he298c

2021-08-11

State whether each statement is true or false and briefly explain why. If the statement is false, try to “fix it” by modifying the given statement to a new statement that is true. For exponential growth and decay, the rate of change is constant.

averes8

In the case of exponential growth and decay, the rate of change is not constant.Thus, the statemant: For exponential growth and decay the rate of change is contant is false.
The general solution of differential equation${y}^{\prime }\left(t\right)=ky\left(t\right)$With constant of proportionality k is calculated by separation of variables asThat is
$\mathrm{ln}|y|+{c}_{1}=kt+{c}_{2}$
$\mathrm{ln}|y|=kt+c$For$y\left(t\right)>0$
$y\left(t\right)=A{e}^{kt}\left(1\right)$
Here$A={e}^{c}$For k>0, equation (1) is exponential growth law and for k<0, equation (1) is exponential decay law.The rate of exponential growth and decay is${y}^{\prime }\left(t\right)=Ak{e}^{kt}$
Therefore, the true statement is: For exponential growth and decay the rate of change is ${y}^{\prime }\left(t\right)=Ak{e}^{kt}$

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