Gonzalo Garcia - iTEC2 - Borrador 1

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

O

iTEC Proyect cycle 2 SEK­Atlántico 4º ESO ­ A 1)

Lesson objectives

Teachers' notes

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Subject: Math Topic: Algebra: equations Grade(s): 12 Prior knowledge: Concepts and practice about equations Cross­curricular link(s): Science, History, Geography

Lesson notes: This lesson activity focuses on students' knowledge of Pages ?, ? and ? use the SMART Response system.

Lesson objectives

Teachers' notes

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

First of all, we have to know what's a function A function is a relation or expression involving one or more variables. Example:

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Funcon graphic representaon: To represent a funcon we have to calculate its "principal elements” following the next steps: 1.‐Funcon’s command and trajectory 2.‐ Symmetries and periodicity 3.‐ Cuts with axis 4.‐ 1ª derivave’s study 5.‐ 2ª derivave’s study 6.‐ Asymptotes 7.‐ Significant values

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

1.‐Funcon’s command and trajectory

Function's command are all the values that forgive the function exist, that can be calculated : }{)()(xfxxDomf

=. Function's trajectory

is formed by all the values that the function can adopt. 2.­Symetries and periodicity

We'll study when the function is symetric respect Y axis (pair) and respect X axis (odd) . 3.­Cuts with axis Cuts with OX's axis: we'll do 0 = y and solve the ecuation. Cuts with OY's axis: we'll do 0 = x and replace it in the function to calculate y 4.­1ª derivave’s study We'll study function's growing and it's relative extremes. We'll derivate function and calculate ) ( xf Function can be crescent or decreasing. Then, we derivate the function and calculate: ­Function will be crescent in points when xf > 0 ­Function will be crescent in points when xf < 0) 5.­2ª derivative's study We calculate the second derivative and then we observe what's happening with the points wich cancelled 1ª derivative 6.‐ Asymptotes Asymptotes are lines that are very close to the funcon but they can't touch them. Asymptotes can be vercals, horizontals or obliques

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Secondly, it is necessary: 1.‐ To write a set of values of the funcon and its argument in a table 2.‐ To transfer the coordinates of the funcon points from the table to a coordinate system, 3.‐ Joining marked points A, B, C, etc by a smooth curve, we receive a graph of the given funconal dependence.

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Types of functions Lineal functions: y = m x + b Quadratic functions: y = a x ^2+ b x + c,

Power functions: y = a x b Polynomial functions:y = an ·x^ n + an −1 · x^ n −1 + … + a2 · x^ 2 + a1 · x+a0

Rational functions. These functions are the ratio of two polynomials

Exponential functions:y = a b x Logarithmic functions: y = a ln (x) + b,

Sinusoidal functions:y = a sin (b x + c)

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Ecuations and natural environment: The general natural environment ecuation is S­­>e*S* and we can use it to represent different forms that are present in natural environment like trees, stars, mountains and clouds. The next images are some examples:

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

Botanic curves: There is one general ecuation ( )We can aproach to vegetal growing mysteries using curves and ecuations: n bigger than 1: ­Case a=b: Simple petal: n=5/2 Ecuation:

Simple petal (with central Ecuation: circle): n=5/2

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

­Case 0 < b < a: Simple petal: n=7/2 Ecuation:

Simple petal (with central Ecuation: circle): n=7/2

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

­Case b > a: Simple petal: n=7/2 Ecuation:

1 bigger than n: ­Case b=a: Ecuation:

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

­Case b > a: Ecuation: Ecuation:

­Case 0 < b < a: Ecuation:

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GRUPO 3­ TRABAJO 2.notebook

May 28, 2012

The next ecuations are some of the most important in natural environment: Hubbel's law for Universe expansion:V = H. d Gibbs' ecuation: Energie's conservation law: Newton's gravitation law: Ecuation for equivalence between energy and matter: Radium of a black hole: Fermat's ecuation: V = H. d

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