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Calculus

Differentiability

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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