How does the author determine what the first equation should be? What about the second equation? How are these examples similar? How are they different?
Author of what? These sound like followup questions to some reading assignment.
Well, when I taught callucus, I told my students to use Wolfram Alpha whenever they couldn't figure out how to do the problem by hand. I told them to try figuring out the steps on the website, then write the solution in their own words, then write with help from WA next to their answer. They seemed happy to do that, and they seemed to instinctively know that if WA did it for them, they weren't exactly on top of things.Homework was not a big part of the grade, though, and if it were I think there would have been more motivation to use WA as a way to fake-it. Not sure what I would have done in that case. I'm also not sure what to do in the cases where the students view the whole class as an obstacle (like remedial math classes), except to say that I can tell when you've done all your homework by computer (the steps are often overly detailed or technically awkward).
To determine what the first equation should be, the author likely considers the problem or scenario they are trying to represent mathematically. They may also take into account any given information or constraints mentioned in the problem. The first equation may be derived from analyzing relationships between quantities, identifying patterns, or applying mathematical principles.
Similarly, the author determines the second equation based on the same process – analyzing the problem, considering relevant information, and applying mathematical concepts. The second equation could represent a different aspect or an extension of the problem compared to the first equation.
These examples are similar in that they both involve the formulation of equations based on a given problem or scenario. The author likely considers the same mathematical principles when creating both equations, such as algebraic operations, variables, and relationships between quantities.
However, there might be differences in the specific variables, coefficients, or constants used in the equations. The second equation may involve additional parameters or adjust the existing components to represent a distinct aspect or variation of the problem. These differences can arise due to the nature of the problem being solved or the specific information provided in each scenario.