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Week 10 – Review: Infinite Geometric Series

keishan2015

This week in Pre-calculus 11 we reviewed the first four units in preparation for the midterm. Though it was review, one thing that I had to go back and take a look at was lesson 1.5 [Infinite Geometric Series].

what to know:

  • There are two types of geometric series, diverging [no sum] / converging [sum].

Diverging (no sum):

  • when a geometric series diverges that means that the numbers are getting bigger
  • If a geometric series diverges that means:
  •  r>1
  • r<-1
  • If a geometric series diverges then it also means that it has no sum

Converging (sum):

  • When a geometric series converges that means that the numbers are getting closer together
  • If a geometric series converges that means:
  • 0<r<1
  • -1<r<0
  • If a geometric series converges then that means that you can find its infinite sum: s_\infty=\frac{a}{1-r}

Example:

  • First, we need to find the common ratio to determine if this geometric series converges or diverges. 
  • To find the common ratio, you take one of the terms and divide it by the preceding term.
  • Once you find the common ratio compare it to the restrictions of both a converging and diverging geometric series. The common ratio in this equation is r = 4, 4>1 and therefore diverges. Since this geometric series diverges we can not find its infinite sum.

Example:

  • First, we need to find the common ratio to determine if this geometric series converges or diverges. 
  • To find the common ratio, you take one of the terms and divide it by the preceding term.
  • Once you find the common ratio compare it to the restrictions of both a converging and diverging geometric series. The common ratio in this equation is r= 0.25 or \frac{1}{4}. since 0<\frac{1}{4}<1, this geometric series converges and therefore has an infinite sum.
  • to calculate the infinite sum of this geometric series we use the formula:  s_\infty=\frac{a}{1-r} and fill it is with what we know. Note: a is used to represent t_1.
  • Once you have solved the equation you should be left with the geometric series infinite sum.

Week 4 – Multiplying and Dividing Radicals

keishan2015

This week in Pre-Calculus 11 we learned how to simplify expressions involving radicals. We learned how to add/subtract and multiply/divide radicals.

Multiplication:

When multiplying radicals all you need to do is multiply the coefficients of the radicals by each other and the radicands by each other. After doing that you check to see if the radicand can be simplified, and simplify if possible.

Example:

  1. First, in this expression you just multiply the coefficients and radicands together.
  2. You then simplify the radicand if possible. Since the radicand is 32, we look for any perfect squares that could be a factor of it. In this case 3\sqrt{32} is the same as (\sqrt{16})(\sqrt{2})
  3. If we know that 16 goes into 32 we take the square root of 16 (4) and then multiply it by the coefficient in front of the radicand. 12(4) = 48.

Example:

  1. First, you need to simplify the expression that is in the brackets before multiplying. To do this all the radicands need to be the same.
  2. Both 3\sqrt{24} and \sqrt{54} can be simplified. 3\sqrt{24} can be simplified to 6\sqrt{6} (the perfect square 4 goes into 24 six times). \sqrt{54} can be simplified to 3\sqrt{6} (the perfect square 9 goes into 54 six times).
  3. Since all the radicands in the brackets are the same to simplify all we do is add the like terms. We add or subtract the coefficients and keep the radicands the same (6+3-8 = 1).
  4. We then multiply the two radicals together and simplify.

Division:

When dividing radicals, the only rule is that you CAN’T leave a radical in the denominator as well as a negative.

Conjugate: A conjugate is formed by changing the sign in between two numbers. For example, the conjugate of x + y would be x – y.

Example:

  1. First, we need to understand that there is a radical in the denominator.
  2. To get rid of it we multiply both the numerator and the denominator by the conjugate which in this case would be \sqrt{5}+\sqrt{3} which we write as a fraction. For the fraction to be accurate it should be equivalent to 1.
  3. We the multiply the radicals together and add like terms. In the denominator when you multiply the radicals together there will be some like terms that cancel out ie. +\sqrt{15} and -\sqrt{15}.
  4. After we are left with a fraction that has no radical in its denominator. Though the expression may look like it can’t be further simplified it still can be. The coefficients can still be simplified since they are both divisible by the denominator.

Note: if the denominator was a negative number and the coefficients weren’t divisible by it, then we would divide everything by -1 to get rid of the negative sign.

 

Chapitre 3 et 4 Étudier

keishan2015

Pour chapitre 3, je trouve les expressions avec faire et des prepositions difficile.

les expressions avec faire:

des prepositions:

 

Pour chapitre 4, je trouve les Adjectifs qui précèdent le nom et Verbes pronominaux difficile.

les Adjectifs qui précèdent le nom:

 

Verbes pronominaux:

Chapitre 3 et 4 Revision

keishan2015

A. Complète les phrases avec des activités differentes.

  1. Quand il fait chaud je fais de la natation.
  2. Quand il neige tu fais du ski.
  3. Au printemps nous faisons du vélo.
  4. En automne vous faites des randonnées.
  5. Quand il pleut elles lisent des livres.

B. Choisi la bonne préposition.

  1. Il a passé un mois en Colombie Britannique puis en Floride.
  2. Nous allons en Ontario pour rendre visite à notre famille.
  3. Le match de hockey est joué au Danemark.
  4. Ils sont invités aux Philippines à un mariage.
  5. Il part aux États-Unis pour visiter ses amis.
  6. Je viens de Chine et elle vient de France.
  7. Mes parents reviennent aux Bahamas.
  8. Ma meilleure amie reside à Dublin.
  9. Mon amie Charlotte habite en Belgique.
  10. Ma soeur est Américaine. Elle vient à New York.

C. Ecris 10 phrases en utilisant 10 verbes pronominaux différents. Utilise tous les pronoms (je, tu, il, elle, on, nous, vous, ils, elles)

  1. je me reveille à huit heures du matin.
  2. Tu te brosses les cheveux dans la salle de bain.
  3. il s’habille pour la fête.
  4. elle se lave les mains.
  5. on ne s’amuse pas en classe.
  6. nous nous’assoyons à la table.
  7. vous ne vous rasez pas en hiver.
  8. ils se couchent à minuit.
  9. elles se prominent avec leurs amies.
  10. je me maquille toujours pour l’école.

D. Place l’adjectif correctement et accorde le. 

  1. (ville (f)/petit) La petite ville.
  2. (acteurs/drole) Les acteurs drôles.
  3. (femme/beau) La belle femme.
  4. (artistes/creatif) Les artistes créatives.
  5. (homme/beau) Le beau homme.

E. Choisis un acteur/actrice ou un chanteur/chanteuse. Décris cette personne, utilise 3 traits physiques et 3 adjectifs. N’oublie pas d’accorder.

C’est Selena Gomez. Elle est une actrice et une chanteuse. Elle a les cheveux bruns et courts. Elle a le visage rond est elle a nez retroussé. Selens Gomez est gentile. Elle est généreuse. Elle est trés travailleuse.