NAME: _____________________________________________
Math 2015 Test 3
Fall, 2007
Instructions: On your opscan sheet, enter your name, student ID number and the form designation (A). Fill in
beneath your ID number and form letter.
Circle the answers to the fifteen multiple choice questions if no opscan is provided.
Answer the 15 multiple choice questions using a #2 pencil on rows 1-15 of the opscan sheet.
Answer the free response in the space provided showing all work to receive full credit.
Write your name at the top of the page, and sign the pledge.
I will neither give nor receive unauthorized assistance on this exam.
Signature: ____________________________________
Free Response:
1.(12 pts) Consider z = 2x 2 !on!0 ! x ! 2 .
(a) Set up ONLY the integral to find the volume resulting from rotating the region between the curve and the
z-axis around the z-axis.
(b) Include a labeled picture. Label and number both axes and show the resulting “solid” figure.
2
" z%
V = ( !$
' dz =
0
# 2&
8
z
! dz
0
2
(
8
(c) Set up ONLY the integral to find the volume resulting from rotating the region between the curve and the
x-axis around the x-axis.
(d) Include a labeled picture. Label and number both axes and show the resulting “solid” figure.
V=
"
2
0
! (2x 2 )2 dx =
"
2
0
4! x 4 dx
2.(10 pts) Let p(t) = 0.1e!.1t be the density function for the waiting time at a subway stop, with t in minutes,
0 ! t ! 60 and 0 elsewhere.
(a) Set up ONLY the integral to calculate the median of this distribution.
"
Median:
M
0
0.1e!0.1t dt = 0.5
(b) Set up ONLY the integral to calculate the mean of this distribution.
Mean =
"
60
0
t(0.1e!0.1t )dt =
"
60
0
0.1te!0.1t dt
3.(9 pts) Derive, using calculus, the volume of a unit ball. A unit ball is a sphere with radius 1 unit.
Hint: Begin with a hemisphere.
x2 + z2 = 1 ! x = 1 " z2 = r
1
V = 2$ #
0
!!!!= 2#
(
1" z
2
)
2
1
%
z3 (
1.
+
dz = 2# $ 1 " z dz = 2# ' z " * = 2# - 1 " 0
0
,
3 )0
3/
&
1
2
2 4
= # !!units 3
3 3
4.(9 pts) Suppose p(x) is the density function given below. Graph the corresponding cumulative distribution
function. Find the median of this distribution using either graph.
Area must be divided in half.
This happens at x = 1.5.
Draw line off the y-axis at 0.5 to the curve.
Draw straight down to the x-axis. It will be 1.5.
BONUS
3 Bonus points: Find the volume of the solid generated by rotating the area between
y = x!!and!!y = x 2 !!for!!0 ! x ! 1!!about!the!x " axis .
[Hint: First draw a picture. What is the volume of the section that has been “removed”?]
V =!
( " (x) dx # " (x ) dx ) = 215! $ 0.419
1
0
2
1
2 2
0
2 Bonus points: Suppose a normal distribution is given by the density function p(x) =
What are the mean and standard deviation of the distribution?
Mean = 15; standard deviation = 1.
1
2!
e
"( x "15)2
2
.
MULTIPLE CHOICE: (4 pts each) Answers: 1, 4, 3, 2, 2, 4, 2, 3, 2, 3, 2, 4, 1, 3, 2
1. Find the volume resulting from rotating the region bounded by e x ,!0 ! x ! 3 , and the x-axis about the x-axis.
!
! 6
(1) (e6 " 1)
(2)
(3) ! (e6 " 1)
(4) ! e3
e
2
2
2. A solid rises from z = 0!to!z = 4 . Cross-sections perpendicular to the z-axis are rectangles with width 3 and
length 4z, at position z on the axis. Which of the following integrals will give the volume of the solid?
(1)
"
4
0
144! dz
(2)
!
4
0
144z 2 dz
(3)
"
4
0
12! z 2 dz
(4)
!
4
0
12z dz
3. For a complicated figure, we have determined that the cross-sectional area at height z can be found by a
function h(z) . Which integral will give the volume of this solid between z = 3!!and!!z = 7 ?
7 1
7
7
7
(1) " ! h(z)dz
(2) " ! (h(z))2 dz
(3) ! h(z)dz
(4) ! (g(z))2 dz
3 4
3
3
3
4. Given: A density function: p(t) =
3
! t !!!on!0 " t " 2 and 0 otherwise.
2
Find the mean of this population.
(1) 0
(2) 0.3
(3) 0.5
(4) 1
5. The radius of a newly planted tree, in inches, is measured every 12 inches up the tree to it’s total height of
60 inches. Using methods we used in class estimate the volume of the tree in cubic inches.
Height
Radius
(1) 444.85
0
6
12
4
24 36
2.5 1.3
48
0.8
60
0.4
(2) 1608.24
(3) 1516.76
6. Suppose that the cumulative distribution for a variable is P(x) =
interval the density function p(x) =
x4
(1)
32
(2) 0
x2
(3)
8
(4) 511.92
x3
!!for!!!0 ! x ! 2 . Then on the same
8
(4)
3 2
x
8
7. Suppose that x measures the time in hours, it takes for a student to complete an exam. All students are done
within 2 hours. The density function for x is given below. Compute the median for this distribution.
! x3
# !!!!!if!0 < x < 2
p(x) = " 4
#0!!!!!!!otherwise
$
(1) 1.6
(2) 1.6 8
(3) 0.0625
(4) 1
In problems 8, 9, 10 and 11 assume p(x) is the density function
"1
$ ,!!!if !1 ! x ! 5
p(x) = # 4
$%0,!!!if !x < 1!!or!!x > 5
Let P(x) represent the corresponding Cumulative Distribution function.
8. Find the fraction of the population that is more than 2 units.
3
2
3
(1)
(2)
(3)
2
3
4
(4)
1
2
9. For the density function given above, what is the cumulative distribution function P(x) for values of x
between 1 and 5? P(x) =
1
1
1
1
(1) x
(2) x !
(3) x 2
(4) 1
4
4
4
8
10. For the same cumulative distribution function P(x) discussed in the previous problem, P(7) =
3
(1) 0
(2) 2
(3) 1
(4)
2
11. Choose the correct pair of graphs for p(x)!and!P(x) described above.
(1)
(2)
(3)
(4)
12. Suppose p(x) is the density function graphed beside, where
p(x)!is!positive!on![0,!5] and p(x) = 0!for!x!outside!of![0,!5] .
What is the height h?
(1) 1
(3)
1
5
1
2
2
(4)
5
(2)
13. Given p(x) a probability density function. Which of the following would give the probability that
0 ! x ! 4?
4
1
(1) ! p(x)dx
(2) p(4) ! p(0)
(3)
(4) p(4)
p(4)2
0
2
14. Suppose p(x) is a density function for the distance in miles that an employee at a company travels to work.
Which of the following MUST be a correct interpretation of
!
30
20
p(x)dx = 0.5 ?
(1) The mean distance traveled by employees is between 20 and 30 miles.
(2) The median distance traveled is 30 miles.
(3) Half the employees travel between 20 and 30 miles to get to work.
(4) Half the employees travel more than 30 miles and half travel less than 20 miles to get to work.
15. Suppose we measure how late the Emporium Shuttle bus to the Math Emporium is, and come up with a
Cumulative Distribution function for the number of minutes the bus is late. The table is given:
Minutes Late
Percent of busses less late
0
15%
2
38%
4
50%
6
61%
8
75%
What percent of the busses are between 6 and 12 minutes late?
(1) 61%
(2) 39%
(3) 100%
(4) 50%
10
91%
12
100%