How do you determine all values of c that satisfy the mean value theorem on the interval [pi/2, 3pi/2] for # f(x) = sin(x/2)#?
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To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ( \left[\frac{\pi}{2}, \frac{3\pi}{2}\right] ) for ( f(x) = \sin\left(\frac{x}{2}\right) ), we first find the average rate of change of ( f(x) ) over the interval. Then, we find the derivative of ( f(x) ) and set it equal to the average rate of change to solve for ( c ).
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Average rate of change: [ \text{Average rate of change} = \frac{f\left(\frac{3\pi}{2}\right) - f\left(\frac{\pi}{2}\right)}{\frac{3\pi}{2} - \frac{\pi}{2}} ]
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Derivative of ( f(x) ): [ f'(x) = \frac{1}{2} \cos\left(\frac{x}{2}\right) ]
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Set ( f'(c) ) equal to the average rate of change: [ f'(c) = \frac{f\left(\frac{3\pi}{2}\right) - f\left(\frac{\pi}{2}\right)}{\frac{3\pi}{2} - \frac{\pi}{2}} ]
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Solve for ( c ).
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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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