functions
DESCRIPTION
I, Hrishikesh Choudhari, gave this presentation, the second one to have my Zen nature, for my MTH-111 course. It is very unique in its content. Even though it is about the basic concept of a 'function' in mathematics, I have made it very visual and it keeps you awake till the end. My professor, Mr. George Thankachan, said that this was the best presentation he had seen in years. That pumped me up, and set the bar for all my future presentations.TRANSCRIPT
Chapter One :
Functions, Graphs and Graphers.
The idea of a function.
The distance, in miles, is 1/5 of the time, in seconds.
15 seconds --> 3 miles
x yOrdered pairs
(x,y)Correspondenc
e
0 0 (0,0) 0 0
1 1/5 (1,1/5) 1 1/5
5 1 (5.1) 5 1
10 2 (10,2) 10 2
15 3 (15,3) 15 3
y = (1/5)x
Distance is a function of time.
A function is a correspondence between a first set, called
domain, and a second set, called the co-domain, and
only the elements which have a pre-image in the domain, make
up the range.
Notation –
inputs (members of the domain) are values of x
outputs (members of the range) are values of y
function f
input x
corresponding output f(x)
f(x) does not mean f*x
?
Vertical Line Test
If it is possible for a vertical line to cross a graph more than
once, then the graph is not a graph of a function.
Finding the domain –
The input should give real output
f(x)=3x
1
3}xR,x|{xdomain
Applications of Functions
Speed of Sound
245724575t
1087.7S(t)
t=temperature in Fahrenheit; S=speed of sound in feet/sec
boeing.com/commercial/industry_info.html
X Y
-400 469.099
-300 678.823
-200 837.599
-100 970.737
0 1087.700
100 1193.292
200 1290.197
300 1380.511
400 1464.967
500 1544.955
Temp
Sp
eed
Temp
Speed
Temp
Sp
eed
Temp
Temp
491.4t
24575t2457
24575t0
245724575t
1087.70
Anthropology
humerus
X F(x)
26 142.98
29 151.23
22 131.98
Humerus
Heig
ht
Heig
ht
y = 2.5x + 71.48
Heig
ht
Humerus
Civil Engineering
runrise
slope
For a run of hundred feet, a rise of twenty feet is permitted.
5x
f(x)
x = run f(x) = rise
Meteorology
Tem
p in
F
Month
polynomial function
0.2607392x2 + 3.620379x + 72.090909
400 ft 120t
d
Express d as a function of t
120(400)d
t
(400)(120t)d
22
22
Population
rt0eN N
N0 is the starting population N is the population after a certain time,
t , has elapsed, r is the rate of natural increase (birth rate - death rate)
and e is the constant 2.71828...
Microbiologists
Time Cells
0 1
1 2
2 4
3 8
4 16
5 32
6 64
7 128
8 256
9 512
10 1024
11 2048
Time (sec)
Cells
x2f(x)
A cardioid is a curve that can be produced as the path (or locus) of a point on the circumference of a circle as that circle rolls around another fixed circle with the same radius. wiki
Mathworld.wolfram.com
Cos = redExp = black Mathworld.wolfram.com
Hoffman-Singleton graph
Mathworld.wolfram.com
Variation
Direct variation
Inverse variation