What is a Function? What is a Function? Mathematical Skills: Functions. The Function Machine. The Function Machine. Examples.

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1 What is a Function? Mathematical Skills: Functions A mathematical function is a process that converts one set of numbers into another. F example: Doubling Doubling Function Input 3 4 Output What is a Function? A mathematical function is a process that converts one set of numbers into another. F example: Doubling The Function Machine Doubling Function Input Output (x) () = X x = x Imptant: F each input, there is onl one possible output! function The Function Machine Examples = x function = x

2 Examples = x Function Notation Variables Dependent Variable ()( Independent Variable (x)( Constants = x = x + f(x)= x = x + z f(x,z) ) = x + z Examples of Functions Circumference of a circle: Circumference = X π X radius f(r) ) = πr Area of a circle: Area = π X radius f(r) ) = πr Volume of a sphere Volume = 4/3 π X radius 3 f(r) ) = 4/3 πr 3 Graphs of functions Functions generate pairs of numbers These can be used as co-dinates dinates to draw graphs Graphical displa of function Dependent Variable () Independent Variable (x) function Graphs of functions Functions generate pairs of numbers These can be used as co-dinates dinates to draw graphs Graphical displa of function Dependent Variable () Independent Variable (x) = x Rate of change gradient = ( change in ) ( change in x ) Change in x Change in

3 Change in x & between two points gives an approximate value (x, ) (x, ) Change in x Change in Change in x & between two points gives an approximate value Using smaller changes increases accurac (x, ) (x, ) Change in Change in x dx d As change in x and approach zero, line becomes tangent Ver small change in x and known as dx and d point (x, ) Gradient of curve at point (x, ) is the gradient of the tangent at that point As change in x and approach zero, line becomes tangent Ver small change in x and known as dx and d Gradient of curve at point (x, ) is the gradient of the tangent at that point dx d d Gradient at point (x, ) = dx Gradient of curve at point (x, ) is the gradient of the tangent at that point Exponentials & Logarithms 3

4 Exponentials 0 =0 0 =0 X 0 = =0 X 0 X 0 =, =0 X 0 X 0 X 0 = 0, =0 X 0 X 0 X 0 X 0 = 00,000 Exponentials 0 - = =0 0 = =0 0 0 = = = = = Logarithms Reverse of an exponential Logarithms Examples log a x a ( ) = x Log 0 (00) = Log (8) = Log 3 (9) = The number e & natural logarithms e =.788 Natural Log = Log Natural Log = Log e Usuall written as ln The number e & natural logarithms Euler's number e is a unique number The value of the slope of f (x) =e x f an value of x is equal to the value of f (x). 4

5 Changing Base Changing Bases log ( x ) = a log log b b ( x ) ( a ) Example: Convert Log 0 (x) to base e loge ( x ) log 0 ( x ) = = log (0) e ln(x).305 General Fm = a x f (x) = a x Exponential Relationships Arise when growth deca of a substance is proptional to iginal amount of substance Examples? a is a constant and called the base It can be an positive real number Example: Exponential Growth A particular bacteria doubles ever da. If the initial number of bacteria (N o ) is 00. After ONE da there are 00 bacteria (N=00) After TWO das there are 400 bacteria (N=400) After THREE das there are 800 bacteria (N=800) And so on... How man bacteria are there after 8 das? How man bacteria are there after 000 das? Need to create mathematical function Example: Exponential Growth A particular bacteria doubles ever da. If the initial number of bacteria (N o ) is 00. After ONE da After TWO das After THREE das After FOUR das N = N o X N = N o X X N = N o X X X N = N o X X X X 5

6 Example: Exponential Growth A particular bacteria doubles ever da. If the initial number of bacteria (N o ) is 00. Example: Exponential Growth A particular bacteria doubles ever da. If the initial number of bacteria (N o ) is 00. After ONE da After TWO das After THREE das After FOUR das F n number of das N = N o X N = N o X N = N o X 3 N = N o X 4 N = N o X n F n number of das After 360 das x = 360 Number of bacteria, N = N o X x N = 00 X 360, N = 00 X.3 X 0 08 N =.3 X 0 0 N = N o X n N =.3 X 0 0 N =.3 X Example: Exponential Deca Deca of a radionuclide. If the initial number of atoms of the nuclide is N o After ONE half-life life N=N o / After TWO half- N=N o / 4 After THREE half- N=N o / 8 And so on... How man are there after 8 half-?? How man are there after 000 half-?? -> > Mathematical function Example: Exponential Deca Deca of a radionuclide. If the initial number of atoms of the nuclide is N o After ONE half-life life After TWO half- After THREE half- After FOUR half- N = N o / N = N o /4 N = N o /8 N = N o /6 6

7 Example: Exponential Deca Deca of a radionuclide. If the initial number of atoms of the nuclide is N o After ONE half-life life After TWO half- After THREE half- After FOUR half- F n number of das N = N o / N = N o /4 N = N o /8 N = N o /6 N = N o X -n Definition of an Exponential Relationship A A quantit is said to var exponentiall with x if equal changes in x produce equal fractional changes in I.e. fractional change in is proptional to change in x The increase/decrease in is often written as d Therefe the fractional changes in is d/ Constant of proptionalit, k Otherwise known as growth/deca constant d = k dx = 0 e kx d = k dx = 0 e kx Relationship between deca constant & half value = e = e ln = kx x = ln / k k = ln()/ x 0 = e kx o kx kx Example: Radioactive deca The half-life life of a particular radionuclide is 8 das. Calculate the deca constant? Example: Radioactive deca A = A o n A = A o e λt 7

8 Trigonometric Functions Trigonometric Functions Sine = sin (x) f(x) ) = sin (x) Cosine = cos (x) f(x) ) = cos (x) Tangent = tan (x) f(x) ) = tan (x) Trigonometric Functions Definition of sine function. The unit circle is the circle with its centre at the igin and a radius of. Angle x is fmed b rotating OA about the igin to OP. Then the -codinate of point P is sin (x). Trigonometric Functions Definition of cosine function. The unit circle is the circle with its center at the igin and a radius of. Angle x is fmed b rotating OA about the igin to OP. Then the x-codinate of point P is cos (x). Function: = sin (x) Function: = cos (x) x x Trigonometric Functions Definition of tangent function. The unit circle is the circle with its centre at the igin and a radius of. Angle x is fmed b rotating OA about the igin to OP. Point Q is the intersection of line OP and x=. Then the -codinate of point Q is tan x. Function: = tan (x) Example Questions. Show that the deca constant is λ = ln()/t/. The half-life of Iodine 3 is eight das. Calculate the deca constant in (a) das -, (b) seconds The initial activit of a radionuclide is MBq. What is it s half-life if after 4 hours the activit has dropped to,00bq? 4. Without the use of a calculat, calculate, a. log9 (8) b. log7 (49) c. ln (e ) 8