Continuous growth and decay

Continuous growth and decay

We now have a better understanding of how the compounding frequency will affect the amount we wish to grow or decay. But what if we are dealing with something, say, that compounds every minute, second, or even millisecond? This concept is also known as continuous compounding. In this section, we will see a slight variation of an exponential growth and decay formula that models continuous exponential growth/decay.
Basic concepts: Natural log: ln, Solving exponential equations with logarithms,
Related concepts: Derivative of inverse trigonometric functions, Derivative of logarithmic functions,

Lessons

Continuous Growth/Decay: Af=Aiert { A_f = A_i e^{rt}}

Af {A_f} : final amount

Ai {A_i} : initial amount

e {e }
: constant = 2.718…

r {r }
: rate of growth/decay
• growth rate of 7% →r=7100=0.07 \to {r = {7\over100} = 0.07}
• growth rate of 15%→r=−15100=−0.15 \to {r = - {15\over100} = - 0.15}

t {t }
: total time given
  • 1.
    On Aiden’s 10-year-old birthday, he deposited $20 in a savings account that
    offered an interest rate of 4% compounded continuously. How much money
    will Aiden have in the account when he retires at the age of 60?
  • 2.
    A radioactive substance decays continuously. If the half-life of the substance
    is 5 years, determine the rate of decay.
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40.
Applications of Exponential and Logarithmic Functions
40.1
Exponential growth and decay by a factor
40.2
Exponential decay: Half-life
40.3
Exponential growth and decay by percentage
40.4
Finance: Compound interest
40.5
Continuous growth and decay
40.6
Logarithmic scale: Richter scale (earthquake)
40.7
Logarithmic scale: pH scale
40.8
Logarithmic scale: dB scale
40.9
Finance: Future value and present value
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