Problem of the Week: That’s a good exam!
Editor’s Note: The Department of Mathematics at Gettysburg College hosts a problem of the week challenge to determine each semester’s Paul Mugabi problem-solving award recipient(s). Each week’s entries are scored by a panel of faculty judges, and winner(s) from each week will receive a Problem Of the Week (P.O.W.) button. The Gettysburgian is not involved in or responsible for accepting or evaluating students’ submissions to this contest.
The contest is open to all Gettysburg College students. Up to three people may work together on a submission. Make sure your name is on your submission and that any sources are properly cited! Send solutions to firstname.lastname@example.org or put solutions in the marked envelope in the hallway outside Glatfelter 215. This problem was posted on Friday, September 28 and solutions are due on Friday October 5 by 5:00 p.m.
THE PROBLEM: That’s a good exam!
On her most recent calculus exam, Professor Toughgrade chose two particular functions f(x) and g(x), and asked students to find the derivative of the product f(x)g(x). One of Professor Toughgrade’s students had not studied very well and used the following (very wrong) formula:
(f(x)g(x))′ = f′(x)g′(x).
Nevertheless, even using this wrong formula, the student got the right answer for the derivative of f(x)g(x)!
On Professor Toughgrade’s exam, the function g(x) was g(x) = xex – ex, and the function f(x) satisfied f(0) = 3. What was f(x)?