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:

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:

How many ways can the letters in the word mathematics be arranged?

11!/(2!2!2!)=11!/8=4989600

 Divide by two factorial for each of the three letters which appear twice: m,a,t.

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:

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:

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)=b−af(b)−f(a)

 

    Mean value theorem:

  Finding where the derivative is equal to the average change:

Local linearization

Seems to be the same as Euler's method




    Local linearization:

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:

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:

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.

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

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.

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:

Determining the line of reflection

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


    Determining the line of reflection:

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:

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:

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:

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.




5×180∘−360∘
=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

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:

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): 

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

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:

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

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:

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: