Exercise · ZW-EXPONENTIAL-GROWTH-AND-DECAY-006 · ★★☆ Intermediate
In \(y'=ky\) with \(k<0\) and \(y(0)>0\), what happens to \(y(t)\) as \(t\to\infty\)?
Choose one option.
Show answer
- C. It tends to \(0\) without reaching it
Explanation
Since \(e^{kt}>0\) and tends to \(0\), the quantity decays but never vanishes or changes sign.
Review the concept: Exponential growth and decay
A quantity \(y\) with initial value \(y_0\) grows or decays exponentially when its rate of change is proportional to itself, with constant \(k\): it grows if \(k>0\), decays if \(k<0\).
\[y'=ky\;\Longrightarrow\; y(t)=y_0\,e^{kt}\]
More practice problems: exponential growth and decay
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-001 Solve y'=3y with y(0)=2.
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- ZW-EXPONENTIAL-GROWTH-AND-DECAY-003 A population satisfies y'=ky, with y(0)=100 and y(2)=200. What is y(6)?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-004 A substance has a half-life of 5 years and satisfies y'=ky. What is k?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-005 A capital grows with y'=0.04y. After roughly how long does it double?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-007 A 100 mg isotope keeps 80 mg after 3 years. With exponential decay, how many mg remain after 9 year…
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-008 With y'=0.1y, about when does the quantity triple?
- ZW-EXPONENTIAL-GROWTH-AND-DECAY-009 If y'=ky with y(1)=6 and y(3)=24, what is y(0)?
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