Question

Find the Taylor series for f(x) = 4 + x * e^(6x) centered at a = 0.

Answer 1

The Taylor series for f(x) = 4 + x * e^(6x) centered at a = 0 is:

`\[ f(x) = 4 + x + (6x^2)/(2!) + (6^2x^3)/(3!) + (6^3x^4)/(4!) + \dots \]`

Step-by-step explanation:

The Taylor series for the function
`\(f(x) = 4 + x \cdot e^(6x)\) centered at \(a = 0\)` is an infinite sum that represents the function's values using the derivatives of the function evaluated at the center. In this case, the center is
`\(a = 0\).`The general form of the Taylor series is given by the formula:

`\[ f(x) = f(a) + f'(a)(x-a) + (f''(a))/(2!)(x-a)^2 + (f'''(a))/(3!)(x-a)^3 + \dots \]`

For the given function, the derivatives at
`\(a = 0\)`are straightforward to calculate. The zeroth-order derivative
`(\(f(0)\)) is 4`, the first-order derivative
`(\(f'(0)\))` is 1, and the higher-order derivatives follow a pattern related to the exponential function. The Taylor series is then constructed by plugging these values into the formula.

The terms involving higher-order derivatives result in powers of
`\(6x\)` in the series. As the series continues, each term involves a higher power of x and a higher power of 6, reflecting the behavior of the exponential term
`\(e^(6x)\)`. Therefore, the final answer is the Taylor series representation of
`\(f(x) = 4 + x \cdot e^(6x)\) centered at \(a = 0\).`