Arab Open University

Unit 12 Growth And Decay

Section 1 Exponential Growth Aims This section aims to introduce a number of examples of exponential growth and decay, and show how to describe this kind of change mathematically

Examples of Exponential growth Chain letter Example The number of letters sent out in chain letters grows exponentially each generation. Population: The set of letters sent out in each generation Generation: the level of the chain letter. Growth factor: 5 ( The number of letters sent by one person.)

Greenfly Example The number of offspring produced by greenflies grows exponentially each day. Population: The set of young offspring produced per day. Generation: The day number after starting the counting. Growth factor: 3 ( Number of offspring produced by each greenfly)

Tree’s Example The number of twigs developed each spring in a tree grows exponentially. Population: The set of twigs produced each spring Generation: The year’s number after starting the counting. Growth factor: Number of buds produced per twig each Autumn.

Compound interest Example The sum of money saved with compound interest grows exponentially each year ( or period). Population: Amounts of money saved at the end of each year (or period) Generation: Year’s (or period’s) number after the deposition of the money. Growth factor: (1 + i ) where i is the annual rate of interest.

The Sierpinski carpet Example Step 0

Step 1

Step 2

Step 3

The Sierpinski carpet Example The black area in the Sierpinski carpet is decreasing exponentially in each step of the construction of the carpet. Population: The set of black areas in each step. Generation: The step number. Growth factor : 8/9

Fixed features of an exponential growth Multiplicative factor is positive. If the multiplicative factor is Greater than 1 , the exponential change is an exponential Growth. If the multiplicative factor is less than 1 , the exponential change is an exponential Decay Population size of one generation = Multiplicative factor  Population size of the previous generation. P (n + 1) = b  P( n )

Formulas for exponential growth Formula Where the initial value of the population is 1 P( n ) = b n ; where: b is the growth factor n is the generation number. P( n ) is the population size in the nth generation P( 0 ) is the initial population ( here: P( 0 ) = 1). Examples In the Chain letter example : P( n ) = 5 n ; P( 0 ) =1 The number of Ancestors of one person is : P( n ) = 2n ; P( 0 ) = 1

The black area in the Sierpinski carpet is : P( n ) = ( 8 / 9 ) n ; P( 0 ) =1 Formula where the initial size of the population is a P ( n ) = a  b n ; P ( 0 ) = a Examples Saving account with compound interest P( n) = C (1 + i ) n ; P ( 0 ) = C Greenfly population: P( n ) = 17  3 n ; P ( 0 ) = 17

Snowflake Curve Step 0

Step 1

Step 2

Step 3

Snowflake Curve Population: The set of Perimeters of the curve at each step Generation: Step number of the construction process Growth factor: 4/3. Initial value: P( 0 ) = L. Formula: P ( n ) = L ( 4/3 )n

Collared Doves Population: Adult female collared doves breeding each year Generation: The year number after starting the counting. Proportion of the survival adult female to the next year: k Number of young doves produced by each adult female: m Proportion of survival of produced young female to the next year: L Growth factor: b = k + (m/2).L ( number of young female is assumed to be half of the total number of young doves). Formula : P ( n ) = P( 0 ) ( k + m L/ 2 )n = a  b n Example: k = 86 % ; L = 60 % ; m = 4 ; then b = 0.86 + 2(0.6) = 2.06 ; and P ( n ) = P( 0 )  (2.06)n.

Variations on the theme Section 2 Variations on the theme Aims This section aims to help you find out about a type of growth which is different from, but related to, exponential growth, namely the accumulation of the results of exponential growth.

Accumulation of the results of exponential growth Let S( n ) = a + a b + a b2 …………+ a bn-1 S ( n ) is the total size of the populations of all generations up to the (n – 1)th generation. Then S ( n + 1 ) = ( a + a b + a b2 ……..+ a bn-1) + a b n = a + b ( a + a b + a b2……..+ a bn-1) Then: S( n ) + a b n = a + b S ( n). Therefore: S ( n ) = N.B. a is the population of the initial generation a bn-1 is the population of the (n – 1)th generation.

Examples on accumulation of exponential growth Queen Calcula’s gift S( n ) = 1 + 2 + 22……..+ 2n-1 = The total amount of money that the Queen would pay up to the nth day of the month is : 2n – 1 Compound interest with withdrawals Let a be the constant amount withdrawal each year. End of 1st year: C(1 + i ) – a End of 2nd year: C(1 + i )2 – a(1 + i ) – a End of the nth year: C(1 + i )n - a (1 + i )n-1-…..- a(1 + i ) – a = C(1 + i )n – a [ (1 + i )n-1+ …… +(1 + i ) + 1]

Snowflakes Area Initial area : A Area after step 1: Area after step 2: Area after step 3: Area after step n-1:

Section 3 Exponential Graphs Aims This section aims to help you become familiar with the graphs of exponential functions.

Y=2x Y=x2 Y=2x+1

Y=2x Y=x2 Y=2x+1

Y = 5x Y = 3x Y = 4x

Y = 5x Y = 4x Y = 3x

Y = 0.125x Y = 0.25x Y = 0.5x

Y = (½)x Y = 2x

Y = 3(2x) Y = 4 (2x) Y = 0.3(2x)

Y = 3(2x) Y = -3(2x)

Y = 3+(0.5)x Y = 2+ (0.5)x Y = 1+(0.5)x

Section 4 Exponents Aims This section aims to introduce some important rules for operating with and simplifying expressions involving exponents

Particular powers of 10 1 kilo = 1000 units = 103 units. 1 mega = 1000 kilos = 106 units. 1 giga = 1000 mega = 109 units. 1 tera = 1000 giga = 1012 units. Rules for exponents Multiplication rule: Division rule: Power rule: N.B. a0 = 1 for every value of a  0

Negative power Example: Fractional power: Example: Particular exponential function : Natural exponential The natural exponential function is the function where the multiplicative factor is the special number e  2.7 The rules for the natural exponential function are the same as for any other exponential function

Using rules for exponents Express a number as a power of 10: Write 5 as a power of 10 We need for that to find a number x such that: Using the calculator we graph the function y = 10x and we find the value of x for which the graph cuts the line of equation “ y = 5” The value of x is found to be 0.6990; thus : 5 = 100.6990 N.B. Find x here is “undoing” the process of exponentiation: It’s called doing the logarithm. is equivalent to ( log is the logarithm with base 10 ) Undoing the natural exponential is equivalent to apply the Natural logarithm denoted by L n ( x ) ( logarithm with base e )

Graphs of “ y = 10x “ and “ y = log x “ are symmetrical with respect to the first bisector. Graphs of “ y = ex “ and “ y = L n x “ are symmetrical with respect to the first bisector.

Logarithmic rules Inverse rule: Multiplication rule: Division rule: Power rule: Natural and decimal Logarithmic rules Inverse rule: And Multiplication rule: And Division rule: And Power rule: And Change of base rule:

Doubling time and half-life State of the problem While investing £200 at an interest of 10 %, in an account in which the interest is compounded, how long will it take for the balance to increase to £400? Pose the equation: 200 ( 1.1 )x = 400 which is equivalent to ( 1.1 )x = 2 Solving the equation: Step 1: Apply log function to both sides of the equality log (1.1)x = log 2 Step 2: Apply the power rule: xlog(1.1) = log 2 Step 3: solve for x:

Equation of the form : a x = 2 This equation is used to solve doubling time problems. The general solution is : Similarly , the half-life of an exponential decaying population is the time that will take this population to become half of its original size Example The half-life of radioactive carbon is 5730 years. In the formula: y is the number of radiocarbon atoms at a time t t is the time measured in half-life and x is the time measured in years a is the initial number of radiocarbon atoms

The annual percentage rate ( APR ) State the problem Find the APR if the monthly rate is known Formula Where m is the monthly rate of interest. Example: if m = 2 % is the monthly interest rate, then the

State the problem Find the monthly rate if the APR is known: Formula The same formula is used but we make m the subject. Where m is the monthly rate of interest. Example: if the APR = 24 %, the value of the monthly interest rate is :