question 3 on the interval 0, 2, which function is above the other? ( f(x)=e^{x} ) and ( g(x)=ln x ). ( f(x)…

Explanation:

Step1: Analyze domain of $g(x)$

The domain of $g(x)=\ln x$ is $x > 0$. At $x = 0$, $g(x)$ is undefined.

Step2: Evaluate functions at endpoints and analyze in - interval

For $x\in(0,2]$, when $x = 1$, $f(1)=e^{1}=e\approx2.718$ and $g(1)=\ln(1) = 0$. For any $x\in(0,2]$, the exponential function $y = e^{x}$ is always positive and increasing, and the natural - logarithm function $y=\ln x$ has a slow growth rate compared to the exponential function. The derivative of $y = e^{x}$ is $y'=e^{x}$ and the derivative of $y=\ln x$ is $y'=\frac{1}{x}$. We know that $e^{x}>0$ and for $x\in(0,2]$, $e^{x}>\ln x$.

Answer:

$f(x)$ is above $g(x)$