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If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
If you are looking for some problems with solutions you can find some by clicking on the "Practice Problems" link above.
Section 4.11 : Linear Approximations
For problems 1 – 4 find a linear approximation to the function at the given point.
- \(f\left( x \right) = \cos \left( {2x} \right)\) at \(x = \pi \)
- \(h\left( z \right) = \ln \left( {{z^2} + 5} \right)\) at \(z = 2\)
- \(g\left( x \right) = 2 - 9x - 3{x^2} - {x^3}\) at \(x = - 1\)
- \(g\left( t \right) = {{\bf{e}}^{\sin \left( t \right)}}\) at \(t = - 4\)
- 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.
- 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.
- 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.
- \(\ln \left( {1.1} \right)\)
- \(\sqrt {8.9} \)
- \(\sec \left( {0.1} \right)\)