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Section 4-11 : Linear Approximations

For problems 1 & 2 find a linear approximation to the function at the given point.

  1. \(f\left( x \right) = 3x\,{{\bf{e}}^{2x - 10}}\) at \(x = 5\) Solution
  2. \(h\left( t \right) = {t^4} - 6{t^3} + 3t - 7\) at \(t = - 3\) Solution
  3. Find the linear approximation to \(g\left( z \right) = \sqrt[4]{z}\) at \(z = 2\). Use the linear approximation to approximate the value of \(\sqrt[4]{3}\) and \(\sqrt[4]{{10}}\). Compare the approximated values to the exact values. Solution
  4. Find the linear approximation to \(f\left( t \right) = \cos \left( {2t} \right)\) at \(t = \frac{1}{2}\). Use the linear approximation to approximate the value of \(\cos \left( 2 \right)\) and \(\cos \left( 18 \right)\). Compare the approximated values to the exact values. Solution
  5. Without using any kind of computational aid use a linear approximation to estimate the value of \({{\bf{e}}^{0.1}}\). Solution