.jpg)
![2 2
GRAFICO
/2
A = tn (X/2)dx
0
Cambio de variable
亮 = X/2
d亮 = dx
2
2 d亮 = dx
A = 2 tn 亮 d亮
A = 2 Ln [ sec 亮 ]
Devuelvo el cambio de variable
/2
A = 2 Ln [ sec x ]
2 0
A = 2 Ln [ sec - sec 0 ]
4
A = 2 Ln [ 2 - 1]](https://image.slidesharecdn.com/ejerciciosvictor-140904230518-phpapp02/85/Ejercicios-victor-7-320.jpg)
![3.- Hallar la longitud de la curva
a. Y = X3 + 1 X = 1 hasta X = 3
6 2X
b
L = 1 + [f卒(x)]2 dx
a
Y卒 = 3X2 - 1
6 2X2
Y卒 = X2 - 1
2 2X2
Y卒 = X4 - 1
2X2
3
L = 1+ X4 - 1 2 dx
1 2X2
3
L = 1+ [X4 - 1] 2 dx
1 4X4
3
L = 4X4 + X8 - 2X4 + 1 dx
1 4X4
3
L = X8 + 2X4 + 1 dx
1 4X4
3](https://image.slidesharecdn.com/ejerciciosvictor-140904230518-phpapp02/85/Ejercicios-victor-13-320.jpg)
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The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval. The area is calculated as the sum of the areas of each trapezoid multiplied by the width of the sub-interval. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals, giving an approximate result of 1.26953125. The document also provides an example of using the trapezoidal rule with n=8 sub-intervals to estimate the area under the curve of the function y=x from 0 to 3.8.7 numerical integration
8.7 numerical integrationdicosmo178
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The trapezoidal rule is used to approximate the area under a curve by dividing it into trapezoids. It takes the average of the function values at the beginning and end of each sub-interval multiplied by the sub-interval width. The general formula sums these values over all sub-intervals divided by the number of intervals. An example calculates the area under y=1+x^3 from 0 to 1 using n=4 sub-intervals and gets an approximate value of 1.26953125.Form 4 add maths note
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This document provides notes on additional mathematics for Form 4 students. It includes definitions and examples of functions, inverse functions, quadratic equations, and logarithms. Some key points summarized:
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2. To find the roots of a quadratic equation, one can use factorisation, the quadratic formula, or complete the square. The nature of the roots depends on the sign of b2 - 4ac.
3. To solve a system of equations involving one linear and one non-linear equation, one can substitute one equation into the other and solveCbse Class 12 Maths Sample Paper 2013 Model 3
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cbse class 12 maths sample paper 2012-13 model 3 - http://cbse.edurite.com/cbse-sample-papers/cbse-sample-papers-class-12-maths.htmlTugasmatematikakelompok
Tugasmatematikakelompokgundul28
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1. The document is a mathematics assignment on differentiation from pages 33-40. It was prepared by 4 students for their 1st semester class at the Polytechnic Manufacturing State University of Bangka Belitung.
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Tugasmatematikakelompok 150715235527-lva1-app6892drayertaurus
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The document is a mathematics calculus homework assignment in Bahasa Indonesia. It contains 10 calculus problems asking students to find the derivatives of various functions. The functions include exponential functions like f(x) = 20ex, logarithmic functions like g(x) = ln(5x3), and combinations of exponentials, logarithms, and polynomials. The document provides the solutions to each problem in steps showing the use of logarithmic differentiation to find the derivatives.Form 4 Add Maths Note
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This document provides notes on functions and quadratic equations from Additional Mathematics Form 4. It includes:
1) Definitions of functions, including function notation f(x) and the relationship between objects and images.
2) Methods for solving quadratic equations, including factorisation, completing the square, and the quadratic formula.
3) Properties of quadratic functions like finding the maximum/minimum value and sketching the graph.
4) Solving simultaneous equations involving one linear and one non-linear equation through substitution.
5) Conversions between index and logarithmic forms and basic logarithm laws.Form 4-add-maths-note
Form 4-add-maths-notejacey tan
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This document provides notes on key concepts in additional mathematics including:
1) Functions such as f(x) = x + 3 and finding the object and image of a function.
2) Solving quadratic equations using factorisation and the quadratic formula. Types of roots are discussed.
3) Sketching quadratic functions by finding the y-intercept, maximum/minimum values, and a third point. Quadratic inequalities are also covered.
4) Methods for solving simultaneous equations including substitution when one equation is nonlinear.
5) Properties of exponents and logarithms, and how to solve exponential and logarithmic equations. graphs of functions 2
graphs of functions 2larasati06
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Here are the steps to solve this problem:
1) The given equation is: y = x2 + 2x
2) To complete the table, we need to calculate the value of y when x = -3 and when x = 1
3) When x = -3:
y = (-3)2 + 2(-3)
y = 9 - 6
y = 3
4) When x = 1:
y = 12 + 2(1)
y = 1 + 2
y = 3
So the completed table is:
Table 1
x -3 1
y 3 3
(b) Sketch the graph of y = x2 + 2Math06reviewsheet (3)
Math06reviewsheet (3)Angel Baez
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This document provides a review sheet with multiple math problems involving algebra, trigonometry, and calculus concepts. There are 23 problems covering topics such as simplifying rational expressions, solving equations, graphing functions, working with complex numbers, trigonometric identities, and applying trigonometric functions. Answers are provided for selected problems. The review sheet is intended to help students practice and review a wide range of mathematics topics.Tugas matematika kelompok
Tugas matematika kelompokachmadtrybuana
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1. This document provides the solutions to differentiation problems for a group mathematics assignment on pages 33-40 regarding differentiation. It lists the names of the group members and provides the solutions to 10 differentiation problems.
2. The solutions find the derivatives of various functions involving exponents, logarithms, and other operations. Derivatives are found using logarithmic differentiation and power rule.
3. The document is part of a mathematics assignment from the Polytechnic Manufacturing State University of Bangka Belitung in Indonesia for the class and semester noted.Basic m4-2-chapter1
Basic m4-2-chapter1犖犖ム幻犢犖÷肩犖迦牽犖萎犖迦牽犢犖犖朽権犖犖犖項犖犖犖巌犖犖迦肩犖犖犢 犢犖犖犢犖犖朽権犖犖犖伍犖犖犖巌犖犢
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1. The document defines various functions and relations using set-builder and function notation.
2. Examples of linear, quadratic, and polynomial functions are provided with their domain and range restrictions.
3. Common transformations of basic quadratic functions like y=x^2 are demonstrated, such as shifting the graph left or right and changing the sign of coefficients.Modul 3 quadratic function
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This document provides information about quadratic functions including:
1) The general form of a quadratic function is f(x) = ax2 + bx + c, where a, b, c are constants and a 0.
2) Characteristics of quadratic functions include involving one variable only and the highest power of the variable being 2.
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Ejercicios victor
- 2. 1. F(X)=X2-4 g(X)=X-4 X2-4=X-4 X2-4-X+4=0 X2-X=0 X(X-1)=0 X=0 X=1 0.5 (X4) = X2-4 g(X)=X- 4 - 4
- 3. b. Y = X3, Y = 4X X3 = 4X X3 - 4X = 0 X (X2 - 4) = 0 X = 0 X = -2 X = 2 GRAFICO Y = X3
- 4. 0 2 A = (X3 4X)dx + ( 4X - X3)dx -2 0 2 A = 2 ( 4X - X3)dx 0 2 A = 2 [ 4X族 - X4 ] 2 4 0 2 A = 2 [ 2X族 - X4 ] 4 0 2 A = 2 [ (2)族 - (2)4 ] 4 0 A = 2 [ 8 - 4 ] c) X = 12 , X = 0, Y = 1, Y = 亠2 Y GRAFICO A = 8 堂2
- 5. 亠2 A = 12 dy 1 Y 亠2 A = 12 dy 1 Y 亠2 12 Ln Y 1 12 [Ln 亠2 - Ln 1] 12 [ 2 Ln 亠] A = 24 堂2
- 6. d) f (x) = tn x, el eje x y las rectas x = 0, x = 1 C
- 7. 2 2 GRAFICO /2 A = tn (X/2)dx 0 Cambio de variable 亮 = X/2 d亮 = dx 2 2 d亮 = dx A = 2 tn 亮 d亮 A = 2 Ln [ sec 亮 ] Devuelvo el cambio de variable /2 A = 2 Ln [ sec x ] 2 0 A = 2 Ln [ sec - sec 0 ] 4 A = 2 Ln [ 2 - 1]
- 8. 2) Hallar el volumen del solido de revoluci坦n generando por las regi坦n encerrada por las curvas dadas (utilice el m辿todo del disco arandelas y corteza cilindrica) A) un arco de y=cos2x, alrededor del eje x
- 10. b) x=4y, x =y, alrededor de la recta x =8 b. x = 4y, x = 34 64 y2 - 1= 0 4y = 34 64 y2 = 1 (4y)3 = (34)3 y2 = 1 64 64y3 = y 64y - y = 0 y (64 y -1) = 0 1/8 3 2 y = 0 3 3 y = -1 8 y = 1 8
- 13. 3.- Hallar la longitud de la curva a. Y = X3 + 1 X = 1 hasta X = 3 6 2X b L = 1 + [f卒(x)]2 dx a Y卒 = 3X2 - 1 6 2X2 Y卒 = X2 - 1 2 2X2 Y卒 = X4 - 1 2X2 3 L = 1+ X4 - 1 2 dx 1 2X2 3 L = 1+ [X4 - 1] 2 dx 1 4X4 3 L = 4X4 + X8 - 2X4 + 1 dx 1 4X4 3 L = X8 + 2X4 + 1 dx 1 4X4 3
- 14. L = (X4 + 1)2 dx 1 4X4 3 L = X4 + 1 dx 1 2X2 3 L = 1 (X2 + X-2) dx 2 1 3 L = 1 X2 dx + X-2 dx 2 1 3 L = 1 33 - 1 - 1 + 1 2 3 3 3 1 3 L = 1 9 - 2 + 1 2 3 1 3 L = 1 10 - 2 2 3 1 L = 14 3
- 15. B)