Section 4.1: MVT
The Mean Value Theorem
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Which of the following functions satisfy the hypotheses of the Mean Value
Theorem on the given intervals? Explain.
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(a) \( f(x)=\frac{x^2-1}{x-1},\; [0,2] \)
(b) \( f(x)=|x|,\; [-1,1] \)
(c) \( f(x)=\frac{1}{x-1},\; [2,4] \)
(d) \( f(x)=\ln x,\; [-1,2] \)
(e) \( f(x)=\sqrt{x},\; [-1,1] \)
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The Mean Value Theorem guarantees a number \(c\in(0,4)\) for
\( f(x)=\sqrt{x} \). Find the number \(c\).
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The Mean Value Theorem guarantees a number \(c\in(0,16)\) for
\( f(x)=\sqrt{x} \). Find the number \(c\).
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Why is \( f(x)=3x^2+2x+5 \) continuous on \([-1,1]\) and differentiable on
\((-1,1)\)? Find all numbers \(c\in(-1,1)\) satisfying the conclusion of
the Mean Value Theorem.
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Does the function satisfy the hypotheses of the Mean Value Theorem on the
given interval? If it does, find all numbers \(c\) that satisfy the
conclusion.
\( f(x)=\ln(2x),\; [1/2,16] \)
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Given that \( f(x)=\sqrt{2x+1} \) satisfies the hypotheses of the MVT on
\([0,4]\), find the value of \(c\in(0,4)\) satisfying the conclusion.
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Does the function satisfy the hypotheses of the Mean Value Theorem on the
given interval? If it does, find all numbers \(c\) that satisfy the
conclusion.
\( f(x)=\frac{x}{x+9},\; [1,18] \)
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