Name
Student ID #
Class Section
Instructor
Math 1210
Spring 2007
EXAM
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. Integrate the following problems
R ( ( )) ( )
(a)
f
0
g x
g
0
x dx
(b)
R
xcos x
(c)
R
cos x sin sin x
(d)
R
2
3
x x
(e)
R
x
( )
( )
2
dx
(
[ + 5]
3
8
+ +4
x
( ))
dx
(( + 5) )
sin
3
x
9
dx
dx
1
2. These problems involve the second fundamental theorem of calculus.
R
(a)
1
0
xdx
(b)
R
(c)
R (2
0
1
0
( )[
cos x
x
( )]
sin x
2
dx
+ 2)(2 + 4 + 99)
2
x
x
95
2
dx
3. These problems are on the rst fundamental theorem of calculus.
(a) Find
dG
dx
Where ( ) =
R
(b) Find
dG
dx
Where ( ) =
R
(c) Find
dG
dx
Where ( ) =
R
G x
G x
G x
x2
sin(x) cos
( )
(x2 +x+1)2
x
x dx
2
x dx
x2
cos(x2 ) tan
3
( )
x dx
4. Estimate the area under the curve = from = 0 to = 1 with
four rectangles. It may help to draw the picture.
P
5. nd the exact number that the following sum equals Nk (1 5)k
6. These problems are on dierential equations. Either solve the separable
dierential equation or show that is a solution to the dierential
equation.
(a) Show that =
( )+
( ) is a solution to the dierential
equation
+ =0
y
3
x
x
x
=0
y
y
y=sp
(b) Solve
dy
dx
= yx3
(c) Solve
dy
dx
=
k1 cos x
00
k2 sin x
y
2 3
x y
4
=
7. These problems are on the mean value thoerem for derivatives.
(a) State and prove the mean value theorem for derivatives. (It is less
likely that this question will be on the exam)
(b) Let ( ) = nd all values c so that the statement of the mean
value theorem for derivatives holds on [ 1 1]
f x
3
x
;
5
8. This question is on section 3.4 and will come from your homework. In
particular 14, 23, 29,50.
9. Find the maxium and minium value of each of the following functions.
Us the rst or second derivative test to show it is a max or a min.
(a) ( ) = x2x2
f x
+1
(b) ( ) =
f x
3
x
3
(c) ( ) =
4
x
2
(d) ( ) =
2
x
f x
f x
x
x
+ 4 + 4 on [ 10 10]
x
;
6