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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. $$\sec \left( {0.1} \right)$$