The Exponential Function What is to be learned

The Exponential Function

What is to be learned • The number that is vital in any sort of natural growth (or decay)

Continuous Growth Number can be modelled by (1 + 1/n)n n=1 (1 + 1/1)1 =2 n=2 (1 + 1/2)2 = 2. 25 n = 10 (1 + 1/10)10 = 2. 594 n = 1000 → 2. 705 → 2. 717 n = 10000 → 2. 718 n = 100000 → 2. 718 known as exp or e as n → ∞ (1 + 1/n)n → 2. 718281826……

very simplistic to model continual growth - use e 3 years later 2 years later 1 year later Harry’s hedge is 40 cm tall His special fertilizer increases its height by 7% p. a.

The Exponential Function Natural growth or decay functions can be expressed in terms of e Usually something like f(t) = f 0 ekt f 0 - initial value k - a number t - time e - e!!!!!!!!

A Proper Growth Calculation The population P of some bacteria after t days can be calculated using formula P(t) = P 0 e 0. 6 t where P 0 is the original population There are 8 000 bacteria to start with. How many will there be after a week? P = 8 000(e(0. 6 X 7)) = 8 000(e 4. 2) = 533 491 bacteria

The Exponential Function The letter e (or exp) is a value that be used to model growth ( or decay) e = 2. 718 (to 3 d. p. ) usual type of formula f(t) = f 0 ekt f 0 - initial value k - a number t - time e - e – which is a number

The mass of radioactive Goofyonium (M grams) after t years is given by M = M 0 e-0. 02 t If there is originally 80 g of Goofyonium, how much will there be after 100 years? M = 80(e-0. 02(100)) = 10. 8 g careful when using calculator