Question

How do you use a Taylor series to expand: `f(x) = x^2 + 2x + 5` about x = 3?

Answer 1

If `f(x) = x^2+2x+5`
then
`color(white)("XXXXX")` `f'(x) = 2x+2`
`color(white)("XXXXX")` `f''(x)= 2`
`color(white)("XXXXX")` `f'''(x)" and beyond" = 0`

The Taylor Series about `x=3` is
`color(white)("XXXXX")` `f(3)+(f'(3))/(1!)(x-3)+(f''(3))/(2!)(x-3)^2`

`color(white)("XXXXX")` `= 20 + 8(x-3) + (x-3)^2`