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Section 4-11 : Linear Approximations

For problems 1 – 4 find a linear approximation to the function at the given point.

  1. \(f\left( x \right) = \cos \left( {2x} \right)\) at \(x = \pi \)
  2. \(h\left( z \right) = \ln \left( {{z^2} + 5} \right)\) at \(z = 2\)
  3. \(g\left( x \right) = 2 - 9x - 3{x^2} - {x^3}\) at \(x = - 1\)
  4. \(g\left( t \right) = {{\bf{e}}^{\sin \left( t \right)}}\) at \(t = - 4\)
  5. Find the linear approximation to \(h\left( y \right) = \sin \left( {y + 1} \right)\) at \(y = 0\). Use the linear approximation to approximate the value of \(\sin \left( 2 \right)\) and \(\sin \left( {15} \right)\). Compare the approximated values to the exact values.
  6. Find the linear approximation to \(R\left( t \right) = \sqrt[5]{t}\) at \(t = 32\). Use the linear approximation to approximate the value of \(\sqrt[5]{{31}}\) and \(\sqrt[5]{3}\). Compare the approximated values to the exact values.
  7. Find the linear approximation to \(h\left( x \right) = {{\bf{e}}^{1 - x}}\) at \(x = 1\). Use the linear approximation to approximate the value of \({\bf{e}}\) and \({{\bf{e}}^{ - 4}}\). Compare the approximated values to the exact values.

For problems 8 – 10 estimate the given value using a linear approximation and without using any kind of computational aid.

  1. \(\ln \left( {1.1} \right)\)
  2. \(\sqrt {8.9} \)
  3. \(\sec \left( {0.1} \right)\)