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Section 4.11 : Linear Approximations
1. Find a linear approximation to \(f\left( x \right) = 3x\,{{\bf{e}}^{2x - 10}}\) at \(x = 5\).
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Start SolutionWe’ll need the derivative first as well as a couple of function evaluations.
\[f'\left( x \right) = 3{{\bf{e}}^{2x - 10}} + 6x\,{{\bf{e}}^{2x - 10}}\hspace{0.5in}f\left( 5 \right) = 15\hspace{0.5in}f'\left( 5 \right) = 33\] Show Step 2There really isn’t much to do at this point other than write down the linear approximation.
\[\require{bbox} \bbox[2pt,border:1px solid black]{{L\left( x \right) = 15 + 33\left( {x - 5} \right) = 33x - 150}}\]While it wasn’t asked for, here is a quick sketch of the function and the linear approximation.
