Download - Gonzalo Garcia - iTEC2 - Borrador 1
GRUPO 3 TRABAJO 2.notebook
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May 28, 2012
Lesson objectives Teachers' notes
1)
iTEC Proyect cycle 2SEKAtlántico 4º ESO A
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GRUPO 3 TRABAJO 2.notebook
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Teachers' notesLesson objectives
Subject:
Topic:
Grade(s):
Prior knowledge:
Crosscurricular link(s):
Math
Algebra: equations
12
Concepts and practice about equations
Science, History, Geography
Lesson notes:
This lesson activity focuses on students' knowledge of
Pages ?, ? and ? use the SMART Response system.
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First of all, we have to know what's a functionA function is a relation or expression involving one or more variables. Example:
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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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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 > 0Function 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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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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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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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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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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Case 0 < b < a:
Simple petal: n=7/2 Ecuation:
Simple petal (with central Ecuation: circle): n=7/2
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Case b > a:
Simple petal: n=7/2 Ecuation:
1 bigger than n:
Case b=a:
Ecuation:
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Case b > a:
Ecuation: Ecuation:
Case 0 < b < a:
Ecuation:
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The next ecuations are some of the most important in natural environment:
Hubbel's law for Universe expansion:V = H. dGibbs' 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