Show All Notes Hide All Notes

Section 10.16 : 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 \(x = 3\)