How do you find a function f(x), which, when multiplied by its derivative, gives you #x^3#, and for which #f(0) = 4#?
This is a first-order differential equation with separable variables:
Given that the starting circumstance is:
So the function is:
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To find the function ( f(x) ), we need to solve the differential equation
[ f(x) \cdot \frac{{df}}{{dx}} = x^3 ]
Given that ( f(0) = 4 ), we can proceed as follows:
- Separate variables and integrate both sides with respect to ( x ):
[ \int f(x) , df = \int x^3 , dx ]
- Integrate both sides:
[ \frac{1}{2} f(x)^2 = \frac{1}{4} x^4 + C ]
- Solve for ( f(x) ):
[ f(x)^2 = \frac{1}{2} x^4 + C ]
- Apply the initial condition ( f(0) = 4 ) to find the constant ( C ):
[ f(0)^2 = C ] [ 4^2 = C ] [ C = 16 ]
- Substitute ( C = 16 ) back into the equation:
[ f(x)^2 = \frac{1}{2} x^4 + 16 ]
- Take the square root of both sides:
[ f(x) = \pm \sqrt{\frac{1}{2} x^4 + 16} ]
Given ( f(0) = 4 ), we choose the positive square root:
[ f(x) = \sqrt{\frac{1}{2} x^4 + 16} ]
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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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- What is the equation of the normal line of #f(x)= 1+1/(1+1/x)# at #x = 1#?

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