Name
Student ID #
Class Section
Instructor
Math 1210
Fall 2007
EXAM 1
Dept. Use Only
Exam Scores
Problem Points Score
1.
20
2.
20
3.
20
4.
20
5.
20
TOTAL
Show all your work and make sure you justify all your
answers.
Math 1210
Exam
1. I want to warn you in advance that whenever I write my practice exams
there are usually a few typos (to keep you on your toes).
2. Compute the limit below.
(a) limx! x2 x x
+2
2
(b) limx!
2
8
2
x2 +5xcos(x)+1
x+2
(c) limx! 4 (x + x)tan(x)
2
(d) limx! xcot(2x)
0
(e) limx!
0
1
cos(x)
sin(x)
(f) limx! x sin( x ) (hint use the squeeze theorem)
0
2
1
1
3. Compute the six limits below.
(a) limx!1 x3x4 x2
99
+3
2
1
(b) limx!
0
sin(x)2
x2
(c) limx! 1
(Hint what is limx!
0
sin(x)
x
)
2
2x
4x +5
2
1) 21
(d) limx!1 x (x
2
(e) limx!1 x8x8 x
+5
+ +1
(f) Show that limx!
lower limits.
1
0 x
(g) Find limx! + x
3
does not exit by computing the upper and
1
3
2
4. These problems involve continunity.
(a) State the three criterion for a function to be continuous at a point
c.
(b) How would you dene the function f (x) = xx2 so that it is continuous at x = 1, and state why it has a discontinuity at x = 1.
1
1
3
(c) Where is the following function continuous f (x) = xx2 . What
types of discontinunities does the function have and where? (Hint
factor)
1
1
4
5. Do the following problems
(a) Let f (x) = x + x + 2, g(x) = x , h(x) = cos(5x). Find f (g(x),
g(f (x)), and h(f (g(x))).
1
+1
2
(b) state the limit denition of derivative. (see page 100).
5
(c) Use the limit denition of derivative to nd f (x) where f (x) =
x + 5.
0
2
6
6. Do the following problems.
(a) State the intermediate value theorem.
(b) Use the intermediate value theorem to show that the equation
99x x + xcos(x) + sin(x) + 9 = 0 has a real solution.
9
5
7