What is #f(x) = int x-3e^(x)dx# if #f(0)=-2 #?
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To find ( f(x) = \int (x - 3e^x) , dx ) given that ( f(0) = -2 ), you first need to find the antiderivative of ( x - 3e^x ), and then evaluate it at ( x = 0 ) and subtract ( f(0) ) from it.
The antiderivative of ( x - 3e^x ) is ( x - 3e^x + C ), where ( C ) is the constant of integration.
When ( x = 0 ): [ f(0) = (0 - 3e^0) + C = -2 ] [ -3 + C = -2 ] [ C = -2 + 3 = 1 ]
So, the antiderivative is ( x - 3e^x + 1 ).
Therefore, ( f(x) = x - 3e^x + 1 ).
Verify by substituting ( x = 0 ) into ( f(x) ): [ f(0) = 0 - 3e^0 + 1 = -2 ]
Thus, the function ( f(x) = x - 3e^x + 1 ) satisfies ( f(0) = -2 ).
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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.
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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