# How do you determine all values of c that satisfy the mean value theorem on the interval [1, 1.5] for #f(x)=sinx#?

To determine all values of ( c ) that satisfy the Mean Value Theorem on the interval ([1, 1.5]) for ( f(x) = \sin(x) ), you first find the average rate of change of ( f(x) ) over the interval. Then, you find the derivative of ( f(x) ) and evaluate it at some point ( c ) within the interval. Finally, you set the derivative equal to the average rate of change and solve for ( c ).

For ( f(x) = \sin(x) ) on the interval ([1, 1.5]), the average rate of change is ( \frac{\sin(1.5) - \sin(1)}{1.5 - 1} ).

The derivative of ( f(x) = \sin(x) ) is ( f'(x) = \cos(x) ).

Setting ( f'(c) = \frac{\sin(1.5) - \sin(1)}{1.5 - 1} ), we solve for ( c ).

( \cos(c) = \frac{\sin(1.5) - \sin(1)}{1.5 - 1} ).

Then, we find ( c ) such that ( \cos(c) ) equals this value.

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Mean Value Theorem

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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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