Exponential growth and decay practice question 2
A quantity satisfies for all , and . Find .
The rate of change is a fixed multiple of the amount itself. Which family of functions behaves like that?
Every solution of has the form . Use to find .
An exact expression, or a decimal.
Theory UBC Math 100 §11.4.2CLP-1 §3.3Key results
Worked solution
Strategy
The general solution of y′ = ky is an exponential with rate constant k; the initial value fixes the constant in front.
Working
General solution of y′ = 3y.
Initial condition.
Evaluate.
Answer
The answer is 4 e cubed, about 80.3.
Check
y = 4e3t starts at 4 and its derivative 12e3t is 3y, so it is the solution, and 4e³ ≈ 80.3.
Where this goes wrong
Writing y = 4 + 3t, treating the rate as constant. The rate is proportional to y, so it grows as y grows: the solution is exponential, not linear.
The idea behind it
The exponential growth and decay lesson, a figure to move, in a few steps
Leans on
Exponential functions, from Functions, graphs and inverses
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