I did this in many lower-level courses over a period of many years. I've also written extensively about what I did and why, but most of this is in Math Forum discussion groups $\ldots$ (beginning of the first of several comments I made here about 2 weeks ago)
I've finally managed to locate the text of two of the (definitely more than two) posts I mentioned in comments here almost 2 weeks ago. I don't know when they were made (very roughly, probably between 2007 and 2013), nor in what order they were made (the order I've used is simply what seems reasonable to me for here), or even whether they were made within a few weeks/months apart or several years apart. I also don't know where I originally had paragraph separations, but to avoid having huge walls of text, I've put in some paragraph separations here. Finally, nothing has been deleted (except the first sentence or two of the 1st post below) and nothing has been changed, but I have added a few additional words between some square brackets $[\ldots]$ in the first of the posts below.
$[\ldots]$ However, I thought I'd jump in now about a policy I used to have (I no longer teach [since early May 2005]). This policy evolved throughout the 1980s into final form by the mid 1990s. It worked quite well for me, at least when I was able to use it (almost always, the only exception I can think of right now were some courses in my last two years of teaching [2003-2004 and 2004-2005]), but like many teaching methods, what works well for some people is not going to work well for some other people.
Short quizzes -- I gave lots of these, each on a 10-point scale. The sum of a student's top 10 quiz grades becomes one of the student's 4 (or 5) major test grades. I usually guaranteed a minimum of 12 quizzes for the semester, but I often tried to work in 1 or 2 more [and for probably about 30% of the courses, I gave 15 or more quizzes]. The nice thing about this policy is that students WANT you to give more quizzes, especially in the final weeks of the semester when things are usually the most difficult. Sometimes near the end of the semester I'd give an "exam-review quiz", which would be a quiz similar to one already given, usually on a topic they had trouble with and a topic that I think is very important for the course. By the way, depending on the difficulty of the material, the classroom climate, and other things, short quizzes were sometimes "you can use your homework but not your book" quizzes, "you can use your homework and class notes but not your book" quizzes, "you can use you homework, class notes, and book" quizzes, "you can work with a partner" quizzes (who I choose, who they choose, or names were paired randomly), the last type combined with one of the earlier types, and other possibilities I experimented with. My best successes for these alternate methods occurred in calculus 2 classes, and next with calculus 1 classes. In higher level classes it didn't work as well, and in lower level classes it didn't work as well, for whatever that's worth.
Major tests -- 50 to 110 minute tests (depending on whether a MWF class or a TuTh class). As much as possible, the test would be made up of problems very similar to problems already put on quizzes, but in practice I was typically only able to get about 50% of the test problems in this way.
Final Exam -- A major deal. However, those students who had done very well in the class up to that point (mid to high A students, maybe low A students, depending on how the overall grades in the class were -- usually this would be about 3 to 7 students in a class of 30-40 students) got to take a scaled back final exam that consisted only of "recent material" (material covered since the last major test of the semester), about twice the length of a short quiz, and if they did satisfactory on this "recent material exam", they were guaranteed an 'A'. I think I didn't give such a student an 'A' only once in all my teaching. (The student only half-way knew the stuff, and I said they could take the full final exam and probably -- unless they really messed up on the stuff they had aced thus far -- earn an 'A', or take an 'A-'. He/she took the A-, being afraid of messing up on the final since they didn't study for it, plus I think this person had several other final exams he/she was really worried about and needed to leave to study for them.)
The Final Exam also replaced the student's lowest test score (which could be their top 10 quizzes, although this may have never actually happened -- typically their "quiz test grade" was the highest test grade), unless the Final Exam score (as a percent) was lower than each of their major test scores (in which case the final exam still counted a lot, usually about 30% of the final grade, but the damage was restricted to just that amount). By the way, probably about 75% of the students did worse on the Final Exam than their test average up to that point, with about half of those 75% dropping a third of a letter grade or more (using +/- letter grades) and about half of those 75% winding up having their Final Exam grade not differing enough to change the pre-exam average (when rounded to a plus/minus letter grade).
Finally, I had a "last chance" option for those true late bloomers, who in reality are generally quite rare unless you're teaching a lot of non-traditional adult students. Any student getting at least a 'C' on their Final Exam was guaranteed to pass the course, regardless of what their final course average was, and any student getting an 'A' on their Final Exam was guaranteed at least a 'C' in the course (at least, since typically anyone making an 'A' on their final, even when doing D work prior to this, wound up with a course average of higher than 'C' anyway). My guess is that I invoked the "last chance" option about once every 2 to 3 years (perhaps about once out of every 400 to 600 students). I also made it very clear at the beginning of the semester how rarely this happened, but I did say it could happen. Actually, what happened far more often (maybe as much as one student out of 50 or 60 students) was that someone was failing and got a 'C' on the Final Exam, or someone had a 'D' and got an 'A' on the Final Exam (the former was much more likely), and the weight of the Final Exam on their final course average plus the Final Exam replacing their lowest test score was more than enough to push them to a passing grade or a 'C' grade, without my having to invoke the "last chance option".
For what it's worth, I've always found final/mid-term exams fairly easy to write. My policy was to never include anything on a final exam whose "type" wasn't represented by a question on a prior test or short quiz, except for whatever new material we covered between the last test/quiz and the exam. So what I did was to get out copies of all the tests and quizzes (already done, since we quickly looked over these in class the day or two spent reviewing for the exam, with me saying "definitely on the exam", "might be on the exam", and "not on the exam" for each of the question types), along with my notes on what I told the students about the exam, and then I'd start making my question selections for the exam.
Nowadays we don't have to hand-write or old-style typewriter-type exams, so it's just a matter of copying-and-pasting the selected questions onto a "final exam test document", changing numbers or equations or graphs or whatever. Also, I often found that ideas for tweaking already asked questions came to me while I was reviewing in class, during the extra "Friday afternoon review sessions" (probably more of a college thing than a high school thing), working with struggling students in my office, etc. Thus, when it came time to actually write the exam, I usually already knew which question types I wanted to ask and I also had many specifics for them already jotted down in my notes.
On the other hand, GRADING the exams was always a major effort -- typically 6 to 10 hours per class of 30-40 students. That for me was where the real work was during exam periods, grading the exams, not writing the exams.
Incidentally, my experiences in many different settings (pre-algebra to graduate level, lowest performing HS in one state to a nationally known math/science academy in another state, small and medium size private colleges, medium and large size public universities) has been that students almost always perform less well on final exams than on previous tests given throughout the semester/year. There are several factors behind this, such as students being burned out by the time exams are given (everyone's last regular test, last term paper, etc. just occurred, although this may not be an issue in your [= original OP I was replying to] case), all the various exams the students have to take are given within a few days, and the exam is over several months of material instead of 3 to 4 weeks of material. Thus, even being as explicit as I was about what would be on the exam did not inflate student grades. In fact, even with this I found that exam grades, on average, were lower than major test and short quiz grades.
The reason I've gone into such detail is I'm wondering whether others do something like this. I'm guessing you (Kerri) don't, otherwise you wouldn't be so concerned with writing the exams. Also, it seems to me that by using someone else's exam, especially for a non-AP calculus course that is quite likely different in many small ways from teacher to teacher and school to school, one runs the risk of over-emphasis and under-emphasis on the topics actually taught.