Summary
0 The limit laws allow us to evaluate limits algebraically.
- When using the limit laws make sure that the individual limits exist.
0 Evaluation by direct substitution works for polynomials and for rational functions when the denominator is not 0.
Deﬁnitions
- xl'_upw f (x) = L means that f ( x) gets close to L as x gets arbitrarily large in the positive
direction.
- lim f (x) = L means that f (x) gets close to L as x gets arbitrarily large in the negative
x—->-oo
direction.
- These two limits desc
Summary
- To ﬁnd the slope of a tangent line to a graph, build a sequence of approximating secant lines.
0 As the scent lines get very close to the tangent line, their slopes will approach a limiting value
which will be the slope of the tangent.
' Tangent
f (x) tells us the value of the function a_t x.
Limits help us describe how a function f (x)is behaving new; 1:.
Goal: to deﬁne limits and understand them numerically and graphically.
Deﬁnition of the derivative
The (instantaneous) rate of change of a function
y = f ( x) at an .7: value, x = a is called the derivative
offat a and is denoted byf' (a).
f’(a) = rateofchangeoffata
= slope of tangent to f at a
= lim 1(a+h)-Z(a)
h
h—)o
V
Summary
- Average velocity = W = slope of secant line
0 Velocity = limit of average velocities over shorter and shorter time intervals = slope of tangent
line
Read the problem carefully: What is to be optimized? What
is the unknown? What quantities are given? What
conditions or constraints are given?
» JG” 71 “fr.
A function‘r‘Lt) is concave Lip_on an interval
if_/" is increasing on that interval.
A luncti0nr‘(_t) is i-
interval
itj“ is decreasing on that interval.
Direct Substitution
The previous two examples illustrate an important property.
If p(x) is apolynomial, thenxligtap(x) =p(a).
If f (x) = ﬁg— is a rational function; that is. a quotient of polynomials then Jig“ f (x) = f (a)
provided we do not get a zero d
Deﬁnition of two sided limit
The statement xliina f (x) = L means that as x gets close to a from either side but remains different
from a, f (x) gets close to L.
A limit is a statement about how the function f behaves near the value a.
This can be diﬂ'ere
Given a domain D and a function/; if c' is a
runnberWiDthennn
_/(c) is an ABSOLUTE. fit) is a LOCAL.
. Off, 0 off
on the domain D if on the domain D if
f(.‘) 3/0-) for all .\' in D. /('() ' 3/0 )when .\' is
ot/on near(
the domain D if 0 Off
ﬂc’)<fiﬂ foraH
Graded Worksheet B2
Name: _
This needs to be completed and emailed to burelle@hartford.edu by the due date announced in
CourseCompass.
Here are the M-SAT scores and the college grade point average at the end of the freshman year of college for 10 differen
Graded Worksheet B3
Name: _
This needs to be completed and emailed to burelle@hartford.edu by the due date
announced in CourseCompass.
1. At a certain hospital the weights of babies is a mound-shaped distribution with a mean of
110 oz. and a standard devi
MAT 114
Graded Worksheet A3
Name: Sara Karlstad
This needs to be completed and emailed to burelle@hartford.edu by the due date announced in
CourseCompass.
_D_1. A researcher wants to know if Pennsylvania adults favor a higher sales tax if it would mean lo
Graded Worksheet A2
Name: Sara Karlstad
This needs to be completed and emailed to burelle@hartford.edu by the due date
announced in CourseCompass.
Here are the highway gas mileages, as reported by the EPA, for 21 brands of sports car.
24
26
28
13
31
28
25
MAT 114
Graded Worksheet A1
Name: Sara Karlstad
This needs to be completed and emailed to burelle@hartford.edu by the due date announced in
CourseCompass.
1.
At a certain university there are 6345 students.
a. If 11% are computer science majors, how many
Graded Worksheet B1
Name: Sara Karlstad
This needs to be completed and emailed to burelle@hartford.edu by the due date
announced in Course Compass.
The table shows data for sex and opinion about the legalization of marijuana, for a sample of respondents i
MAT 114
Graded Worksheet A4
Name: Sara Karlstad
This needs to be completed and emailed to burelle@hartford.edu by the due date
announced in CourseCompass.
1. A group of male college students are hound to have a mean of 69.5 with a standard deviation
of 2.