Name
Student ID #
Class Section
Instructor
Math 1210
Spring 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. these problems involve the detion of limit.
(a) State the detion of limitx!cf (x) .
(b) Use the denition that you gave in (a) to show that limitx! x
4=0
2
2
(c) Use the denition that you gave in (a) to show that limitx!c ax +
b = ac + b
1
(a) let F (x) and G(x) be functions such that 0 F (x) G(x) for all
x near c except possibly at c . Prove that if limitx!cG(x) = 0 then
limitx!c F (x) = 0. (To get full credit you must use the denition
given in 1 (a))
2. Compute six of the seven limits below. (If you do not clearly indicate
which three you want graded I reserve the right to give you a zero
score.)
(a) limitx! x2x2 x
+
1
2
1
(b) limitx! 2x(x
2
1
1) 12
(c) limitx!
x2 1
1 x2 +1
(d) limitx!
1
x2 +x
x2 1
(e) limitx!
1
(f) limitx!
sin(x)2
x2
0
0
cos(x)
sin(x)
(g) limitx!= xtan(x)
4
2
3. Compute six of the seven limits below. (If you do not clearly indicate
which three you want graded I reserve the right to give you a zero
score.)
(a) limitx!1 x2x2 x
+
2
1
(b) limitx!
4x
2
4
(c) limitx!
x2 1
1 x2 +1
(d) limitx!1 x (x
2
(d) limitx!1 xx36
(f) limitx!
0
1) 21
+x
+1
sin(x)2
x2
(g) limitx!1 cosx2x
( )2
3
4. These problems involve continunity.
(a) State the three criterion for a function to be continuous at a point
c.
(b) Where is the following function continuous f (x) = xx2 . What
types of discontinunity does the function have and where?
3
9
(c) How would you dene the function x cos(1=x) so that it is continuous at x = 0 ?
4
(d) Show that the function jxx j has a non removable discontinunity
at x=5.
5
5
4
5. These problems involve the intermediate value theorem.
(a) State the the intermediate value theorem.
(b) Use the intermediate value theorem to show that the equation
100x 5x + 1 + cos(x) = 0 has a real solution.
9
5
(c) Prove that if f (x) is continuous on [0; 1] and satisies 0 f (x) 1
then f (x) has a xed point (ie there is a number c in [0; 1] such
that f (c) = c.) Hint consider the function g(x) = x f (x) and
apply the intermediate value theorem.
5
6. Suppose that an object is droped from a cli and it's position function
is given by p(t) = 16t + 500 nd the velocity of the object.
2
6