What's the difference between the subject pronouns "On" and "Nous?" When would you use "On" instead of "Nous" and vice versa?
Both subject pronouns "On" and "Nous" mean "We" or "Us" in English. However, we usually use "on" when we are speaking in a general sense or context. For example, it would make more sense to say, "En France, on parle francais," instead of "En France, nous parlons francais." This is because it is a fact that in France, the general population speaks French. If you used "Nous" it technically wouldn't be wrong, but "On" is more convenient in the context.
Which statement is correct and why? "Jon and Rosie decided to go to the gym, because they hadn't worked out in a long time." "Jon and Rosie decided to go to the gym because they hadn't worked out in a long time."
The second statement would be the correct one. If you look at the first sentence, there is the placement of an unnecessary comma between "gym" and "because." A comma indicates a pause between the ending of an introductory word, phrase, or clause, and the beginning of the main part of the sentence. A comma also splits to independent clauses. Notice how the second half of the first statement is a fragment. Removing the comma would make the sentence whole.
Can you use the IMVT (Intermediate Value Theorem) to prove that the equation x^3 = x + 3 is solvable?
Let f(x)= x^3 -x - 3 and let m = 0, where f is continuous for all x-values (it is continuous because f is a polynomial). Note that f(0) = -3 and f(2) = 3 and m = 0 is in between f(0) and f(2), so choose the interval [0,2]. Thus, by IMVT, there is at least one x-value "c" on the closed interval 0<c<2, so that f(c) = m , i.e, c^3-c-3 = 0 and the equation is solvable.