What is #f(x) = int xsqrt(5-2) dx# if #f(2) = 3 #?
The function
First, compute the integral:
Power rule:
If you would like, you can rewrite it as:
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To find ( f(x) = \int x \sqrt{5 - 2x} , dx ) when ( f(2) = 3 ), we need to find the definite integral of the given function and then use the given value to determine the constant of integration.
Since ( f(2) = 3 ), this means:
[ \int_{2}^{2} x \sqrt{5 - 2x} , dx = 3 ]
This integral evaluates to zero since the lower and upper limits of integration are the same.
Now, we have:
[ \int_{2}^{x} x \sqrt{5 - 2x} , dx = f(x) - f(2) ]
We need to evaluate this integral to find ( f(x) ). Then we can use the given information to find the value of ( f(x) ).
After evaluating the integral, we find:
[ f(x) = \frac{5}{12} (5 - 2x)^{3/2} + C ]
Now, we use the given value ( f(2) = 3 ) to find the constant of integration ( C ).
[ 3 = \frac{5}{12} (5 - 2(2))^{3/2} + C ]
[ 3 = \frac{5}{12} (5 - 4)^{3/2} + C ]
[ 3 = \frac{5}{12} (1)^{3/2} + C ]
[ 3 = \frac{5}{12} + C ]
[ C = 3 - \frac{5}{12} ]
[ C = \frac{31}{12} ]
Therefore, the function ( f(x) ) is:
[ f(x) = \frac{5}{12} (5 - 2x)^{3/2} + \frac{31}{12} ]
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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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