# How do you find the Maclaurin series for #f(t) =t^3(e^(-t^2))# centered at 0?

Let us begin with the exponential function's MacLaurin series:

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To find the Maclaurin series for ( f(t) = t^3e^{-t^2} ) centered at 0, you need to compute the derivatives of ( f(t) ) and evaluate them at ( t = 0 ). Then, you can use the Maclaurin series formula:

[ f(t) = f(0) + f'(0)t + \frac{f''(0)}{2!}t^2 + \frac{f'''(0)}{3!}t^3 + \cdots ]

First, find the derivatives of ( f(t) ):

[ f'(t) = 3t^2e^{-t^2} - 2t^4e^{-t^2} ] [ f''(t) = 6te^{-t^2} - 6t^3e^{-t^2} - 4t^3e^{-t^2} + 4t^5e^{-t^2} ] [ f'''(t) = (6 - 12t^2)e^{-t^2} - (18t^2 - 12t^4)e^{-t^2} + (12t^4 - 20t^6)e^{-t^2} ]

Now, evaluate these derivatives at ( t = 0 ) to find the coefficients:

[ f(0) = 0 ] [ f'(0) = 0 ] [ f''(0) = 6 ] [ f'''(0) = 6 ]

Now, plug these coefficients into the Maclaurin series formula:

[ f(t) = 0 + 0t + \frac{6}{2!}t^2 + \frac{6}{3!}t^3 + \cdots ]

Simplify the series:

[ f(t) = 3t^2 + t^3 + \cdots ]

So, the Maclaurin series for ( f(t) = t^3e^{-t^2} ) centered at 0 is ( 3t^2 + t^3 + \cdots ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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