Tuesday, January 10, 2012
Old Student
hey mr.p. i dont know if you remember me but i was just curious if you have an email account that i can reach you at. its about a college professor and its pretty funny. you can reach me at G_C_1993@hotmail.com .
Monday, September 26, 2011
GRAPHING CIRCULAR FUNCTIONS:
HI..guys....muskan here.... we learned about graphing circular functions couple of days ago...lets take a quick rewiew.
1.Graphing on a cartesian plane is known as ``unrolling``
2.We will plot θ values on x-axis and the trignometric function values at θ on the axis y-axis.
3. One cycle is a portion of the graph from one point to another at which time the graph begins to repeat itself.
4. One period is the lenght of one cycle in either degrees or radians. The period for a sin x or cos x function is 2π/|b| . The period for a tan x is π/|b| .
5.The amplitude is the distance from the middle axis to the highest or lowest point for a sin x or cos x function. A change in amplitude vertically stretches or compresses the basic shape of the curve. The amplitude for sin x or cos x function |a|. The amplitude for a tan x function is unlimited or infinite.
6. Basic equations: y = a sin bx y = a cos bx y = a tan bx
7. x represents θ values for x when graphing sin x or cos x.
8. Use quadrant values for x when graphing sin x or cos x.
9. Use quadrant and π/4 values for x when graphing tan x.
10.The tan x function will have asymptotes at quadrantal values where tan x is undefined.
The illustration above only draws the graph of the sine or cosine function for one revolution. However, the domain of these functions is all real numbers. The picture below shows the graphs of sine (orange) and cosine (green) on a larger domain (-3
to 3).
HI..guys....muskan here.... we learned about graphing circular functions couple of days ago...lets take a quick rewiew.
1.Graphing on a cartesian plane is known as ``unrolling``
2.We will plot θ values on x-axis and the trignometric function values at θ on the axis y-axis.
3. One cycle is a portion of the graph from one point to another at which time the graph begins to repeat itself.
4. One period is the lenght of one cycle in either degrees or radians. The period for a sin x or cos x function is 2π/|b| . The period for a tan x is π/|b| .
5.The amplitude is the distance from the middle axis to the highest or lowest point for a sin x or cos x function. A change in amplitude vertically stretches or compresses the basic shape of the curve. The amplitude for sin x or cos x function |a|. The amplitude for a tan x function is unlimited or infinite.
6. Basic equations: y = a sin bx y = a cos bx y = a tan bx
7. x represents θ values for x when graphing sin x or cos x.
8. Use quadrant values for x when graphing sin x or cos x.
9. Use quadrant and π/4 values for x when graphing tan x.
10.The tan x function will have asymptotes at quadrantal values where tan x is undefined.
The illustration above only draws the graph of the sine or cosine function for one revolution. However, the domain of these functions is all real numbers. The picture below shows the graphs of sine (orange) and cosine (green) on a larger domain (-3
to 3).
May be this video help to understand better:
http://youtu.be/uKyBl1FOaks thanks :))
Wednesday, June 8, 2011
Probability Using Permutations and Combinations
Hello everyone !!! Guess what?? This is the LAST blog for this course !!! hahaha
But as Mr. P said ,
we can ALWAYS use this blog to communicate with our peers, free to ask questions,
and answer some questions (be SURE to give the right answer though JK) !!
We all worked hard to pass this course and finally its ALMOST over :)
GREAT JOB EVERYONE !!!
Study well on the last test guys and GOOD LUCK for the
PROVINCIAL EXAM !!!
. . . LET'S START . . .
But as Mr. P said ,
we can ALWAYS use this blog to communicate with our peers, free to ask questions,
and answer some questions (be SURE to give the right answer though JK) !!
We all worked hard to pass this course and finally its ALMOST over :)
GREAT JOB EVERYONE !!!
Study well on the last test guys and GOOD LUCK for the
PROVINCIAL EXAM !!!
. . . LET'S START . . .
EXAMPLE 1
EXAMPLE 2
EXAMPLE 3
We can find more examples on the site below:http://staff.argyll.epsb.ca/jreed/math30p/perms_combs/probability.htm
Thursday, June 2, 2011
Probability of independent and dependent events
Mutually Exclusive Events
Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.
If two events are disjoint, then the probability of them both occurring at the same time is 0.
Disjoint: P(A and B) = 0
If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
Specific Addition Rule
Only valid when the events are mutually exclusive.
P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events
In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.
General Addition Rule
Always valid.
P(A or B) = P(A) + P(B) - P(A and B)
Independent Events
Two events are independent if the occurrence of one does not change the probability of the other occurring.
An example would be rolling a 2 on a die and flipping a head on a coin. Rolling the 2 does not affect the probability of flipping the head.
If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring.
Specific Multiplication Rule
Only valid for independent events
P(A and B) = P(A) * P(B)
Dependent Events
If the occurrence of one event does affect the probability of the other occurring, then the events are dependent.
Conditional Probability
The probability of event B occurring that event A has already occurred is read "the probability of B given A" and is written: P(BA)
General Multiplication Rule
Always works.
P(A and B) = P(A) * P(BA)
EXAMPLES
A fair die is tossed twice. Find the probability of getting a 4 or 5 on the first toss and a 1, 2, or 3 in the second toss.
Answer
P(E1) = P(4 or 5) = 2/6 = 1/3
P(E2) = P(1, 2 or 3) = 3/6 = 1/2
They are independent events, so
P(E1 and E2) = P(E1) × P(E2) = 1/3 × 1/2 = 1/6
EXAMPLE -
Two balls are drawn successively without replacement from a box which contains 4 white balls and 3 red balls. Find the probability that
(a) the first ball drawn is white and the second is red;
(b) both balls are red.
(a) The second event is dependent on the first.
P(E1) = P(white) = 4/7
There are 6 balls left and out of those 6, three of them are red. So the probability that the second one is red is given by:
P(E2 E1) = P(red) = 3/6 = 1/2
Dependent events, so
P(E1 and E2) = P(E1) × P(E2 E1) = 4/7 × 1/2 = 2/7
--------------------------------------------------------------------------------
(b) Also dependent events. Using similar reasoning, but realising there will be 2 red balls on the second draw, we have:
P(RR)=3/7*2/6=1/7
EXAMPLE-
A bag contains 5 white marbles, 3 black marbles and 2 green marbles. In each draw, a marble is drawn from the bag and not replaced. In three draws, find the probability of obtaining white, black and green in that order.
Answer
We have 3 dependent events.
P(W1)*P(B2W1)*P(G3B2 AND W1)
=5/10*3/9*2/8
=1/24
Two events are mutually exclusive if they cannot occur at the same time. Another word that means mutually exclusive is disjoint.
If two events are disjoint, then the probability of them both occurring at the same time is 0.
Disjoint: P(A and B) = 0
If two events are mutually exclusive, then the probability of either occurring is the sum of the probabilities of each occurring.
Specific Addition Rule
Only valid when the events are mutually exclusive.
P(A or B) = P(A) + P(B)
Non-Mutually Exclusive Events
In events which aren't mutually exclusive, there is some overlap. When P(A) and P(B) are added, the probability of the intersection (and) is added twice. To compensate for that double addition, the intersection needs to be subtracted.
General Addition Rule
Always valid.
P(A or B) = P(A) + P(B) - P(A and B)
Independent Events
Two events are independent if the occurrence of one does not change the probability of the other occurring.
An example would be rolling a 2 on a die and flipping a head on a coin. Rolling the 2 does not affect the probability of flipping the head.
If events are independent, then the probability of them both occurring is the product of the probabilities of each occurring.
Specific Multiplication Rule
Only valid for independent events
P(A and B) = P(A) * P(B)
Dependent Events
If the occurrence of one event does affect the probability of the other occurring, then the events are dependent.
Conditional Probability
The probability of event B occurring that event A has already occurred is read "the probability of B given A" and is written: P(BA)
General Multiplication Rule
Always works.
P(A and B) = P(A) * P(BA)
EXAMPLES
A fair die is tossed twice. Find the probability of getting a 4 or 5 on the first toss and a 1, 2, or 3 in the second toss.
Answer
P(E1) = P(4 or 5) = 2/6 = 1/3
P(E2) = P(1, 2 or 3) = 3/6 = 1/2
They are independent events, so
P(E1 and E2) = P(E1) × P(E2) = 1/3 × 1/2 = 1/6
EXAMPLE -
Two balls are drawn successively without replacement from a box which contains 4 white balls and 3 red balls. Find the probability that
(a) the first ball drawn is white and the second is red;
(b) both balls are red.
(a) The second event is dependent on the first.
P(E1) = P(white) = 4/7
There are 6 balls left and out of those 6, three of them are red. So the probability that the second one is red is given by:
P(E2 E1) = P(red) = 3/6 = 1/2
Dependent events, so
P(E1 and E2) = P(E1) × P(E2 E1) = 4/7 × 1/2 = 2/7
--------------------------------------------------------------------------------
(b) Also dependent events. Using similar reasoning, but realising there will be 2 red balls on the second draw, we have:
P(RR)=3/7*2/6=1/7
EXAMPLE-
A bag contains 5 white marbles, 3 black marbles and 2 green marbles. In each draw, a marble is drawn from the bag and not replaced. In three draws, find the probability of obtaining white, black and green in that order.
Answer
We have 3 dependent events.
P(W1)*P(B2W1)*P(G3B2 AND W1)
=5/10*3/9*2/8
=1/24
Tuesday, May 31, 2011
Probability
Sample space - is the set of all possible outcomes of an experiment.
Tree diagrams are ordered pairs are often use to list samples space.
The probability that a specific event will occur can be described as P(E)=success / sample space.
Example 1
a. Tossing 3 coins
b. Rolling a pair of 6 sided dice.
Sample Space = 36
Example 2
A box contains 13 orange marbles and 10 green marbles.If one marble is drawn, what is the probability it will be orange?
P(orange marble) = success
sample space
P(orange marble) = 13
23
Probability of Independent and Dependent Events
Complementary Events - are mutually exclusive and their sum exhaust the sample space
ex. Tossing a coin and rolling a die.
Independent Events - two events occurs so that neither one effects the probability of the others.
ex. Drawing two cards from a deck without replacement
Example 1: Comparing Independent and Dependent Events
You randomly select two marbles from a bag contains 14 green, 7 blue and 9 red marbles. What is the probability that the first marble is blue and the second marble is not blue if:
a. You replace the first marble before selecting the second
b. You do not replaced the first marble
Example 2
A single card is drawn from deck of 52 cards. What is the Probability that it is either Ace or Jack?
Tree diagrams are ordered pairs are often use to list samples space.
The probability that a specific event will occur can be described as P(E)=success / sample space.
Example 1
a. Tossing 3 coins
b. Rolling a pair of 6 sided dice.
Sample Space = 36
Example 2
A box contains 13 orange marbles and 10 green marbles.If one marble is drawn, what is the probability it will be orange?
P(orange marble) = success
sample space
P(orange marble) = 13
23
Probability of Independent and Dependent Events
Complementary Events - are mutually exclusive and their sum exhaust the sample space
ex. Tossing a coin and rolling a die.
Independent Events - two events occurs so that neither one effects the probability of the others.
ex. Drawing two cards from a deck without replacement
Example 1: Comparing Independent and Dependent Events
You randomly select two marbles from a bag contains 14 green, 7 blue and 9 red marbles. What is the probability that the first marble is blue and the second marble is not blue if:
a. You replace the first marble before selecting the second
b. You do not replaced the first marble
Example 2
A single card is drawn from deck of 52 cards. What is the Probability that it is either Ace or Jack?
Thursday, May 26, 2011
Hyperbolas on a Coordinate Plane
Hello everyone! I'm posting again, this time on hyperbolas.
Vertical Hyperbola
Transverse Axis Length= 2a
Let's look at an example now.
The hyperbola is the set of all points in a plane such that the difference of the distances from two fixed points (the foci) is constant.
There are two different types.
Horizontal Hyperbola
Vertical HyperbolaTransverse Axis Length= 2aConjugate Axis Length=2b
Note: The a2 value is always the first denominator (not necessarily the larger) and the variable above the a2 indicates if the transverse axis is horizontal or vertical.
Here is the standard form for both a Horizontal Hyperbola and Vertical Hyperbola. For the horizontal hyperbola, the horizontal tranverse axis opens sideways. The vertical hyperbola, the vertical tranverse axis opens up and down.
The General Form of a Hyperbola: Ax2-By2+Cx+Dy+E=0
You could tell that this is a hyperbola because of the negative sign between the A and B unlike the parabola where the sign instead is positive.
The equation for the asymptotes is:
Let's look at an example now.1) Let's graph 9x2 - 16y2 = 144.
First, multiply each side of the equation by 1/144 to put it in standard form.
x2 y2
-- - -- = 1
16 9
First, multiply each side of the equation by 1/144 to put it in standard form.
x2 y2
-- - -- = 1
16 9
The direction of the hyperbola is horizontal. We now know that a = 4 and b = 3. The vertices are at (±4, 0). (Since we know the center is at the origin, we know the vertices are on the x axis.)
The easiest way to graph a hyperbola is to draw a box using the vertices and b, which is on the y-axis.
Draw the asymptotes through opposite corners of the box. Then draw the hyperbola.
The figure below is the graph of 9x2 - 16y2 = 144.
The figure below is the graph of 9x2 - 16y2 = 144.
It's a lot more simple than it actually looks. Okay that's it! I didn't really do anything too fancy for this post so I hope that it's easy to understand. See you all tomorrow. :)
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