state an antiderivative F of the function f(x)=e^-x^2 that has the property that F(-3)=1. write one sentence to explain how you determined the formula

An antiderivative \( F(x) \) of the function \( f(x) = e^{-x^2} \) is given by:

\[
F(x) = \int_{-3}^{x} e^{-t^2} \, dt + 1
\]

I determined this formula by using the fact that \( F(-3) = 1 \) to define the antiderivative as a definite integral with the lower limit \( -3 \), ensuring the condition \( F(-3) = 1 \) is satisfied by adding 1 to the integral.

Would you like more details or have any questions?

Here are 5 related questions you might explore:
1. How can we approximate the integral \( \int e^{-x^2} dx \) using numerical methods?
2. What is the error function (erf), and how is it related to integrals of \( e^{-x^2} \)?
3. What is the significance of Gaussian integrals in probability theory?
4. How can we express an indefinite integral of \( e^{-x^2} \) in terms of special functions?
5. How does the antiderivative of \( e^{-x^2} \) behave as \( x \to \infty \)?

**Tip:** The function \( e^{-x^2} \) does not have an elementary antiderivative, so integrals involving it often involve special functions or numerical approaches.

State an antiderivative F of the function f(x) = e^-x^2 that has the property that F(-3) = 1. Write one sentence to explain how you determined the formula.

Explore how to find an antiderivative of the function f(x) = e^-x^2 that satisfies the condition F(-3) = 1. This problem delves into key calculus concepts, including the use of definite integrals and the Fundamental Theorem of Calculus, suitable for students in Grades 11-12.

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