Monday, September 29, 2014

Range of a function

Use differentiation, to find the vertex.

plot2d in Maxima can also be used to find the shape of the graph which can help to find the range.

    Domain and range of a function:




Saturday, September 27, 2014

Extreme value theorem

The absolute maximum or minimum value could also be at the beginning or end of the function.

 plot2d in Maxima  can be used to solve some of the problems for this topic.

    Extreme value theorem:






Thursday, September 25, 2014

Wednesday, September 24, 2014

When is a particle speeding up

A particle is speeding up when the velocity and acceleration are  in the same direction (have the same sign).

For this, drawing a sketch of the graph of velocity against time helps. To do this, find  the velocity when t=0, and then find the times when the velocity is zero. Also find the time when acceleration is zero.


    When is a particle speeding up:




Tuesday, September 23, 2014

Applications of derivatives: Motion along a line

Total distance is different from displacement. Velocity (differentiation of displacement)=0 helps to find the total distance.

    Total distance traveled by a particle:







    Analyzing particle movement based on graphs:





Sunday, September 21, 2014

Empirical Rule

Divide by 2 to get values from the origin
If z is less than 2 then the answer will be 0.5+ 95/2=97.5%


    ck12.org normal distribution problems: Empirical rule: Using the empirical rule (or 68-95-99.7 rule) to estimate probabilities for normal distributions




Mean value theorem

At some point, the instantaneous rate of change is the same as the 
average rate of change.


f(c)=baf(b)f(a)
 

    Mean value theorem:






  Finding where the derivative is equal to the average change:




Saturday, September 20, 2014

Friday, September 19, 2014

Finding the best deal

In this case, the amount of concentration per bottle needs to be calculated  first = bottle size x concentration
Then, divide by the price to find the amount of concentration per dollar.

    Finding the best deal on pesticides:




Thursday, September 18, 2014

Sum of an infinite geometric series



 The sum of an infinite geometric sequence, but only in the special circumstance that the common ratio r is between –1 and 1; that is, r | < 1 is  found by the formula S= a/[1-r]


    Sum of an infinite geometric series





Constructing probability distributions

Just have to  count for the fair probability distribution but for the unfair one there may need to be some calculation.


    Constructing a probability distribution for random variable:




Wednesday, September 17, 2014

Sample and population variance

Divide by n-1 instead of n for sample variance


    Sample variance: Thinking about how we can estimate the variance of a population by looking at the data in a sample.




Tuesday, September 16, 2014

Constructing a box and whisker plot



First, find the median which is the middle of all the terms arranged in order.
Then use a similar method to find the first and third quartile.
http://www.physics.csbsju.edu/stats/box2.html
 
    Constructing a box and whisker plot: Here's a word problem that's perfectly suited for a box and whiskers plot to help analyze data. Let's construct one together, shall we?







Mean Absolute Deviation

http://www.mathsisfun.com/data/mean-deviation.html   more or less explains what it means

Find the mean 

Find the distance of each value from that mean

Then find the mean of those distances

Validity


when no info given, the claim is usually not valid

random sample and exact sample can be a valid claim




    Reasonable samples: To make a valid conclusion you'll need a representaive, not skewed, sample





Monday, September 15, 2014

Interpreting a scale drawing

Scale factor of length ×Scale factor of width = Scale factor of 
area.  eg 40x40=1600
This is to get scale factor of area, not for the video below.




    Interpreting a scale drawing: Understand how a scale drawing is converted into real numbers using the scale factor.





Sunday, September 14, 2014

Specifying planes in three dimensions

Two points can define a line but cannot define a plane.



    Specifying planes in three dimensions:




Lines, line segments, and rays


The ray looks  somewhere between a line and line segment.
 
Lines, line segments, and rays: Difference between lines, line segments, and rays




Understanding linear and exponential models


If you increase by a constant number then it is linear but if you increase by a factor such as 5% then it is  exponential.


    Understanding linear and exponential models:




Saturday, September 13, 2014

Determining the line of reflection

The midpoints of ST and HG are on the line of reflection.


    Determining the line of reflection:




Thursday, September 11, 2014

Constructing circle inscribing triangle

Draw the angle bisectors of the three points of the triangle.  This can be done by drawing two circles (compass)of the same radius and putting the edge of them, to the point of the triangle. Then draw a line through their intersecting points.

The angle bisectors will meet at a point which is the center of the circle inscribing the triangle.


    Constructing circle inscribing triangle:




Wednesday, September 10, 2014

Horizontal and vertical asymptotes of function

Solve where function  is undefined or infinity  - denominator is  zero for this case. This helps to find x.  Can check this by using slightly higher or lower values than the one found.

To find the  horizontal asymptote, solve when x is infinity or negative infinity. This helps to find  y.

    Horizontal and vertical asymptotes of function:




Tuesday, September 9, 2014

Scaling down a triangle by half

For this example, points are  half the distance from the center of dilation (the origin in  this case).

    Scaling down a triangle by half:




Monday, September 8, 2014

Absolute value equation example

The value on the right hand side should be positive or negative towards the end.


    Absolute value equation example:




What is the sum of this polygon's interior angles?

What is the sum of this polygon's interior angles?
There are a couple ways to approach this problem.
 there are 180 degrees in a triangle.
Since this polygon has 5 sides, we can draw 5 triangles that all meet in the center.
Subtract 360 degrees because the circle in the middle is extra.

5×180360
=540

An alternative approach is shown below.
make 3 triangle from the corner of the polygon
  3 x 180=540

540
540
    Sum of interior angles of a polygon: Showing a generalized way to find the sum of the interior angles of any polygon


Sunday, September 7, 2014

Constructing a geometric series for new users

 Start, added and end are the ones used in this example but it seems start and end is good enough for some problems.


   Constructing a geometric series for new users:




Saturday, September 6, 2014

Cool unit circle tool from Khan Academy

I remember learning  " All Silly Tom Cats" some time ago but I like this cool tool from Khan Academy.

https://www.khanacademy.org/math/trigonometry/unit-circle-trig-func/Trig-unit-circle/e/unit_circle

Average rate of change (example 1)

The average rate of change over an interval is the total change in the value of y(x) divided by the change in x.



    Average rate of change (example 1)





Friday, September 5, 2014

Graphing inequalities

When there is a dash line, it is less than or more than - there is no equality together with them.

The direction of inequality is changed when multiplying or diving both sides by a negative number.

    Graphing inequalities: Graphing Inequalities




Thursday, September 4, 2014

Reading stem and leaf plots


The 10 to the power numbers are in the first column


    Reading stem and leaf plots:




Modeling annual temperature variation with trigonometry

http://www.purplemath.com/modules/grphtrig.htm

Find out whether sine or cosine function should be used. Amplitude is the peak value. 2*pi/period  is the second value needed to be found which is the argument. The third value is the mid line.

    Modeling annual temperature variation with trigonometry:




Wednesday, September 3, 2014

Congruence postulates

SSS and AAS are congruent. SSA is not congruent


    Congruent triangles and SSS: What it means for triangles to be congruent








    Other triangle congruence postulates: SSS, SAS, ASA and AAS postulates for congruent triangles.  Showing AAA is only good for similarity and SSA is good for neither




Tuesday, September 2, 2014

Constructing equilateral triangle inscribed in circle

First construct a circle of equal radius. Make the centres sit on each other.

    Constructing equilateral triangle inscribed in circle:




Monday, September 1, 2014

Constructing circumscribing circle


Center the circle at the perpendicular bisectors of the sides.


    Constructing circumscribing circle:




Proof: Invertibility implies a unique solution to f(x)=y

In mathematics, an inverse function is a function that "reverses" another function: if the function f applied to an input x gives a result of y, then applying its inverse function g to y gives the result x, and vice versa. i.e., f(x) = yif and only if g(y) = x. Referred from Wikipedia.



    Proof: Invertibility implies a unique solution to f(x)=y: Proof: Invertibility implies a unique solution to f(x)=y for all y in co-domain of f.




Graphs of absolute value functions

vertex is when absolute value=0, so x can be found.


    Graphs of absolute value functions: