In the first Mean Value Theorem #f(b)=f(a)+(ba)f'(c), a<c<b, f(x) =log_2 x, a=1 and f'(c)=1. How do you find b and c?
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To find (b) and (c) in the first Mean Value Theorem equation (f(b) = f(a) + (b  a)f'(c)) where (a < c < b), and given that (f(x) = \log_2 x), (a = 1), and (f'(c) = 1), you can follow these steps:

Substitute the given values into the equation: (f(a) = \log_2 1 = 0) since (\log_2 1 = 0). (f'(c) = 1).

Solve the equation for (b): (f(b) = f(a) + (b  a)f'(c)) (f(b) = 0 + (b  1)(1)) (f(b) = b  1)

Since (f(b) = \log_2 b), equate it to (b  1): (\log_2 b = b  1)

Solve for (b) by finding the value that satisfies this equation.

Once you find the value of (b), substitute it back into the equation to find (c).

Given that (a = 1) and (b) is the value you found, choose (c) such that (1 < c < b).
This will provide the values of (b) and (c) that satisfy the conditions of the first Mean Value Theorem for the given function (f(x) = \log_2 x) and (f'(c) = 1).
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When evaluating a onesided 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 onesided 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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