Over that interval, on average, every time x increases byġ, y is increasing by 4. So that would be ourĪverage rate of change. So what is our average rate of change? Well, it's going to be our change in y, or our change in x, which is equal to 8 overĢ, which is equal to 4. When x increased by 2 from 1 to 3, y increases by 8, so it's Y over the same interval? Our change in y is equal to. What's my change in x?" Well, we could see very clearly that our change in x over this interval is equal to positive 2. And to figure out theĪverage rate of change of y with respect to x, you say, "Okay, well And so you can see when x is equal to 3, y is equal to 9. And when x is equal to 3, y is equal to 3 squared, Is y is equal to x squared, when x is equal to 1, y If I were to just make a table here, where, if this is x, and this And that's a closed interval, where x could be 1, and We want to know the average rate of change of y with respect to x over the interval from And the first thing I'd like to tackle is think about the average rate of change of y with respect to x over the interval from xĮqualing 1 to x equaling 3. Is equal to x squared, or at least part of the graph So right over here we have the graph of y There are plenty of things in mathematics that have no real world application, though they are studied nonetheless. Lastly, "not having a purpose" (which is not the case with secant lines and average rates of change) is a poor argument for neglecting to study anything – especially in mathematics. Furthermore, if you are looking at discrete data (as is the case in every real world observation), there is no way to get an instantaneous rate of change from that data because it is not continuous. In particular, in physics, there are a lot of phenomena that occur that have to deal with average rates of change instead of instantaneous rates of change. As for "not bothering with the secant", there is no way to explain the process of finding instantaneous change without explaining how to find average change for the reason you just identified: the principles involved in finding average rate of change are part of finding instantaneous rate of change.Īlso, average rates of change have advantages in their own right. As you say, the process of finding the slope of the tangent line descends from the process of finding the slope of a secant line (the only difference is that a certain limit is taken of the difference quotient which is the expression for the slope of a general secant). The break even point is to sell 150 hot dogs.Calculating average change using secant lines is actually an important intermediate step to finding instantaneous change via tangent lines (and thus also derivatives). Variable cost = fee charged for 1 hot dog × number of hot dogs sold R = selling price of 1 hot dog × number of hot dogs sold Let R be the revenue made for selling x hot dogs Let C be the cost of buying and selling x hot dogs However, the city hall charges him 1 dollar for each hot dog soldĬalculate the break even point if the price he charges for 1 hot dog is $1.50 The city allows him to sell his hot dogs somewhere near the city hall. It costs a man 75 dollars to buy the things that he needs to make hot dogs. The break even point is to sell 10000 books. Variable cost = fee charged for 1 book × number of books sold R = selling price of 1 book × number of books sold Let R be the revenue made for selling x books Let C be the cost of producing and selling x books If the company charges 9 dollars per book, how many books should they sell to break even? To help the publishing company sell the books, a marketing company charges 4 dollars for each book sold. The 50,000 is a fixed cost or a cost that cannot change. It costs a publishing company 50,000 dollars to make books.
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