Name:
Limit Laws
One of these limits can be evaluated using substitution; for one of these limits substitution doesn't
work. Which can you evaluate using substitution and for which does substitution not work? Explain
how you decided and find the limits if
Name:
Derivatives and Rates of Change
1. Write two different limit expressions that can be used to determine the slope of the tangent line
to
at
.
f(x) x2 1 x 3
2. TRUE or FALSE: The derivative of velocity is position. Explain your choice.
3. Given that
a
Name: _
TRUE OR FALSE
Continuity
a. All polynomials are continuous at all points. _
b. All rational functions are continuous at all points. _
1. Sketch a function that is continuous from the left at
removable discontinuity at
x 5
.
x 3
but not from the ri
Name:_
For the
the function
function
For
values:
x
x
-3.1
-2.1
-3.01
-2.01
-3
-2
Intro Limits
complete the
the table
table of
of values:
, complete
x
4
g(x) x 9
g(x)
x32
x
g(x)
g(x)
2
2
x
x
-2.9
-1.9
-2.99
-1.99
-3
-2
Based on the table approximate
Based
Name: _
1. a. Explain in words what is meant by the statement
Limits involving infinity
lim f (x)
.
x2
b. Sketch a graph that illustrates this statement.
2. a. Explain in words what is meant by the statement
lim f(x) 4
x
b. Sketch a graph that illustrate
Name:_
Tangent Lines
Name:_
Tangent Lines
ximating the slope of a tangent line to a function f (x) at a
point x = a involves using _ on the
1. The strategy for approximating the slope of a tangent line
1. T
to a function f (x) at a point x = a involves us
Calculus II
Week One Notes
Calculus I Review
Function: relationship between an independent and dependent variables. Every independent has 1
dependent
Derivative: rate of change (slope) at any point of a line
Y= f(x)
Derivative can be represented by: dy/dx
Calculus II
Week Two Notes
Dean Hadad
Partial Fractions,
Partial fractions: integrals with fractions with a product as a denominator may be seperable into two
separate integrals using the method of partial fractions.
1
x
x ( x+ 1 )
Separate to sum of two
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 [email protected] 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 [email protected] 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