Author Topic: Q5 TUT 0102  (Read 163 times)

Victor Ivrii

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Q5 TUT 0102
« on: November 02, 2018, 03:29:27 PM »
Give the order of each of the zeros of the given function:
$$
e^{2z}-3e^z-4.
$$

Xiting Kuang

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Re: Q5 TUT 0102
« Reply #1 on: November 02, 2018, 03:39:32 PM »
Please see the attachment below.
« Last Edit: November 02, 2018, 03:44:49 PM by Xiting Kuang »

Yuechen Huang

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Re: Q5 TUT 0102
« Reply #2 on: November 02, 2018, 04:36:44 PM »
\begin{equation}
f(z) = e^z - 3e^z - 4 = 0
\end{equation}
Let $w=e^z$, then
\begin{equation}
w^2 - 3w - 4 = 0 \\
(w-4)(w+1) = 0 \Rightarrow w = 4 \space or \space w = 1 \\
e^z = 4 \space or \space e^z = -1 \\
z = \log4 \space or \space z = \log(-1) \\
\end{equation}
When $e^z = 4$, the order is 1
\begin{equation}
f'(z) =2e^{2z} - 3e^{z} = 2 \times 4^2 - 3 \times 4 \neq 0
\end{equation}
When $e^z = -1$, the order is 1
\begin{equation}
f'(z) =2e^{2z} - 3e^{z} = 2 \times (-1)^2 - 3 \times (-1) \neq 0
\end{equation}

Victor Ivrii

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Re: Q5 TUT 0102
« Reply #3 on: November 04, 2018, 09:36:39 PM »
Yuechen
you need to write what is $\log(4)$ and $\log(-1)$