NOTE: The Spring 2008 final exam is identical to the Fall 2005 final exam.
There are slight differences in how the two exams are typeset,
but the problems are verbatim identical.
(In posting solutions, we simply took the Fall 2005 solutions
and replaced the cover page by the Spring 2008 cover page.
So the typesetting of the
Spring 2008 final exam
and the typesetting of the
Spring 2008 final exam with solutions
will differ slightly.)
Errata for Spring 2008 final exam (same as for the Fall 2005 final exam):
Problem 3 is ambiguous. I interpreted it to ask for upper and lower bounds on sin^2 on \R, which then yield upper and lower bounds on the integrand, which are then integrated from 1 to 2 to give upper and lower bounds on the integral. That coarse approximation yields answer B. A numerical integrator shows that the integral, to three decimals, is given by 3.754, so a highly refined approximation using Riemann sums would show that A, B or C could be correct.
Part (A) of Problem 6: Of course, if the function's domain is bigger than [-1,2], there is no reason that the absolute max of f (across its entire domain) would necessarily occur in [-1,2], or even necessarily exist at all. It would be clearer if, somewhere earlier in the problem, the author had stated that the domain of f is [-1,2] and that f is differentiable on the open interval (-1,2), instead of specifying differentiability on [-1,2].
To do Problem 10 as stated, one needs to consider rectangles whose sides are not necessarily parallel to the legs of the right triangle. It is true that there is a maximal-area inscribed rectangle two of whose sides are contained in the legs. However, proving this seems difficult, especially since there is also a maximal-area inscribed rectangle one of whose sides is contained in the hypotenuse.
Problem 20 refers to both the disk and washer methods, but it's the
washer method that's used.
NOTE for Problem 20: The same volume is computed, using the
shell method, in Problem 18 of the
Spring 2005 final exam with solutions.
Errata for Fall 2007 final exam:
In Part (C) of Problem 17, "Choose (a)" should be "Choose (A)".
Part (e) of Problem 18 asks us to sketch a graph "in the grid below", but no grid appears.
Errata for Spring 2007 final exam:
In Problem 2, the domain for the correct answer is: the intersection of (x>-1) and (x≠0). However, the domain for Answer (A) is: the intersection of (x≠-1) and (x≠0), which is slightly different. So Answer (A) is technically slightly incorrect, but is the best of the answers given.
In Problem 15, it would be clearer to say "the region below y=sqrt{5-x}, above y=sqrt{x+1}, to the right of the y-axis and to the left of x=2". These lines and curves split the plane into many regions, but this is the only one that is bounded.
In Problem 20, the instructions say, "Be sure to simplify your answer." I don't see any obvious simplifications to make.
Errata for Fall 2005 final exam (same as for the Spring 2008 final exam):
Problem 3 is ambiguous. I interpreted it to ask for upper and lower bounds on sin^2 on \R, which then yield upper and lower bounds on the integrand, which are then integrated from 1 to 2 to give upper and lower bounds on the integral. That coarse approximation yields answer B. A numerical integrator shows that the integral, to three decimals, is given by 3.754, so a highly refined approximation using Riemann sums would show that A, B or C could be correct.
Part (A) of Problem 6: Of course, if the function's domain is bigger than [-1,2], there is no reason that the absolute max of f (across its entire domain) would necessarily occur in [-1,2], or even necessarily exist at all. It would be clearer if, somewhere earlier in the problem, the author had stated that the domain of f is [-1,2] and that f is differentiable on the open interval (-1,2), instead of specifying differentiability on [-1,2].
To do Problem 10 as stated, one needs to consider rectangles whose sides are not necessarily parallel to the legs of the right triangle. It is true that there is a maximal-area inscribed rectangle two of whose sides are contained in the legs. However, proving this seems difficult, especially since there is also a maximal-area inscribed rectangle one of whose sides is contained in the hypotenuse.
Problem 20 refers to both the disk and washer methods, but it's the
washer method that's used.
NOTE for Problem 20: The same volume is computed, using the
shell method, in Problem 18 of the
Spring 2005 final exam with solutions.
NOTE for Problem 18: The same volume is computed, using the washer
method, in Problem 20 of the
Fall 2005 final exam with solutions
and in Problem 20 of the
Spring 2008 final exam with solutions.
Errata for Spring 2005 final exam:
In Problem 3, the correct answer is that f is
concave down on (-∞,-1] and on [1,∞),
and is therefore
concave down on (-∞,-1) and on (1,∞).
The set
{ x such that |x| > 1 }
is sometimes abbreviated
{ |x| > 1 }
and is equal to
the union of (-∞,-1) and (1,∞),
and so Answer B is the best answer.
However, f is not, in fact concave down on that union,
even though it IS concave down on each of the two intervals.
So, technically, none of the stated answers is correct.
In Problem 9, the limit as x ---> 3 of f(x) is -∞. In my opinion, that limit *does* exist, even though it's not finite. However, some might disagree.
In Problem 10, I assume that f(x) continues to decrease on x > 8, even though we can't see the graph much to the right of x = 8.
NOTE: The Fall 2004 final exam is identical to the Spring 2004 final exam except:
1. Problem 15 is different; and
2. the Spring 2004 final exam has a Problem 18, but the Fall 2004 final exam does not.
Errata for Spring 2003 final exam:
In Problem 9, the integral is missing a "dt".
The start of Problem 17 reads, "Find the following limit or derivatives", but Part (a) is an averaging problem, and so involves neither limits nor derivatives; instead, it involves integration.
In Problem 19, "dimension" should be "dimensions".
Errata for Fall 2001 final exam:
None noted yet.
Errata for Fall 2000 final exam:
None noted yet.