In this and following sections, we will ask you to approximate the value of several constants, such as \(\log(0.93)\text{.}\) A valid question to consider is why we would ask for approximations of these constants that take lots of time, and are less accurate than what you get from a calculator.

One answer to this question is historical: people were approximating logarithms before they had calculators, and these are some of the ways they did that. Pretend you're on a desert island without any of your usual devices and that you want to make a number of quick and dirty approximate evaluations.

Another reason to make these approximations is technical: how does the *calculator* get such a good approximation of \(\log(0.93)\text{?}\) The techniques you will learn later on in this chapter give very accurate formulas for approximating functions like \(\log x\) and \(\sin x\text{,}\) which are sometimes used in calculators.

A third reason to make simple approximations of expressions that a calculator could evaluate is to provide a reality check. If you have a ballpark guess for your answer, and your calculator gives you something wildly different, you know to double-check that you typed everything in correctly.

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2

Use a constant approximation to estimate the value of \(\log(x)\) when \(x=0.93\text{.}\) Sketch the curve \(y=f(x)\) and your constant approximation.

(Remember that in CLP-1 we use \(\log x\) to mean the natural logarithm of \(x\text{,}\) \(\log_e x\text{.}\))

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3

Use a constant approximation to estimate \(\arcsin(0.1)\text{.}\)

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4

Use a constant approximation to estimate \(\sqrt{3}\tan(1)\text{.}\)