Function 1: \(f(x) = 1 - x\)

Given function: \(f(x) = 1 - x\), with \(c = 0\)

Derivatives:

  1. \(f(x) = 1 - x\)
  2. \(f'(x) = -1\)
  3. \(f''(x) = 0\)
  4. \(f'''(x) = 0\)

Taylor Series Expansion:

\[ f(x) = 1 - x \]

Conclusion:

The Taylor series expansion for \(f(x) = 1 - x\) about \(c = 0\) is \(f(x) = 1 - x\).

Function 2: \(f(x) = e^x\)

Given function: \(f(x) = e^x\), with \(c = 0\)

Derivatives:

  1. \(f(x) = e^x\)
  2. \(f'(x) = e^x\)
  3. \(f''(x) = e^x\)
  4. \(f'''(x) = e^x\)

Taylor Series Expansion:

\[ f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots \]

Conclusion:

The Taylor series expansion for \(f(x) = e^x\) about \(c = 0\) is \(f(x) = 1 + x + \frac{x^2}{2!} + \frac{x^3}{3!} + \ldots\).

Function 3: \(f(x) = \ln(1 + x)\)

Given function: \(f(x) = \ln(1 + x)\), with \(c = 0\)

Derivatives:

  1. \(f(x) = \ln(1 + x)\)
  2. \(f'(x) = \frac{1}{1 + x}\)
  3. \(f''(x) = -\frac{1}{(1 + x)^2}\)
  4. \(f'''(x) = \frac{2}{(1 + x)^3}\)

Taylor Series Expansion:

\[ f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots \]

Conclusion:

The Taylor series expansion for \(f(x) = \ln(1 + x)\) about \(c = 0\) is \(f(x) = x - \frac{x^2}{2} + \frac{x^3}{3} - \frac{x^4}{4} + \ldots\).

Function 4: \(f(x) = x^{1/2}\)

Given function: \(f(x) = x^{1/2}\), with \(c = 0\)

Derivatives:

  1. \(f(x) = x^{1/2}\)
  2. \(f'(x) = \frac{1}{2\sqrt{x}}\)
  3. \(f''(x) = -\frac{1}{4x^{3/2}}\)
  4. \(f'''(x) = \frac{3}{8x^{5/2}}\)

Taylor Series Expansion:

\[ f(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \ldots \]

Conclusion:

The Taylor series expansion for \(f(x) = x^{1/2}\) about \(c = 0\) is \(f(x) = 1 + \frac{1}{2}x - \frac{1}{8}x^2 + \frac{1}{16}x^3 - \ldots\).