Critical Numbers and f following g

How to find THE DERIVATIVE

Suppose a driver is on a highway with mile markers.
And also suppose that we would be able to know
the mile position that paired up with the elapsed
time on the highway.
For example:

after 1 hour the mile marker was 111
at 2 hours the marker was 171
at 1.1 hours the marker was 117

Moving into the world of Math:
( 1, 111 )
( 1.1, 117 )
( 2, 171 )

What y value would you assume
went with x = 0 ?
( 0, __?__ )

How about y = 51?
What assumption are you making?
(The speed was a constant 60 mph.)
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Given a function f(x)

Suppose you want to know the INSTANTANEOUS
RATE OF CHANGE in the function at a particular
value of x. Let's say when x = a.
This is called the DERIVATIVE of f(x) @ x=a.
f ' (a) = ?

If f(x) is a LINEAR FUNCTION, then the slope
of the line gives the RATE OF CHANGE of y with
respect to a change in x. This ratio (Delta y)/ (Delta x)
is the constant m in the form
y = m x + b
You can check my blog for more on SLOPE!

In general, the SLOPE OF THE TANGENT
LINE TO THE CURVE at the particular value
of x ( x = a ) gives the INSTANTANEOUS
RATE OF CHANGE at this point.

In CALCULUS we look at the AVERAGE RATE
OF CHANGE to find the INSTANTANEOUS.

The AVERAGE RATE OF CHANGE is found by picking
two points on the curve and finding the slope
of the SECANT LINE that passes thru them.
Suppose we are looking for the SLOPE of the TANGENT
LINE (INSTANTANEOUS RATE OF CHANGE)
at the point on the curve where x=a. Let's call it point P.
Using one point on the curve as P ( a , f(a) )
We choose a point very close to P calling it Q.
Q can and should be allowed to be on either side of P.

After we find the SECANT SLOPE we make point Q
get very close to point P. As this happens we look for
the limit of these SECANT SLOPES.



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LINEAR EQUATIONS

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Instead of memorizing the formulas for Centigrade to Fehrenheit ----

why not use the fact that there exists
a linear relationship.
By forming the ordered pairs for

the Freezing and Boiling
Temperatures for WATER we get:

(0 , 32) and (100 , 212)

OR
(32 , 0 ) and (212 , 100)

Depending on which pair you use,

the slope will be either 9/5 or 5/9

CALCULUS - graphing f' and f"

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Comparing f to f' to f"

******************

Example:
If y = Parabola
then y ' = Straight Line

******************
f = position at time T

f ' = Velocity

f " = Acceleration

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If you are given the Velocity

and asked to find the Distance Traveled

from Time t1 to t2, then INTEGRATE

the Velocity from t1 to t2.

AREA UNDER the VELOCITY!

Sample Problems

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SLOPE (of a LINE)

FINDING the EQUATION of a LINE

What is this SLOPE THING?

Suppose we want to buy some T-shirts.

We are told that each T-shirt costs the

same amount. There is a charge

(an initial fee)

for setting up the logo for the order.

The price for 10 shirts would be $93
and 100 shirts would cost $633.

What is the COST for each shirt?

Dollars per Shirt!

(The SLOPE OF A LINE)

The difference in cost 633 minus 93 is $540.

$540 divided by 90 shirts gives

a RATE (Cost per Shirt) of $6.00 per shirt.

Why then do 10 shirts at $6 each cost $93?

Because there is a set-up fee of $33.

The Cost formula that Mr. Smith uses is:

Total Cost = ( $6 ) times # of shirts PLUS $33

We could represent this with the equation:

y = 6 x + 33

(Note: y depends on x for its value!)

$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$$

Take a look at the set of ordered pairs:

{ (3,11) (4,13) (5,15) (6,17)}

We notice that when X changes one unit

(as it does from 3 to 4), Y changes 2 units

(11 to 13). As X changes 2 units (3 to 5),

Y changes 4 units (11 to 15).

NOTE:

We are not saying that the values for y

are twice as big as the values for x.

We are talking about changes in y and x.

Y's change is 2 units for each 1 unit of change in x.
We call this the RATE OF CHANGE OF Y

WITH RESPECT TO X. Graphing these points

onto the xy-plane and connecting the points

with a line we have what is called a line

with SLOPE of 2. The EQUATION that these

points SATISFY is y = 2x + 5. Notice the

SLOPE of 2 is situated next to x.

Picking a point on the graph and moving

3 units to the RIGHT (RUN)

and then moving

6 units UP (RISE) we will land

us back onto the line.

The RISE is 6 and the RUN is 3.

the FRACTION (RATIO)

RISE/RUN EQUALS 2.

One definition for the slope of a LINE

is the RATIO

m = RISE/RUN.

How about the graph of the equation

y = (3/4)x - 1

Some of the ordered pairs that satisfy

this equation are {(4,2) (8,5) (12,8) ...}

When x changes 4 units, y changes 3 units.

The RATIO of 3 to 4 represents the

change in y with respect to the change in x.

THE CHANGE IN Y over the CHANGE in x is 3/4.

The slope of the graph of this equation is 3/4.

The SLOPE of the graph of the solutions to:

(i) a LINEAR FUNCTION f(x) = mx + b

(ii) an equation in the form Ax + By = C

(iii) or in the form y = mx + b

..........SLOPE is the rate that y changes

with respect to a change in x

..........In other words,

m = (y2-y1) / (x2-x1)

or m = change of y divided by the change in x

-----the values for x change by (x2-x1)

and the corresponding y's change by (y2-y1)

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Matrix Multiplication Example

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Logarithms / Exponential Functions

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Exponential Functions and Logarithmics Functions
are INVERSES of eachother.
y = log base 10 of x

MEANS:
y is the power on 10
that gives x as an
answer.

RULES which can be used with logs:

Some examples follow:

Puzzles - using for extra credit

 

I would like to share some puzzles with the Math Teacher Community.
Please feel free to use any or all of the problems.

I believe my unique method used to award the points will be of interest. When I gave any extra credit points for a puzzle I would offer the same number of points as there was students in the class. (25 students could get 25 points.) Too many points to offer? Well there was a catch. Everyone who got the question correct would share equally in the total number of points available. (Thus, if only 2 got the question correct, then they each got about 25/2 [12 or 13] points. If 5 students were correct, then each student would get only 5 points each. If 13 students were right, then each one got about 2 points. This wrinkle in the distribution of points slowed down the sharing of correct answers within my class!

Puzzle #1
Sometimes called a Word Arithmetic Problem:
You must find the distinct digit that replaces a letter, each time that it appears,
so as to satisfy the arithmetic problem presented.
Given:
SMAX + SMAX + SMAX + SMAX = XAMS
What will "XMAS" equal?

Puzzle #2
The sum of all the positive integer factors of
a certain prime number is 32.
What is this prime number?

Puzzle #3
If a quart and a half of egg nog costs $6.30,
how much would two pints cost?

Puzzle #4
Find the SUM of the following ROMAN NUMERALS:
VII + XIV
Give your answer in Roman Numerals.

Puzzle #5
A REGION bounded by the X-AXIS and the
LINES Y = M X + 4 , and X = 1 and X = 4
has an AREA of 7 square units.
What must be the value for M?

Puzzle #6
Successive DISCOUNTS of 10% and then 20%
are equivalent to what SINGLE DISCOUNT?

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ANSWERS:
1) 8172
2) 31
3) $4.20
4) XXI
5) -2/3
6) 28 %
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Puzzle #7
A father has a son.
As of now, their ages total 100.
The father is three times as old as
the son was when the father was
ten years older than the son is now.
How old is the father?
.........................................................
Answer: The father is now 63.
(The best method for solving this
problem is "Guess and Check".
The equations that evolve using
an Algebraic approach are very
messy!)
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Here is good Christmas Handout:
Separate the SNOWMEN!
(Use only 6 straight lines.)

LOOK BELOW FOR THE SOLUTION

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Math in Nature - Fibonacci

Fibonacci Sequence:

0,1,1,2,3,5,8,13,21,34,...

(add two adjacent terms to get the following term!)

There is a great website to see

Fibonacci Numbers in Nature:

http://britton.disted.camosun.bc.ca/fibslide/jbfibslide.htm

1) Is 13 really that unlucky?

2) Why do we consider 21 as ADULTHOOD?

3) Why do we use 3 by 5 cards for notes?

4) Is a 5 by 8 inch photo visually appealing to you?

5) The Mountain Laurel Blossom (found in the Smokies)

has the shape of a Pentagon! (5 sided)

Maybe SIGNS are not considered Nature,
but we find them in our world.
First let's review the names of polygons:

NOW A FEW SIGN EXAMPLES:

Below is the bottom of a stalk of CELERY.
(Check out the SYMMETRY!)

Fibonacci Numbers and Nature :

http://www.maths.surrey.ac.uk/hosted-sites/R.Knott/Fibonacci/fibnat.html

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