MATH 1 LEC 02: FUNCTION Instructor: Dr. Nguyen Quoc Lan (October, 2007)

1. HUT – DEPARTMENT OF MATH. APPLIED-------------------------------------------------------------------------------------------------------- MATH 1 LEC 02: FUNCTION Instructor: Dr. Nguyen Quoc Lan (October, 2007)

2. NOTION OF FUNCTION ----------------------------------------------------------------------------------------------------------------------------------- Some quantity A changes and depends on another quantity B  Function: A = f(B). Example: The human pouplation depends on the time

3. HISTORY ----------------------------------------------------------------------------------------------------------------------------------- 1786, Scotland: The Commercial an Political Atlas, Playfair. Graph used to compare exports, imports by England to Denmark … Mid – eighteenth century, Euler: By alphabet  y = f(x)

4. MATHEMATICAL DEFINITION ----------------------------------------------------------------------------------------------------------------------------------- A function y = f(x): X  R Y  R: Rule associates each x  X  unique output (exactly one) y  Y. Variablex, function (value) y. One x  Two different y: It’s not a function  Vertical test (for a graph) Domain D = {x| f(x) defined} Range Imf: y =f(x), xDf  y = sinx  D= R, Imf = [–1, 1]

5. THE VERTICAL LINE TEST ----------------------------------------------------------------------------------------------------------------------------------- The Vertical Line Test: a curve in the xy-plane is the graph of a function of x if and only if no vertical line intersects the curve more than once x=a x=a (a,c) (a,b) (a,b) a a This is the graph of a function of x This is not the graph of a function of x

6. GIVE A FUNCTION: TABLE OF ITS VALUE ----------------------------------------------------------------------------------------------------------------------------------- Verbally and table of it values: The natural way to represent the function C(w) expressing the cost of mailling first class letter is using a table of values

7. GIVE A FUNCTION: GRAPH ----------------------------------------------------------------------------------------------------------------------------------- Plot a picture (graph): For the function P(t) expressing the dependence of human population in time, one can express it by table of values, then construct a graph The graph of this function is a scatter plot

8. FUNCTION DEFINED BY ALGEBRAIC FORMULA ------------------------------------------------------------------------------------------------------------------------------------------- Explicit form: y = f(x) Example: y = x2, elementary functions : 1 t  1 (x, y) Formula: Example: x = 1 + t, y = 1 – t  Line Example: x = acost, y = asint  Circle Implicit form F(x, y) = 0  y = f(x) Example: x2 + y2 – 4 = 0,

9. MAPLE ----------------------------------------------------------------------------------------------------------------------------------- • (Declare a function) p := x^3 + x^2 + 1; • (Evalue its value) subs(x=1, p); • (Evalue its limit) limit( sin(2*x)/x, x = 0) ; • (Evalue its derivative) diff(p, x) ; (2nd order) diff(p,x$2) • (Graph) plot(sin(x), x = 0..Pi); (Many graphs) plot( [sin(x),cos(x)],x = 0..2*Pi, color = [red,blue]); • (Parametric curve) plot( [31*cos(t)-7*cos(31*t/7), 31*sin(t)-7*sin(31*t/7), t = 0..14*Pi] ); • plot( [17*cos(t)+7*cos(17*t/7), 17*sin(t)- …, t = 0..14*Pi] );

10. MATHEMATICAL MODEL: RADIOACTIVE DECAY -------------------------------------------------------------------------------------------------------------------------------- Radioactive elements disintegrate continuously in a process called radioactive decay. Experimentation has shown that the rate of disintegration is proportional to the amount of the element present. Find the rule of the radioactive decay process Solution: Suppose that m = m(t): the amount at time t  The rate of disintegration is dm/dt. The assumption above gives:

11. MATHEMATICAL MODEL: RADIOACTIVE DECAY -------------------------------------------------------------------------------------------------------------------------------- By the expression of m(t), all radioactive elements have a common special property: after some constant period of time, its original amount will reduce by half  the half – life. Every radioactive element has a specific half – life, and it depends only on decay rate, not on the initial amount The half – life of radioactive carbon C – 14 is about 5730 years. Find the expression m(t) of C – 14? Solution: T – the half – life  The amount: m0/2 at time T:

12. THE SHROUD OF TURIN -------------------------------------------------------------------------------------------------------------------------------- In 1988 the Vatican authorized the British Museum to date a cloth relic called the Shroud of Turin, possibly the burial shroud of Jesus of Nazareth. This cloth founded in 1356 and contains the negative image of a human body. From the British Museum: the fibers in the cloth contained between 92% and 93% of their original carbon C – 14. Conclusion? Solution: From R/R0: 0.92  0.93  The test: 1988  The shroud was between 600 – 688 years old!