Name: ____________________
Exam 4
1. Write the Riemann sum for an approximation of
endpoints.
!
2
2. Estimate the integral
# f(x)dx using the
"1
graph to the right. Use six intervals and the
right endpoints of the intervals. Draw in the
rectangles on the graph.
!
n
3. Find the value of the sum
# (3i + 2)(2i " 6) .
i=1
!
4. If g(x) =
!
"
x3
x
2t sintdt , find g"(x) .
!
"
7
2
sin xdx using n intervals and left
#1" x & 2
5. Evaluate ) %
( dx .
$ x '
!
#/3
6. Evaluate
$ cos(tan" )sin(tan" )sec
2
"d" .
0
!
7. Using Newton’s method and the initial approximation x1 = 1, find x3, the third approximation
to the root of the equation x3 + 2x – 4 = 0. Give your answer to four decimal places.
8. If 1200 cm2 of material is available to make a box with a square base and an open top, find
the largest possible volume of the box.
9. Without using the Fundamental Theorem of Calculus, explain why if f(x) ≥ m on an interval
[a,b], then
b
" f(x)dx # m(b $ a) .
a
!
10. Let g(x) =
#
x
"1
(1" 2t + 3t 2 )dt . Find the formula for g(x) so that g(b) =
any choice of b.
!
!
#
b
"2
(1" 2t + 3t 2 )dt for