Math 227504 Quiz #2 (Sections 12.412.5) Instructor; Yunrong Zhu
Name:
1. [th] Find a vector which is perpendieular to both a = i ~ j k and b = (2, 17 1) .
2 V35/=/1"':~/~le11:
LII ' 7
Maw; 4/5: H ;:/
= leek7;-
3. [2pt] Find a vector equation of the li
Math 227504 Quiz #1 (Sections 12.112.3) Instructor: Yunrong Zhu
Name:
M
1. [3pt] Show the equation :32 + 312 + Z2 23: 4y + 62 = 15 represents a sphere, and nd its
center and radius.
xizx + y4y +2362 :15
(mt: war-4 + (283? 5
2. [3pt] Let a = i + 2j k, an
MATH 3352
Name:
Quiz #02
06/02/2016
ID:
1. (5 points) An urn contains 4 red and 6 black balls. Players A and B withdraw balls
from the urn consecutively until a red ball is selected. Find the probability that A
selects the red ball. (Player A draws the fi
Chapter 13: Curve Sketching
13.1 Relative Extrema
Increasing and Decreasing
Definition:
1. A function f is increasing on the interval I if, for any two numbers x1 , x 2 in I, where
x1 x2 , then f ( x1 ) f ( x 2 ) .
2. A function f is decreasing on the int
Chapter 13: Curve Sketching
13.6 Applied Maxima and Minima
Recall that:
1. Total revenue function = (number of units) * (price of each unit)
r q. p
2. Profit function = (total revenue function) (total cost function)
P r c
3. Total cost function = (number
Chapter 13: Curve Sketching
13.3 Concavity
Concavity
Rule1: Let f be differentiable on the interval ( a, b) , if f ( x) 0 for all
x in (a, b) , then f is
concave up on ( a, b) . If f ( x) 0 for all x in ( a, b) , then f is concave down on ( a, b) .
Exampl
Chapter 12: Additional Differentiation Topics
12.5 Logarithmic Differentiation
To differentiate y f (x) ,
1. Take the natural logarithm of both sides ( Lny Ln f (x ) ).
2. Simplify Ln f (x ) by using properties of the logarithms.
3. Differentiate both sid
Chapter 12: Additional Differentiation Topics
12.7 Higher Order Derivatives
Recall that we denoted for the first derivative of y f (x ) by y ,
dy
or f (x) .
dx
d2y
dx 2
th
(n )
or f (x ) . In the similar way, we can find any order derivatives. We denote f
Chapter 12: Additional Differentiation Topics
12.4 Implicit Differentiation
To find
dy
for an equation;
dx
1. Differentiate both sides of the equation with respect to x .
dy
on one side of the equation and collect the other terms on the other
dx
2. Collec
Chapter 11: Differentiation
11.2 Rules for Differentiation
We will denote for the derivative of a function y f (x ) by one of the following:
y ,
dy
d
, Dx y, f ( x), f ( x) or D x f (x)
dx
dx
The derivative of y f (x ) at x1 is denoted by
f ( x1 ) or
dy
d
Chapter 12: Additional Differentiation Topics
12.2 Derivatives of Exponential Functions
As farther differentiated formulas, we offer in this section the differentiating of exponential functions.
Rule1: If y e
f ( x)
, then
dy
f ( x)e f ( x ) .
dx
Example
Chapter 11: Differentiation
11.5 The Chain Rule and the Power Rule
Chain Rule: If y g (u ) and u h(x) , then
dy dy du
.
dx du dx
dy
for the following:
dx
2
1) y u and u 3x 1
Example: Find
2) y 3u 5u 6 and u 4 x
2
2
3) y 3 s and s x 7
3
4) y t 1 and t ( x
Chapter 12: Additional Differentiation Topics
12.3 Elasticity of Demand
Definition:
Elasticity of demand is a means by which economists measure how a change in the price of a
product will affect the quantity demanded.
If p f (q ) is a differentiable dem
Chapter 11: Differentiation
11.3 The Derivative as a Rate of Change
We will denote for the total cost function by c f (q ) , where q is the number of marketed units.
The marginal cost function is the rate of change of the total cost function, that is
Marg
Chapter 3: Lines, Parabolas and Systems
3.6 Applications of Systems of Equations
The point of Equilibrium:
When demand and supply curves of a product are represented on the same plane, then the intersection
point ( m, n ) is called the point of equilibriu
Chapter 11: Differentiation
11.4 Product and Quotient Rules
Product Rule: If f ( x) g ( x)h( x) , then f ( x) g ( x )h( x ) g ( x )h ( x ) .
3
2
Example: If f ( x) ( x 5x)(7 x 3) , find f (x) ?
Example: If y ( x 7)(3
dy
2
,
find
)
dx
x2
?
x 1
2
Example:
Chapter 12: Additional Differentiation Topics
12.1 Derivatives of Logarithmic Functions
In this section, we develop formulas for differentiating logarithmic functions. Before we begin this
section, we will recall the most important logarithmic properties,
Chapter 3: Lines, Parabolas and Systems
3.5 Nonlinear Systems
Definition: A nonlinear system is a system of equations in which at least one equation is not linear.
Example: Solve the following nonlinear systems:
y 2 x 2 5
x y 1
y 3 x 2
y x 4
Page 1 of
PORTNEUF RIVER
PEDESTRIAN ACCESS (PRPA)
By: Kyle Parker, Mohammad Alraei, Fahad
Bin Madhi, Nael Alnasr
Update for November 4, 2015
Kyle Parker
Existing Channel Conditions
Kyle Parker
Introduction
The Portneuf River runs in a concrete channel from the
Hal
Lecture 1 Notes: Hull Resistance
2
R c1vs
physics
PER vs
PE= effective_power
P c
v
1
E
(3.1)
(3.2)
defined
(3.3)
substitution
3
s
c yc v
1
0 s
physics
(3.4)
y = f( fouling displacement_variations sea_state water_depth)
essentially time and operations
(3.5
Lecture 5 Notes: First Law
pg 83 van Wylen & Sonntag Fundamentals of Classical Thermodynamics 3rd Edition SI Version
first law for cycle
(5.2)
1 dQ = 1 dW
The net energy interaction between a system and its environment is zero for a cycle executed by the
Lecture 9 Notes: Reliability and Availablity
Including and improving reliability of propulsion (and other) systems is a challenging goal for system
designers. An approach has developed to tackle this challenge:
1. a design and development philosophy
2. a
Lecture 7 Notes:
Electrical Overview
Ref: Woud 2.3
C1C
Q = charge
Q
I= t
much of what is is the text
s1s
A1A
I = current
work done per unit charge = potential difference two
points aka electromotive force (EMF)
V1V
U = volts
1V1A 1 watt
1V 1A 1 W
Power =
Lecture 3 Notes: KT and KQ
we have seen in general the development of the Wageningen B series. The performance curves are available either in chart form or can be generated
from polynomials:
regression coeff. Re=2*10^6
polynomial representation
use in des
Lecture 8 Notes:
Electric Motors
M = KmI
from electrcal overview
Lorentz force.
Ref: Chapter 9
Km= constant_for_given_motor
(ref: 2.93)
M = torque
Nm
= magnetic_flux
Wb 1 weber
I = current
A 1 amp
(9.1)
1Wb1A 1 N
when rotating, electromotive force induce
Lecture 2 Notes: Propeller Testing
Screw propeller replaced paddle wheel ~1845 in Great Britain (vessel) Brunel In test;
independent variables are
VA
velocity of advance
shaft rotation speed
n (rev/sec), N (rev/min)
dependent variables are:
torque
Q
thrus
Lecture 1 Notes: Convexity
2
Topics
To understand better what is going on, we will embark in a journey to learn a wide variety of methods
used to approach these problems. Some of our stops along the way will include:
Linear optimization, second order con