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Chapter 1: Problem 51

In your own words, state the properties of the natural exponential function.

Short Answer

Expert verified
The key properties of the natural exponential function are: its asymptotic behaviour towards 0 as \( x \) approaches negative infinity, the derivative of the function is the function itself, the function increases rapidly for positive \( x \) and decreases rapidly for negative \( x \), and the function intercepts the y-axis at \( y=1 \), but has no x-intercepts.

Step by step solution

01

Identify the Asymptotic Behavior

The natural exponential function \( e^x \) has a horizontal asymptote at \( y=0 \). This means that as \( x \) approaches negative infinity, the function \( e^x \) approaches 0 but never reaches it.
02

Identify the Derivative

The derivative of the natural exponential function \( e^x \) is unique in that it is the function itself, i.e., the derivative of \( e^x \) is \( e^x \) again.
03

Identify the Rate of Change

The natural exponential function \( e^x \) increases rapidly for positive \( x \) (since the rate of change equals the function itself), and decreases rapidly (tending to zero) for negative \( x \).
04

Identify the Intercepts

The graph of \( y=e^x \) intersects the y-axis at \( y=1 \), i.e., when \( x=0, y=1 \). As far as x-intercepts are concerned, there are none, as the function never touches the x-axis.

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Most popular questions from this chapter

(a) Let \(f_{1}(x)\) and \(f_{2}(x)\) be continuous on the closed interval \([a, b]\). If \(f_{1}(a)f_{2}(b),\) prove that there exists \(c\) between \(a\) and \(b\) such that \(f_{1}(c)=f_{2}(c)\). (b) Show that there exists \(c\) in \(\left[0, \frac{\pi}{2}\right]\) such that \(\cos x=x\). Use a graphing utility to approximate \(c\) to three decimal places.

Find two functions \(f\) and \(g\) such that \(\lim _{x \rightarrow 0} f(x)\) and \(\lim _{x \rightarrow 0} g(x)\) do not exist, but \(\lim _{x \rightarrow 0}[f(x)+g(x)]\) does exist.

Sketch the graph of the function. Use a graphing utility to verify your graph. $$ f(x)=\arccos \frac{x}{4} $$

Find the point of intersection of the graphs of the functions. $$ \begin{array}{l} y=\arcsin x \\ y=\arccos x \end{array} $$

Prove that if \(\lim _{x \rightarrow c} f(x)=0\) and \(|g(x)| \leq M\) for a fixed number \(M\) and all \(x \neq c,\) then \(\lim _{x \rightarrow c} f(x) g(x)=0\).
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