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 derivatives
(a) State the product rule.
( ) ( + 1)
(b) Dierentiate ( ) =
cos x
(c) Dierentiate ( ) =
cos x
f x
f x
2
x
( ) ( + 1) 2
x
( )
sin x
(d) State the chain rule.
(e) Dierentiate ( ) =
f x
(( + 1) cos
2
x
( ))
sin x
(f) Dierentiate ( ) = sin(cos(sin ( )) ( + 1) sin( ))
2
f x
x
(g) State the quotient rule.
(h) Dierentiate ( ) =
f x
x 2 )
cos(sin(x2 +1)
cos(x)
1
2
x
x
2. These problems involve implicit derivatives. Find
equations.
(a) tan(cos(sin( ) ) ( + 1) + = 0
x y
(b) ( ) +
xy
(c)
3
x
5
3
y
4
4
x
( ) + ( + 1) = 0
sec xy
+ +
y
3
2
x y
+
x y
2
y x
6
5
=( + )
2
x
y
2
2 3
dy
dx
for the following
3. This question will be on related rates and the test problem will be one
the homework problems which I assigned. Also note that we discussed
the solutions to all of these problems in class.
4. This problem will be on dierentials and approximations. Dene what
a dierential is, Study example 3 on page 144, problem 26, 23,17. The
test problem will be extreemly similar to one of two of these problems.
5. Find the maxium and minium value of each of the following functions
if they exist. If they exist you must state why it exists and if it does
not exist you must state why it does not exist.
(a) ( ) = x2x2 x on [ 1 1]
+
2
+1
f x
;
(b) ( ) = x on [0 1]
f x
2
4
(c) ( ) =
cos x
(d) ( ) =
2
x
f x
f x
;
( ) on [ 2 1]
;
+ 4 + 4 on [ 10 10]
x
;
(e) ( ) = x on [ 1 1]
f x
1
;
3
6. Find where the functions are increasing, decreasing, concave up and
concave down.
(a) ( ) = + 3
12
f x
(b) ( ) =
f x
3
x
x
2
3
x
+ 12
x
4
df
df
exists and is continuous on the interval I, and if dx
is
7. Prove that if dx
nonzero for all points in I then either ( ) is increasing or decreasing
on I. (Hint what does the intermediate value theorem say?)
x
f x
5