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PSET 3 - Applications of Differentiation exercises: What do the numbers in parentheses refer to?
This problem set appears to have been used in another course in which the students had a text or handout that listed equations or formulas designated by numbers in parentheses. For at least some of them we can figure out what they're referring to. Recall from the lectures that we can do approximations two different ways. One is by using the general formula, which for linear approximation is \[f(x)\approx f(x_{0})+f'(x_0)(x-x_0)\]The other method is to use shortcut formulas for x near 0, such as\[(1+x)^r \approx 1+rx\]In exercise 2A-3 you're asked to find the linearization of a formula by using basic approximation formulas, like the second one above, or by using (2), which must be the general formula given first above. Exercise 2A-1 also refers to (2), and tells us not to confuse the variable a appearing in the exercise with the variable a in (2), which has the value 0. From this we can deduce that formula (2) used in connection with these exercises had was like the first one above but with the variable a in place of x sub 0. Because exercise 2A-1 asks for linearization at 0, the value of x sub 0 would be 0 for purposes of that exercise.