Let g(x)=√(x) for x ≥ 0 a. Find the average rate of change...

Let $g(x)=\sqrt{x}$ for $x \geq 0$ a. Find the average rate of change of $g(x)$ with respect to $x$ over the intervals [1,2],[1,1.5] and $[1,1+h]$ b. Make a table of values of the average rate of change of $g$ with respect to $x$ over the interval $[1,1+h]$ for some values of $h$ approaching zero, say $h=0.1,0.01,0.001,0.0001,0.00001$ and 0.000001 c. What does your table indicate is the rate of change of $g(x)$ with respect to $x$ at $x=1 ?$ d. Calculate the limit as $h$ approaches zero of the average rate of change of $g(x)$ with respect to $x$ over the interval $[1,1+h]$

Transcript

00:01 In this question we have given that z of x is equal to root x.

00:05 We know that the average rate of change of g of x on the interval a comma b is g of b minus g of a divided by b minus a.

00:16 Let us solve part a.

00:19 The average rate of change for 1 comma 2 is given by g of 2 minus g of 1 divided by 2 minus 1.

00:30 This is equal to root 2 minus 1 divided by 1.

00:36 Simplifying this, we get 0 .414 -213.

00:43 Next, the average rate of change for 1 comma 1 .5 is given by g of 1 .5 minus g of 1 divided by 1 .5 minus 1 .1.

00:57 This is equal to square root of 1 .5 minus 1 divided by 0 .5.

01:05 Simplifying this, we get 0 .449.

01:13 Next, for 1 comma 1 plus h, it is given by square root of 1 plus h minus 1 divided by h.

01:26 We now move on to solve part b the average rate of change of g for h is equal to 0 .1 is given by square root of 1 .1 minus 1 divided by 0 .1.

01:48 Simplifying this we get 0 .488.

01:52 088.

01:55 Next for hs.

01:58 Equal to 0 .01, we get square root of 1 .01 minus 1 divided by 0 .01.

02:10 Simplifying this, we get 0 .49756.

02:16 Next, for hs equal to 0 .001, we get square to 0 .001, we get square root of 0 .49756.

02:22 Next, we get square root of, 1 .001 minus 1 divided by 0 .001.

02:31 Simplifying this we get 0 .49 75.

02:38 Next for hs sql to 0 .001.

02:43 We get square root of 1 .0 .0 .1 minus 1 divided by 0 .000 0 .000...

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