[Technology Exercise] 22. Make a table of values for the funct...

Question

[Technology Exercise] 22. Make a table of values for the function at the points x=1.2, x=11/10, x=101/100, x=1001/1000, x=10001/10000, and x = 1.

a. Find the average rate of change of F(x) over the intervals [1,x] for each x≠1 in your table.

b. Extending the table if necessary, try to determine the rate of change of F(x) at x = 1.

Accepted Answer

Welcome back, everyone. For the function G of X equals X + 3 divided by X minus 3, calculate the average rate of change over the intervals 2 to X inclusive for X equals 2.2, X equals 21/10. 2010, 2010 and 20,000 1 10,000 rounder answer to three decimal places. What is the plausible rate of change of geo X at X equals 2? Now what are we trying to figure out here? Well, it's basically two things, and, and one leads to the other. First, we want to calculate the average rate of change over the intervals for different values of X, and then we want to look for a pattern in these rates of change to determine a plausible rate of change of G of X at X equals 2, OK? So generally, OK, we are trying to find the rate of change here, but now what do we know about the rate of change? Well, we know that the rate of change, OK. For geopics over these intervals. Is going to be equal to the change in G divided by the change in X. Now our interval is always 2 to something else. So that means it's gonna be G of X minus G of 2, OK, divided by X minus 2. GF 2 is going to be constant, so we can calculate that early on. And the G of 2 is gonna be equal to 2 + 3 divided by 2 minus 3. That's gonna be 5 divided by -1, which equals -5. So now, let's use this idea that we found for the rate of change for G of X, OK? And let's go ahead and do it for our intervals and thus determine the plausible rate of change when X equals 2. Let's start with when X equals 2.2, OK? No, when X equals 2.2, that means our change in G divided by a change in X is going to be G of 2.2 minus G of 2 divided by X, well, we know X is 2.2, so that's divided by 2.2 minus 2. Now, what is G of 2.2? And here, if I put that right beside when X equals 2.2, G of 2.2 is gonna be equal to um 2.2 + 3. Divided by 2.2 minus 3, which would be equal to -6.5. So no, that means then the change in G divided by the change in X is going to be equal to. 6.5 minus -5, OK, divided by 2.2 minus 2. Which is going to be equal to -7.500 to 3 decimal places. No, we're going to repeat the same for the next intervals. OK? So no. Then if we go to. When X equals 210 or 2.1, OK. Then our change in G divided by a change in X is gonna be equal to G of 2.1 minus G of 2, divided by 2.1 minus 2. And again, all we need to do is to put is to figure out G of 2.1. So that's gonna be 2.1 + 3 divided by 2.1 minus 3. Which is gonna be equal to -17/3. So our change in G is gonna be negative 173 minus -5 divided by 2.1 minus 2, which equals -23, OK, or approximately 6.67. Next, OK, when X equals 2.01, that's 201,100, OK. And our change and and and by now you realize that we're basically doing the same thing. So we're just substituting the values of X into G, OK, and putting that result into our formula here. So I mean, we, we could go ahead and continue to do that, OK, so. In this case, this is going to be um G of. 2.01, OK, which I only write that out. It's gonna be equal to -167, 303. So this is gonna be -167303 minus -5. Divided by 2.01 minus 2, which equals -233s or approximately 6.061. Next, For 20,001,000. Or 2.001. Then our average rate of change. Is gonna be equal to G of 2.001 minus GR 2 divided by 2.001 minus 2, which I mean by no, you probably get the hang of it. So this is no -1,667, 303 minus negative5, that's the value you should have got. Divided by our denominator here. Which is going to be equal to -2,333 or approximately 6.006. And then when X equals 2.0001, 20,000 and 110,000. Then our rate of change, our average rate of change, is gonna be equal to 2.0001 minus G of 2. Divided by 2.0001 minus 2. That's negative 16,667 divided by 3300. 33 minus -5. Divided by 2.0001 minus 2, which equals negative 20,300, sorry. 3,333, which is approximately 6.001. Now, we found our average rates of changes for our intervals. OK, so we have the values for our intervals. So what do we do with these? Well, no, we need to figure out if it can help us to find a plausible rate of change when X equals 2. So let's just quickly, let's put these in a table, right? So now we have our intervals. And we have our rates of change. And let's see if we can find some type of pattern here, which you may already see. You, you might already see it coming out. So now, like we said, for the interval, 2 to 2.2, the rate of change was -7.500. Between 2 to 2.1, it was -6.667. 2 to 2.01. It was -6.061. 2 to 2.001, it was -6.006. And from 2 to 2.0001, it was -6.001. If that's the case, then what do you think it's going to be as X approaches two? Well, notice here that it, as it gets closer to 2, the rates of change gets closer to 6, OK. Thus, the plausible rate of change of G of X at X equals 2, OK. The rate of change, plausible rate of change. Uh, GFX. Is -6. Thanks a lot for watching everyone. I hope this video helped.

Key Concepts

Average Rate of Change

Limit and Instantaneous Rate of Change

Function Evaluation

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