Exercise 1.4.25

Question

Compare the functions $f(x)=x^{10}$ and $g(x)=e^{x}$ by graphing both functions in several viewing rectangles. When does the graph of $g$ finally surpass the graph of $f$?

Khagan Gulusoy · Accepted Answer

\begin{figure}[ht]\centering
\begin{tikzpicture}
\begin{axis}
[axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm},restrict y to domain=-5:5, width=6.2cm]
         \addplot[domain=-1.5:1.5, samples=200, mark=none, color=red, ultra thick] {x^10};
         \addplot[domain=-3:5, samples=500, mark=none, color=blue, ultra thick] {exp(x)};
         \addplot[mark=*, mark options={scale=1, fill=black}] coordinates {(-0.913,0.401)} node[left]{(-0.913, 0.401)};
\addplot[mark=*, mark options={scale=1, fill=black}] coordinates {(1.118,3.06)} node[right]{(1.118, 3.06)};
    \end{axis}    
    \end{tikzpicture}
\begin{tikzpicture}
\begin{axis} [axis lines=middle, xlabel={$x$}, ylabel={$y$}, x label style = {at={(ticklabel cs:0.9)},anchor=south,above=4.8mm}, y label style={anchor=north, above=3mm}, scaled y ticks=false, ymin=0, ymax=1.5*10^16, xmin=30, xmax=40, width=6.2cm]
    \addplot[samples=200, domain=30:40, mark=none, color=red, ultra thick] {x^10};
    \addlegendentry{$f(x)=x^{10}$}
    \addplot[samples=500, domain=30:37, mark=none, color=blue, ultra thick] {exp(x)};
    \addlegendentry{$g(x)=e^x$}
    \addplot[mark=*, mark options={scale=1, fill=black}] coordinates {(35.772,3.4306*10^15)} node[above, 3mm]{$(35.772,3.4306\cdot 10^{15})$};
    \end{axis}    
    \end{tikzpicture}
    \end{figure}
The graphs of the functions cross at the points given on the graph. After $x \approx 35.772$ the graph of $g$ finally surpasses the graph of $f$.

Exercise 1.4.25

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Exercise 1.4.25

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