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### Section 4-16 : Taylor Series

For problems 1 – 3 use one of the Taylor Series derived in the notes to determine the Taylor Series for the given function.

1. $$f\left( x \right) = \sin \left( {{x^4}} \right)$$ about $$x = 0$$
2. $$f\left( x \right) = 9{x^4}{{\bf{e}}^{ - 12x}}$$ about $$x = 0$$
3. $$f\left( x \right) = 6{x^2}\cos \left( {7{x^5}} \right)$$ about $$x = 0$$

For problem 4 – 13 find the Taylor Series for each of the following functions.

1. $$f\left( x \right) = \sin \left( x \right)$$ about $$\displaystyle x = \frac{{3\pi }}{2}$$
2. $$f\left( x \right) = {{\bf{e}}^{1 - 8x}}$$ about $$x = 3$$
3. $$f\left( x \right) = \ln \left( {1 - x} \right)$$ about $$x = - 2$$
4. $$f\left( x \right) = \ln \left( {2 + 9x} \right)$$ about $$x = 1$$
5. $$\displaystyle f\left( x \right) = \frac{1}{{{{\left( {6 - x} \right)}^7}}}$$ about $$x = 4$$
6. $$\displaystyle f\left( x \right) = \frac{1}{{{{\left( {4 + 9x} \right)}^2}}}$$ about $$x = - 2$$
7. $$f\left( x \right) = \sqrt {2 + x}$$ about $$x = 1$$
8. $$f\left( x \right) = \sqrt {1 - 4x}$$ about $$x = - 3$$
9. $$f\left( x \right) = - 3{x^2} - x + 10$$ about $$x = - 8$$
10. $$f\left( x \right) = {x^3} + 9{x^2} - 10x + 2$$ about $$x = 3$$