Taylor and Maclaurin Series
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Questions and Answers

What is the general form of the Taylor series for a function f(x)?

  • f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{6} + \ldots
  • f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2!} + \frac{f'''(0)x^3}{3!} + \ldots (correct)
  • f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2} + \frac{f'''(0)x^3}{6} + \ldots
  • f(x) = f(0) + f'(0)x + \frac{f''(0)x^2}{2} + \frac{f'''(0)x^3}{3} + \ldots

    • What is the Maclaurin series for a function g(x)?

  • g(x) = g(0) + g'(0)x + \frac{g''(x)x^2}{2!} + \frac{g'''(x)x^3}{3!} + \ldots
  • g(x) = g(0) + g'(x)x + \frac{g''(x)x^2}{2!} + \frac{g'''(x)x^3}{3!} + \ldots
  • g(x) = g(0) + g'(0)x + \frac{g''(0)x^2}{2} + \frac{g'''(0)x^3}{3} + \ldots
  • g(x) = g(0) + g'(0)x + \frac{g''(0)x^2}{2!} + \frac{g'''(0)x^3}{3!} + \ldots (correct)

    • What does the Taylor series approximation allow us to do?

  • Approximate a wide range of functions by a polynomial expression (correct)
  • Represent all functions as trigonometric series
  • Find the exact value of any function at a given point
  • Solve differential equations with exact solutions

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