Solutions to Math 41 Exam 1 October 20,
2011
1. (16 points) Find each of the following limits, with justication. If the limit does not exist,
explain why. If there is an innite limit, then explain whether it is or .
( x + x)2
(a) lim
x 1 + x x
Solution 1
Part 1: Integration problems from 2004-05 exams 1. Find each of the following. (a) (b)
2
4t4 t1 +
t2
2 +1
dt x2 + 1 3x dx
sec x tan x sin x + x2 ex dx
0 1
3
(a ) (b )
0
x2 e3x dx
2. (a) Let f (x) = ex . On the graph of f pictured below, draw the approxima
Math 41 Fall 2005 Practice Final Exam
1. Evaluate each of the following limits. If there is an innite limit, then state whether the limit is or (ln x)2 x x t (b) lim 2te (a) lim
t0
(c) lim +
x0
cos x x
2. Dierentiate each of the following functions (a) y
1. Find the absolute maximum and minimum values of the function h(x) = x3 3x2 9x + 19 on the interval [1, 4]. Show all reasoning. 3 2. Consider the function g (x) = x 25 x2 . (a) Find all critical numbers of g . (b) For each critical number c, determine w
Solutions to Math 41 Final Exam December 10, 2012
1. (10 points) Find each of the following limits, with justication. If there is an innite limit, then explain
whether it is or .
x
ln(t + 1) dt
(a) lim 0
x 0
x2
(5 points) We are dealing with a 0 indetermi
Solutions to Math 41 Final Exam December 6, 2010
1. (10 points) Find each of the following limits, with justication. If there is an innite limit, then explain
whether it is or .
(a)
lim x3 +
x
x6 + x3
(5 points) Note that the expression makes sense for x
Solutions to Math 41 Second Exam November 4, 2010
1. (13 points) Dierentiate, using the method of your choice.
(a) p(t) = ln(sec t + tan t) + log2 (2 + t)
(4 points) Using the rule for the derivative of a sum,
p (t) =
d
d
d
(ln(sec t + tan t) + log2 (2 +
Solutions to Math 41 First Exam October 12, 2010
1. (13 points) Find each of the following limits, with justication. If the limit does not exist, explain
why. If there is an innite limit, then explain whether it is or .
x5
ex
(a) lim
2
x
x 6x + 5
x5
(4 po
Solutions to Math 41 First Exam October 18, 2012
1. (12 points) Find each of the following limits, with justication. If the limit does not exist, explain
why. If there is an innite limit, then explain whether it is or .
2
1
(a) lim
21
x 1
x
x1
(4 points)
Solutions to Math 41 Second Exam November 8, 2012
1. (14 points) Dierentiate, using the method of your choice.
(a) f (z ) = sec(2z ) ln(sin2 z )
(5 points) We know that
d
(sec(2z ) = 2 sec(2z ) tan(2z )
dz
and using the Chain Rule we obtain
d
1
(ln(sin2 z
Solutions to Math 41 Final Exam December 12, 2011
1. (10 points) Find each of the following limits, with justication. If there is an innite limit, then explain
whether it is or .
(a) lim
x
1
x
1/ ln(x)
(5 points) First we compute the limit:
lim ln
x
1
x
1
Solutions to Math 41 Exam 2 November 10,
2011
1. (12 points) Find each of the following limits, with justication. If the limit does not exist,
explain why. If there is an innite limit, then explain whether it is or .
ln(x /2)
(a) (4 marks) lim +
x/2
tan(x
Math 41: Calculus Final Exam December 11, 2006
Name : Section Leader (Circle one) : Section Time (Circle one): Chang 11:00 Ivanov 1:15 Mathews Requeijo Segerman
This is a closed-book, closed-notes exam. No calculators or other electronic aids will be per
Math 41: Calculus Second Exam November 16, 2006
Name : Section Leader (Circle one) : Section Time (Circle one): Chang 11:00 Ivanov 1:15 Mathews Requeijo Segerman
This is a closed-book, closed-notes exam. No calculators or other electronic aids will be pe