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Help with numer 5 please. thank you​

Help with numer 5 please. thank you​-example-1
User Ksu
by
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1 Answer

4 votes

Answer:

See Below.

Explanation:

We are given that:


\displaystyle I = I_0 e^(-kt)

Where I₀ and k are constants.

And we want to prove that:


\displaystyle (dI)/(dt)+kI=0

From the original equation, take the derivative of both sides with respect to t. Hence:


\displaystyle (d)/(dt)\left[I\right] = (d)/(dt)\left[I_0e^(-kt)\right]

Differentiate. Since I₀ is a constant:


\displaystyle (dI)/(dt) = I_0\left((d)/(dt)\left[ e^(-kt)\right]\right)

Using the chain rule:


\displaystyle (dI)/(dt) = I_0\left(-ke^(-kt)\right) = -kI_0e^(-kt)

We have:


\displaystyle (dI)/(dt)+kI=0

Substitute:


\displaystyle \left(-kI_0e^(-kt)\right) + k\left(I_0e^(-kt)\right) = 0

Distribute and simplify:


\displaystyle -kI_0e^(-kt) + kI_0e^(-kt) = 0 \stackrel{\checkmark}{=}0

Hence proven.

User OCa
by
7.8k points

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