Basic and Pythagorean Identities
ESSEX): am (I) smix) = CSCICJE)
1 cos x = 1
seclxj .303ij I: ) sec[x)
:03 x = 1 =Cslx) 3mg 2: = 1 : 51?:[xj
[ ) ERICK) EMU) l: :l COEICJE) cosix)
Notice how a "co-(something? trig ratio is always the recip
Recall: First Fundamental Theorem of Calculus (FTC 1)
If f is continuous and F = f , then
b
f (x)dx = F (b) F (a)
a
We can also write that as
b
x=b
f (x)dx
f (x)dx =
x=a
a
Do all continuous functions have antiderivatives? Yes. However.
What about a func
MATH 1A - PROOF OF THE FUNDAMENTAL THEOREM OF CALCULUS
PEYAM RYAN TABRIZIAN
1. T HE F UNDAMENTAL T HEOREM OF C ALCULUS
Theorem 1 (Fundamental Theorem of Calculus - Part I). If f is continuous on [a, b], then
the function g defined by:
Z x
g(x) =
f (t)dt
a
Riemann Sums and definite integrals
(1). Riemann Sums For a function f defined on [a,b], a partition P of [a,b] into a collection of
subintervals
[x0,x1],[x1,x2],[xn1,xn],
and for each i = 1,2,n, a point xi in [xi1,xi], the sum
n
n
Xf(xi)(xi xi1) = Xf(xi)
Riemann Sums and definite integrals
(1). Riemann Sums For a function f defined on [a, b], a partition P of [a, b] into a
collection of subintervals
[x0 , x1 ], [x1 , x2 ], , [xn1 , xn ],
and for each i = 1, 2, , n, a point xi in [xi1 , xi ], the sum
n
X
f
ASSUMPTION OF RISK AND RELEASE WAIVER
By signing the Assumption of Risk and Release, the individual named below wishes to
participate in the Event/Activity described below and recognizes that there are risks of
damage or injury arising from this event or
After completing our evaluation of the 1st circuit we compared our recorded value of voltage V AD
(5.123v) with the sum of voltages V AC (1.0241v) and VCD (4.099v). The sums of these two voltages ends up
to be 5.1231v which is pretty much exactly the same
~ Find the dot product and cross product of two vectors.
~ Find the length (magnitude) of a vector.
~ Find the projection of one vector onto another vector.
Projvu=(
)v
~ Find the angle between two vectors (using the dot product).
~ Find the equation of a
Nate Jackson
10/15/2015
In Dudley Randells poem Ballad of Birmingham, the context of the poem plays a pivotal role in
delivering its heavy message. The poem was written in 1969 but it is set in 1963. The fact that the reader
already knows about the infamo
Bhopal
M.J. Peterson
Bhopal Plant Disaster
Triangle
David Von Drehle
Triangle: The fire that
changed America
SS Sultana
AJ Carr
SS Sultana
Iglesia de la Compania
Launceston Examiner
Panics in Public Places
The Tay Bridge disaster
Unknown author
The Tay Br
In this lab we modeled a tank draining system by using data collected from a two tank draining
system that resembles schematic 1 shown below. We began our experiment by measuring the width and
depth of both tanks along with the diameter of the tanks orifi
Student Solutions Manual
to accompany
Engineering Fluid Mechanics, 7th Edition
Clayton T. Crowe and Donald F. Elger
October 1, 2001
ii
prgmea.com
Contents
1 Introduction
1
2 Fluid Properties
5
3 Fluid Statics
9
4 Fluids in Motion
21
5 Pressure Variation i
Angular acceleration and moment of inertia
This experiment studies the forces applied to a body rotating around a specific axis and its
resulting angular acceleration. The angular acceleration will be found as a function of the force applied
at the radius
In this lab we modeled a tank draining system by using data collected from a two tank draining
system that resembles schematic 1 shown below. We began our experiment by measuring the width and
depth of both tanks along with the diameter of the tanks orifi
NAME: Sela/nib m CM#
E5203-01 Midterm Exam #1 Version B 9/30/2014
50 Minutes Duration.
100 Points Total (Closed book, closed notes, closed MAPLE).
You may use your calculator.
Show your work clearly, and fill in the answer blank, if one is given.
Remember
_SsJLL+f9rL
NAME: CM#
E5203-01,02 Midterm Exam #1 12/19/2013
50 Minutes Duration.
100 Points Total (Closed book, closed notes, closed MAPLE).
You may use your calculator.
Show your work clearly, and ll in the answer blank, if one is given.
Remember, for f
NAME: so lmlwe m CM#
E5203-01 Midterm Exam #1 3/27/2014
50 Minutes Duration.
100 Points Total (Closed book, closed notes, closed MAPLE).
You may use your calculator.
Show your work clearly, and [ill in the answer blank. if one is given.
Remember, for full
NAME:_ CM#_
ES203-01 Midterm Exam #1 Version B 9/30/2014
50 Minutes Duration.
100 Points Total (Closed book, closed notes, closed MAPLE).
You may use your calculator.
Show your work clearly, and fill in the answer blank, if one is given.
Remember, for ful
CALCULUS II
MA112
PART 1: CLOSED BOOKS
FINAL EXAM
FEBRUARY 22, 2012
NAME:
INSTRUCTIONS:
Read each problem CAREFULLY! Please circle your answer. You must show all of your work.
In particular, your answers should retain square roots, fractions, etc. The use
Problem 3.63 Apply the source-superposition method to the circuit in Fig. P3.63
to determine:
(a) Vx , the component of Vx due to the 1-A current source alone.
(b) Vx , the component of Vx due to the 10-V voltage source alone.
(c) Vx , the component of Vx
Problem 3.60 Determine the current Ix in the circuit of Fig. P3.60 by applying the
source-superposition method. Call Ix the component of Ix due to the voltage source
the component due to the current source alone. Show that I = I + I
alone, and Ix
x
x
x
Problem 3.41 Apply the supermesh technique to nd Vx in the circuit of Fig. P3.41.
2 k
6V
+
I3
6 k
_
_
1 k
+
_
5 k
Vx
I1
supermesh
I2
4 k
2 mA
Figure P3.41: Circuit for Problem 3.41.
Solution:
Supermesh:
Mesh 3:
Auxiliary:
103 I1 6 + 6 103 (I2 I3 ) + 4 103
Problem 3.31 Apply mesh analysis to determine the amount of power supplied by
the voltage source in Fig. P3.31.
4A
I3
3
6
2
I1
+
2
_
4
I2
48 V
Figure P3.31: Circuit for Problem 3.31.
Solution:
Mesh 1:
2I1 + 3(I1 I3 ) + 2(I1 I2 ) + 48 = 0
Mesh 2:
48 + 2(I
Problem 3.33 Use the supermesh concept to solve for Ix in the circuit of Fig. P3.33.
12
Ix
supermesh
I4
4A
2A
I1
6
I2
6
I3
3A
Figure P3.33: Circuit for Problem 3.33.
Solution:
Mesh 1:
Supermesh:
Mesh 3:
Auxiliary:
I1 = 2 A.
6(I2 I1 ) + 12I4 + 6(I2 I3 ) =
Problem 3.26 Apply mesh analysis to nd the mesh currents in the circuit of
Fig. P3.26. Use the information to determine the voltage V .
2
+
_
2
4
3
I1
I2
+
16 V
V
_ 12 V
Figure P3.26: Circuit for Problem 3.26.
Solution: Application of KVL to the two loops
Problem 3.38 Repeat Problem 3.37 after replacing the 5- resistor in Fig. P3.37
with a short circuit.
Ix
+
+
+
12.3 V _
10
I1
4
I4
20
I2
2Ix
I3
2
Figure P3.38: Circuit for Problems 3.37 and 3.38.
Solution:
Mesh 1:
Supermesh 2/3:
Mesh 4:
Auxiliary 1:
Auxi
Problem 3.25
Fig. P3.25.
Apply nodal analysis to determine Va , Vb , and Vc in the circuit of
15
2.5
5
3A
+
_
50 V
Vc
10
_
Va
10 V
_
+
Vb
25 V
+
7.5
5
Figure P3.25: Circuit of Problem 3.25.
Va 50 Va Vb Va Vc
+
+
=0
5
2.5
15
Vb 10 Vc
Vb Va
+3+
=0
2.5
1