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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 $$g\left( z \right) = \sqrt{z}$$ at $$z = 2$$. Use the linear approximation to approximate the value of $$\sqrt{3}$$ and $$\sqrt{{10}}$$. 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