2.4 SUMMARY
' Defimuon: f is continuous at 1' = c if 1E1: HI = c]. This means that c] exists, 1'5": 1 cfw_I 3
exists, and they are equal.
* If I" x does not exist, or if it exists hut does not equal c), then f is discontinuous at
II'I:
1' = c.
* If f is c
2.6 SUMNARI
- We sag:r that f is squasssd atx =r: if there exist flmctionsi and I: such that HI) 5 x] 5 um
fmallxscmanopenmtmalfcontamjngc, and
lim HI] = lim an] 2 L
IBE 1E
The Squeeze Theoa'em states that in this case I" fix = L.
- Two impmtaut tfiganome
2.3 SUMMARY
' The Intermediate Value Theorem cfw_Ile says that a continuous function cannot sh values.
' More precisely, if f is continuous on [at b] with ail 9E ll), and if Mr is a number hehareen
u] andb], thenc] = Mfor some c E (a, Err].
' Existence of
2.1 SUMMARY
' Thelj1111t35131111ro3che5a of3 functionfi5 3nun1her thatx) 3ppro3che5, 1f 5uch an L
exi5t5.
' The average 11115 ofchan'ge of y = x] over an interval [14], 11]:
gzm
Average rate of change 2
x 1| In
(II 95 In]
' The instantaneous 11115 of chan
2.? SUMMARY
- Limits as iname
- lirri LUZ = L if Lfix) L| becomes arbitrarily small as 1' increases without hound.
Iim
- lim HI = L if Lx L| becomes arbitrarily small as .1' decreases without bound.
Ibm
lime-1:00 and lim e=cfw_)
xrm 1403
- A horizontal l
3.1 SUMMARY
. The dgfeinnne qnniieni:
fl+h)f[l
h
The difference quotient is the slope of the seeant line through the points P = (n, njj and
Q=n+iLn+irjj onthegraphof
' The dei'ivniivs f'fnj is dened by the following equivalent limits:
r _. fia+hfcfw_a_. r
2.5 SUMMARY
. When f is known to be continuous at I = c, the limit can be evaluated by substitution:
53; no = so.
* We say that x) is indeterminate cfw_or has an indeterminate form) at I = c if the formula for
e) yields an undened expression of the type
I
2.9 SUMMARY
I Infmmallj' speaking, the statement lix = L means that the gap Lx Ll tends to U as I
approaches 6.
- The fmmai denition cfw_called the 55 denition: 1'53 HI = L if, for all 5: III, there exists a
5 := U such that
iflxcl, then |fcfw_x)L|<:E
PROBLEM 15 : The graph of x2 - xy + y2 = 3 is a "tilted" ellipse (See
diagram.). Among all points (x, y) on this graph, find the largest and smallest
values of y . Among all points (x, y) on this graph, find the largest and
smallest values of x .
SOLUTION
Math 151, Sections 23, 24, 25
Workshop #5
Problem 1
The graph of the tilted ellipse 2 + 2 = 3 is shown to the right.
What are the dimensions and the location of the box containing the
ellipse?
Note: The sides of the box are vertical and horizontal and als
Workshop VII Problems
Matt Hohertz
November 2, 2016
Problem I
Euclid taught me that without assumptions there is no proof. Therefore, in any
argument, examine the assumptions. E.T. Bell, The Search for Truth.
Calculator allowed.
In gambling, the house edg
640:152 Calculus II, Midterm Exam #1, Spring 2016
Department of Mathematics, Rutgers University
NAME (PLEASE PRINT):
SIGNATURE:
INSTRUCTOR: Dr. Nikolaev
SECTION:
Instructions:
This is a closed book exam: Turn o and put away all mobile phones, computers,
640:152 Calculus II, Midterm Exam #2, Spring 2016
Department of Mathematics, Rutgers University
NAME (PLEASE PRINT):
SIGNATURE:
INSTRUCTOR: Dr. Nikolaev
SECTION:
Instructions:
This is a closed book exam: Turn o and put away all mobile phones, computers,
Workshop III Solutions
Matt Hohertz
September 21, 2016
Problem I
The set of points (x, y, z) that solve the system of inequalities
!2
2
x + 10 z sin(z )
+
e z
y+
z
!2
1z
1
z2
(1)
(2)
forms a solid in 3-D space. Find the volume of this solid.
Note that a
Workshop VI Problems
Matt Hohertz
October 24, 2016
Problem I
Let Rp be, for p > 0, the region bounded above by y = 1/xp , below by the
x-axis, and on the left by x = 1.
Let Ap designate the solid obtained by rotating Rp around the x-axis and Bp
be that ob
6ba4c2e413f998cbaacf45b1b658b7c6f30484eb.docx
1. When an array reading and storing input runs out of space
a) the program could be recompiled with a bigger size for the array.
b) the array could be "grown" using the growArray method.
c) it automatically r
Instructions for Registering a New Clicker
Create a TurningPoint Cloud Account
1. Log into Blackboard using NetID
2. Go to any course in your list (Using Introduction to Management as an example here)
3. Click Tools
4. Find Turning Account Registration
5.
Introduction to Management 301 Agreement to Syllabus
Now that you have had a chance to review the syllabus and to attend at least one of the classes for
this course you must make a decision whether you wish to continue. In order to continue you must
ackno
Chapter 1 (Algebra and Precalc Review)
NOTE TO SELF: Need more on radians and trig function and graphs
(see syllabus). www.math.rutgers.edu/courses/135/135-s17/
1.2 Preliminaries
Distance and absolute value
Distance on a number line from a to b is |b a| o
Review Exam 2
Implicit Differentiation
Logarithmic Differentiation
Used when the variable is in the base and the exponent
y = xx
dy
y 1 ln x
dx
ln y = ln xx
ln y = x ln x
1 dy
1
x ln x
y dx
x
dy
x x 1 ln x
dx
Related Rates:
STRATEGY
1. Express the g
Syllabus for Math 135
Fall 2014
Prerequisite: Placement into calculus, Rutgers Math 112 or Math 115, or equivalent.
Text: CALCULUS and Its Applications, Custom Edition for Rutgers University, published by Pearson
Custom Publishing. ISBN 0-536-80120-7 or 0
Math 151, Midterm 2 Review
Do not assume that the second midterm will be similar to this review sheet. Your second exam
may contain questions that do not resemble any of the questions on this review.
Textbook Problems
Do not assume that merely completing
6-6.
Vizzini is given some function f(x), and attempts to discover the root(s) of f using Newton's
Method. He (correctly) determines the recursion relation as
xn+1=(5xn2 +18)/(10xn+1)
, and then promptly dies from a lethal dose of iocaine powder.
a) Set x
Math 151, Midterm 1 Review
Do not assume that the first midterm will be similar to this review sheet. Your first exam may
contain questions that do not resemble any of the questions on this review.
Textbook Problems
Do not assume that merely completing al
Math 151, Midterm 2 Answers to some Review Problems
(ln 10) 10arccos x
1. a) f (x) =
1 x2
0
b) g 0 (x) =
c) h0 (x) =
1
(1 +
x2 ) arctan x
3x2 + 1
q
(ln 5) (x3 + x + 1) 1 [log5 (x3 + x + 1)]2
2. a) 4
c) 0
b) 1/3
d) 0
1
1
3. The horizontal asymptotes are y
4-6. A circle with center on the y-axis is tangent to the parabola y=x2 at the points (1,1) and (1,1). Find its center and radius. A diagram is shown to the right. Suggestion: Find the equation of
the normal line to y=x2 at the point (1,1), that is, the l
Problem 4:
A tangent line is drawn to the graph of the function f(x)=16-x2 at some point P in the first
quadrant. Along with the x-and y-axes, this forms a right triangle. Find the smallest possible area
of this triangle.
a) Find the equation of the tange