Name
Student ID #
Class Section
Instructor
Math 1210
Fall 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. I want to warn you there could be a few typos.
2. Integrate the following problems
R
(a) f (g(x))g (x)dx
0
0
(b)
R xcos(x )dx
(c)
R cos(x)sin(sin(x))dx
(d)
R x [x + 5] sin((x + 5) )dx
(e)
R x + x + 4dx
2
2
3
8
3
9
3
1
3. These problems involve the second fundamental theorem of calculus.
R
(a) (2x + 2)(2x + 4x + 99) dx
1
2
95
0
(b)
R
(c)
R
( )[
0
( )] dx
cos x sin x
1
1
2
sin3 (x)cos(sin(x))
dx
1+x2
2
4. These problems are on the rst fundamental theorem of calculus. (you
must show all your work and be explicit as to how you are using the
rst fundamental theorem of calculus.)
(a) Find
dG
dx
Where G(x) =
R
(b) Find
dG
dx
Where G(x) =
R
x2
sin(x) cos sin x dx
(
(x2 +x+1)2
x
3
( ))
x2 dx
5. Estimate the area under the curve y = x from x = 0 to x = 1 with
four rectangles using the left, right or midpoint rule. It may help to
draw the picture.
3
4
6. This problem will come from section 3.4 rewrite your solutions to your
homework set from this section.
7. Use the denition of integral to nd
5
R
1
0
x2 dx
8. These problems are on dierential equations. Either solve the separable
dierential equation or show that y is a solution to the dierential
equation.
(a) Show that y = x + cos(x) is a solution to the dierential equation
xy=sp00 + y x = 1
(b) Solve
dy
dx
= yx3
(c) Solve
dy
dx
= x y given y = 1 when x = 0
2
3
6
9. Find the area bounded by y = x and y = x 12 .
2
7
10. Find the exact number the following sums equal.
P
(a) Nk (1=5)k (derive your result or if you use the formula derive
the formula).
=0
(b)
P
N
2
k=0 k
8
11. Find the local 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) f (x) = x2x2
+1
(b) f (x) = x
3
3x
(c) f (x) = x + 4x + 4
2
9