This is a big question. I can't answer that from the top of my head, but I remember reading a blog entry about it.
- the "zu" is used, when you have to verbs in the sentence in a hierarchic relationship. The first verb will be conjugated, the second one will have "zu" and be in infinitive at the end of the sentence.
Ich versuche, mit dir zu sprechen. I try to speak with you.
'2. the "zu" is not used when the first verb is a modal verb like dürfen, sollen, können, müssen, wollen, möchten (be allowed, should, can, must, want, would like).
Ich darf/soll/kann/muss/will mit dir sprechen. I am allowed to/should/can/must/want to/would like to speak with you.
Notice: not all German modal verbs translate to English modal verbs as some still need to add to (want to, would like to, be allowed to).
'3. the "zu" is not used in constructions that resemble accusative-cum-infinitivum from Latin, where the first verb is usually a verb of perception.
Ich höre dich sprechen. I hear you speak. (very common in English)
'4. the "um...zu" adds intention and purpose. meaning "in order to".
Ich spreche mit dir, um dir zu helfen. I talk to you (in order) to help you.
Here's the blog.
"Der Ansatz" is the noun to the verb "ansetzen". That derives from separable prefix "an-" which means "at, next to" and "setzen" to sit. If you inflict "setzen" on something it means to put and "ansetzen" is to put next to. It can also mean to add to something or to place a tool in a promising spot on a larger structure. Thus, "der Ansatz" can be an approach to a logical problem.
"Der Ansatz" can also be several things that were put together to achieve a certain result, like making beer from barley, yeast and so on.
I don't know how the entry or the departure fit to "der Ansatz". However, there are more combinations with separable prefixes: der Aufsatz (the essay, the landing), der Besatz (the trimming) der Absatz (the heel, the paragraph, the sales), der Aussatz (the break, the leprosy) der Einsatz (the input, the dedication, the insertion, the mission), der Ersatz (the replacement), der Umsatz (the volume of sales), der Zusatz (the additive, addition)
Thank you for the explanation, very helpful. Although I knew that this is about the results or the action of ansetzen, I didn't really think it through and the dictionary translations confused me.
Based on what you said, "entry" in this context probably means an entry made to a guestbook or a database. Departure is less clear, but I can see how it might be a technical term related to logical problem solving in a similar fashion as "approach".
The dictionary where these are from is dict.leo.org, which is usually quite decent, but in the case of this word the translations are all over the place (neck, nose, rudiment, scale, etc.) However, now I can see or at least speculate on the rationale for most of the correspondences.
Combinations of verbs and prefixes generate a multitude of new verbs in German, each of which usually has at least two meaning. If you like to speculate on them, I recommend this blog:
German native speaker here. "Ansatz" could also be the way to form the lips as you said. It could also be putting herbs or fruits into alcohol to produce liquor. But most commonly it is used as mathematical or generally intellectual approach to a problem. I think duolingo doesn't take professional language into account. It would be too much, I guess.
No, "Ansatz" is more static. You put something in place, you don't approach it.
We're now making our approach to the airport.
Wir nähern uns jetzt dem Flughafen. (We now approach the airport.)
The approach to the citadel is heavily guarded.
Die Annährerung an die Zitadelle is stark bewacht.
Nähern is to come closer. Annähern is almost the same, but more tentative and not fully reaching it.
I guess that "approach to the citadel" should be translated as "das Vorfeld der Zitadelle" or even better "das Glacis der Zitadelle". Don't be surprised, if most Germans will stare at you, when you use the word Glacis, though. They'll think that you're a tourist guide, historian or at least someone who regularly solves crossword puzzles.