Introduction:

Note that, without bounds on how big the error might be, an "estimate" is next to useless: if your repairs will cost you an "estimated" $300 but you don't know if this estimate is within an "error" of $25 or $2,500 of the true cost, you may be in for an unpleasant surprise! Conversely, in most situations, we don't know the exact value of the error; otherwise we could add that value to our estimate and get the exact value of what we're estimating -- so there would be no use for an "estimate" at all!

(I) Let g(x)= x = x^1/2.

(1) Then the derivative g'(x) = ______ (In your notation, avoid using fractional or negative powers).

(2) What do you notice about the range of values that g'(x) takes on? __________. We can therefore conclude something about g(x), namely, that g(x) is __________.

(3) Furthermore, notice that g(x)=h^-1(x) where h(x)=x² restricted to x>=0. Since h(x) is increasing and concave up (for all x) and increasing, it follows visually that we know the concavity of g -- g(x) is _______________ for all x.

Give a clear, convincing argument for why your claim in (I)(3) is true, including a graphical component:






(II) Recall that for a general function f(x), the approximate-equality used when we apply the technique of linear approximation is:

	f(x)		f(a)	+	(x-a)f'(a)
Hence, using f(x) = g(x), i.e., f(x) = x1/2 this becomes:
g(x) =	________		________	+	________
Even more specifically, using x=10 and a=9 we get:
g(10) =	________		________	+	________
		=	________	+	________

Your estimate is hence: 10 __________, or in decimal, about __________.

To analyze your estimate more carefully, consider the following three numbers:

(ii) 10