Derivatives, Limits, and Arc Length in Calculus
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Questions and Answers

What is the derivative of $f(x) = e^x imes an(x)$?

  • $e^x imes ( an(x) + rac{1}{ an^2(x) + 1})$ (correct)
  • $e^x imes ( an(x) + rac{1}{ an(x)})$
  • $e^x imes ( an(x) - rac{1}{ an(x)})$
  • $e^x imes ( an(x) - rac{1}{ an^2(x) + 1})$

    • What is the arc length of the curve defined by $y = e^{-x^2}$ from $x = 0$ to $x = 1$?

  • $rac{ ext{erf}(1)}{ ext{erf}(1)^2}$
  • $rac{ ext{erf}(1)}{ ext{erf}(rac{1}{ ext{erf}(1)})}$
  • $ ext{erf}(1)^2$
  • $rac{ ext{erf}(1)}{2}$ (correct)

    • What is the limit of $rac{ an(x)}{x}$ as $x$ approaches 0?

  • $rac{ an(0)}{0}$
  • Undefined
  • 0
  • 1 (correct)

    • What is the integral of $e^{-x^2}$ with respect to $x$?

    <p>$\frac{\sqrt{\pi}}{2}\text{erf}(x) + C$</p> Signup and view all the answers

    What is the Taylor series expansion of $e^{-x^2}$?

    <p>$1 - x^2 + \frac{x^4}{2} - \frac{x^6}{6} + \cdots$</p> Signup and view all the answers

    What is the value of $\lim_{x \to \infty} e^{-x^2}$?

    <p>0</p> Signup and view all the answers

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