The table gives the height as time passes of a typical pine tree grown for lumber at a managed site.
If $ H(t) $ is the height of the tree after $ t $ years, construct a table of estimated values for $ H' $ and sketch its graph.
So in this problem we are given the this data on the height of typical pine tree grown for lumber as a man and a managed site. And so we have these different heights from the Arkansas Forestry Commission for the different ages of the trees. And were asked to determine first of all to construct a table of estimated values for H. Prime the derivative and then sketch the graph of it. So H prime the derivative is the average rate of change for each of the intervals and so H. Prime of 14 okay will be H at 21 -H at 14 Divided by 21 -14 Which will be 54 On its 41 over seven. And this is 13/7. So this one is 13/7. Now To do hpe prime at 21. First of all we'll do the forward interval and then we'll do the reverse interval And averaged the two we already have the forward interval here. So all we need now is hpe prime at 21 on the reverse interval. I'll do it like that. So that's h of 28 Minour age of 21 Divided by 28 -21. and so this is 64 -54 over seven. So this is 10/7 and now we do the average of this. So h prime of 21 is one half times 13/7 plus 10/7. So what is that? That is 23 over 14. So this one is 23 over 14. Okay now Do H private 28 We do the forward interval and then the reverse interval but we already have the forward interval right here so we just need the reverse interval H. prime at 28 on the reverse interval. There's a church at 35 -H. 28 over 35 -28. And so that's 72 -64 over seven Which is 8/7. And so then H. Prime at 28 then is the average of these two so that's one half times 10/7 plus 8/7. Well 10 plus eight is 18 by about two that's nine. So this is 9/7. Okay so put that one up here in our table now 9/7 do the same thing again we're going to the forward interval and the reverse interval and we already have the forward interval right here. So we just need H. Prime at 35 on the reverse interval which is H. At 42 -H. 35 over 40 to -35. So that's 78 minus 72/7. And so that's 6/7 and then doing the average of these two H. prime at 35. Uh huh. Is one half of 8/7 plus 6/7. Late Plus six is 14 14/7 is two Half times two. That's gonna be one. So this is one. Okay then For 42 we do the forward interval and the reverse interval. We already have the forward interval, I have it right here So I just need the reverse interval. So h prime at 42 on the reverse interval Is gonna be a church at 49 -H at 42 over 49 -42. Oh, which is gonna be 83 -78/7, So that's 5/7. And now we average these The age prime at 42 is one half of 6/7 plus 5/7, Which is well six plus 5, that's 11. So this is 11/14. Okay, so this is 11/14 and we've already done the last interval now Right here we got 5/7 right there so that gives us 5/7 and so now we're asked to graph this, so a good or graphing but in a table here, so I have The first tee is at 14 And then 21, 35,- 49. And values for the derivatives we got Or 13/7, 23/14, 9/7, one and 11/14 and right there and 5/7 and so here is the graph we have we're here. Mhm. Okay, let me adjust, it's for a minute. My exes will go from 14 2 49 and my wise will go from We'll say .5 two two let's say right and there we go. There is our data graft up for us, right there for the derivative.