Math 251 (Phillips) Final exam
Information and material covered
The final exam will be on Friday 23 May in class, using the whole period.
No early start
It will allow a standard size page (8.5 by 11 inches) of notes,
written on both sides,
but no calculators or other electronic devices.
It will be cumulative, but with emphasis on course material
since Midterm 2.
A course summary is here.
Besides a general summary, it has a condensed review of all the
methods we have seen of finding limits.
Limit problems on the final exam will not tell
you which method to use.
In particular, trying to use L'Hopital's Rule on limits not
of indeterminate form is a very expensive error, costing many points.
It can be avoided simply by remembering to check the conditions for
L'Hopital's Rule before trying to use it.
New topics since Midterm 2:
-
Concavity and inflection points (in Section 4.5 of the book).
(We stated this before Midterm 2.)
-
Equations of tangent lines and linear approximation.
(Linear approximation is in Section QQQ of the book.)
-
Derivatives as rates of change (Section 3.4 of the book).
-
Limits at infinity (Section 4.6 of the book).
-
L'Hopital's Rule (Section 4.8 of the book).
-
The Mean Value Theorem (Section 4.4 of the book).
We have already been using some of its consequences, such as the
fact that a function with positive derivative is increasing.
The exam will have at least one of each of the following kinds of
problems:
-
Optimization, maybe on a closed bounded interval, maybe not. Also
maybe just a request to set the problem up, without giving a full
solution.
-
Related rates.
-
Something about some of intervals of increase and decrease,
intervals of concavity, local minimums,
local maximums, and inflection points,
possibly with the derivative already given to you.
-
Some differentiation problems, definitely including some
requiring the chain rule, and very likely an implicit differentiation
problem.
-
Some limit problems, definitely including
at least one using L'Hopital's Rule,
and at least one fraction for which
L'Hopital's Rule gives the wrong answer.
-
Very likely something about derivatives as velocities or
other rates of change, including average velocity
or average rate of change.
-
Very likely something involving linear approximation.
Review session
The review session is Thursday 5 June 6--8:50 pm,
105 Fenton.
Sample Problems
There are seven primary sources for sample problems for the final exam:
-
Written homework.
These, and their solutions, are linked on the web pages for
weeks 1--10 of the course, accessible from
here.
-
WeBWorK
homework.
Answers to WeBWorK problems are available on WeBWorK
after the due date of each assignment.
-
Additional practice
WeBWorK
problem sets.
Again, answers are available on WeBWorK
after the "due date" of each assignment.
-
Quizzes.
These, and their solutions, are linked on the web pages for
weeks 1--10 of the course, accessible from
here.
-
In class worksheets.
Warning: The hints on the worksheets won't be provided on the midterm.
These, and their solutions, are linked on the web pages for
weeks 1--10 of the course, accessible from
here.
-
Sample final exam (pdf).
Solutions (pdf).
-
Worksheet
used at the review session.
Solutions
to the worksheet used at the review session.
Minor corrections made 8 June 2025: correction to the solution to
Problem 1 (near the end), and more details in the solution
to Problem 2c.
Changes are shown
in the difference file (pdf);
and the reverse difference file (pdf).
The difference file has
new text wavy underlined in blue and old text crossed out in red;
the reverse difference file reverses the color scheme.
More
information.
-
Everything posted for
Midterm 1
and Midterm 2,
including the real and retake real Midterm 1 and the real Midterm 2
themselves.
Of course, you should also know how to do all the
Midterm 0 problems.
Note, though, that the the final exam problems will be in format much
more like the problems in the first list above.
Real final exam
The real exam and its solutions will not be posted until after
the exam is administered.
Return to main Math 251 page.
This page maintained by
N. Christopher Phillips,
email.
Last significant change 8 June 2025.