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Differentiability

Those friendly functions that don't contain breaks, bends or cusps are "differentiable". Take their derivative, or just infer some facts about them from the Mean Value Theorem.

Rolle's Theorem

         

Suppose \( f \) and \( g \) are differentiable functions on the interval \( [ 8 , 42 ] \) such that \( f \left( 8\right) = g \left( 8\right) \) and \( f' \left( x \right) < g' \left( x \right) \) for all \( x \) on \( ( 8 , 42 ) .\) If \( f \left( 42\right) = 48 \) , which of the following values NOT possible for \( g \left( 42\right)? \)

Suppose \(f(x)\) is a differentiable function on the interval \( \left [6, 9 \right ]\) such that \(f(6) = 18\) and \(f(9) = 27 .\) Which of the following values must be contained in \( f'(\left [6, 9 \right ])?\)

What is the value of \(\displaystyle{\lim_{x \rightarrow 0^{+} }\frac{e^{7\sin x}-e^{7x}}{\sin x - x} }?\)

How many real solutions are there to the following equation: \[ \frac{4}{3}x^3 + 7x + 2=0? \]

Suppose \(f(x) \) is a differentiable function on \(\left [ 6, 13 \right ] \) such that \( (2x-19)(f(x)-6) \neq f'(x)(x^2 - 19x + 78)\) for any \(x \in \left [ 6, 13 \right ]. \) Which of the following values must be contained in \(f(\left [ 6, 13 \right ])?\)

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