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Apply the Mean Value Theorem to f(t) = ln(1 + t) on [0, x] for x > 0 to show that x/(1 + x) < ln(1…

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Apply the Mean Value Theorem to $f(t) = \ln(1 + t)$ on $[0, x]$ for $x > 0$ to show that $\dfrac{x}{1 + x} < \ln(1 + x) < x$ . Use $x = 0.2$ to bound $\ln 1.2$ . Give the lower bound, the upper bound, and the exact value of $\ln 1.2$ to four decimal places.

Need a reminder? The key ideaShow

Mean Value Theorem: if $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$ , then $f'(c) = \dfrac{f(b) - f(a)}{b - a}$ for some $c$ in $(a, b)$ . It gives bounds: if $m \le f'(x) \le M$ then $m(b - a) \le f(b) - f(a) \le M(b - a)$ .

The worked solution

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Work it out on paper first. It is fine to get stuck or make a mistake: that is how we learn. When you are ready, go through the solution one step at a time, and check each step against your own.

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Apply the Mean Value Theorem to f(t) = ln(1 + t) on [0, x] for x > 0 to show that x/(1 + x) < ln(1…

The worked solution

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