END BEHAVIOR MODELS:
For numerically large values of x, we can sometimes model the behavior of
a complicated function by a simpler one that acts virtually the same.
Example 6 Modeling Functions for x large:
Let f (x) = 3x − 2x + 3x − 5x + 6 and f (x) = 3x . Show that while f and g are
quite different for numerically small values of x, they are virtually identical
for x large.
4
3
2
4
**If one function provides both a left and right end behavior model, it is
simply called an end behavior model. (Example 6)
“The Dance”
Example End Behavior Models:
Find an end behavior model for
f (x) =
3x 2 − 2x + 5
4x 2 + 7
Example 7 Finding End Behavior Models:
Find an end behavior model for
a)
f (x) =
2x 5 + x 4 − x 2 + 1
3x 2 − 5x + 7
b)
g(x) =
2x 3 − x 2 + x − 1
5x 3 + x 2 + x − 5
5
Pg. 76 #35-38
Examples End Behavior:
(Similar to #17-19 on MML)
a) Find a power function end behavior model for f
b) Identify any horizontal asymptotes
1) f (x) = 3x − 2x + 1
2
3)
2)
f (x) =
f (x) =
x−2
2x + 3x − 5
2
4x 3 − 2x + 1
x−2
6
Try It Limits as x → ±∞ :
(Similar to #20 on MML)
Pg. 76 #53
2.3 CONTINUITY
Continuity at a Point:
Any function y = f(x) whose graph can be sketched in one continuous
motion without _____________________________is an example of a
continuous function.
Objective: To determine continuity at a point and continuity on an open
interval.
Exploration: Show a quick sketch of your graph on the given interval and
discuss the continuity.
a) y = x + 1
2
c)
y=
sin x
x
b)
d)
y=
y=
1
x−2
x2 − 4
x+2
7
Three Conditions exist for which the graph of f is not continuous at x =c
1) The function is not defined at x = c
2) The limit of f(x) does not exist at x = c.
3) The limit of f(x) exists at x = c, but it is not equal to f(c).
Definition of Continuity:
Continuity at a Point: A function is ________________________________
If the following three conditions are met.
1) f(c) is ______________________________.
2)
lim f (x)
_____________________________.
3)
lim f (x)
_____________________________.
x→c
x→c
Continuity on An Open Interval: A function is continuous on an open interval
(a, b) if its continuous at each point in the ____INTERVAL___________. A
function that is continuous on the entire real line (−∞,∞) is
________EVERYWHERE________ continuous.
Two categories of Discontinuities:
1) Removable: A discontinuity at c is called removable if f can be made
continuous by appropriately
________________________ or ____________________ f(c).
2) Nonremovable: can not define ____________________(Limit DNE)
3 types:
1) _______________________
2) _______________________
3) _______________________
8
Label the following discontinuities on the given graphs.
Example: Continuity of a Function:
Discuss the continuity of each function. Start by drawing a graph and
identify the domain.
1)
f (x) =
1
x
2)
3)
g(x) =
x2 − 1
x −1
4) y = sinx
x + 1, x < 0
h(x)=
x2 , x > 0
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DEFINITION CONTINUITY AT A POINT
Interior Point: A function y = f(x) is continuous at an interior point c of it
domain if
_______________________
Endpoint: A function y = f(x) is continuous at a left endpoint a or is
continuous at a right endpoint b of its domain if
_______________________ or __________________________
If a function f is not continuous at a point c, we say that f is discontinuous at
c and c is a point of discontinuity of f. Note that c need not be in the
domain of f.
Example 1 Investigating Continuity:
Find the points at which the function f in Figure 2.18 is continuous, and the
points at which f is discontinuous.
Figure 2.18
Try It Continuity at a Point:
Find the points at which the given function is continuous and the points at
which it is discontinuous.
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Example 2 Finding Points of Continuity and Discontinuity:
Find the points of discontinuity of the greatest integer function.
Try It Continuity: (MML #1-4)
Find the points of continuity and the points of discontinuity of the function.
Identify each type of discontinuity.
a)
y=
1
(x + 2)2
b) y = x − 1
3
c) y = 2x − 1
CONTINUOUS FUNCTIONS
A function is continuous on an interval iff it is continuous at every point of
the interval. A continuous function is one that is continuous at every point
of its domain. A continuous function need not be continuous on every
interval.
COMPOSITES:
All composites of continuous functions are continuous.
THEOREM 7 Composite of Continuous Functions
If f is continuous at c and g is continuous at f(c), then the composite
_______________________ is continuous at c.
The Limit of a Composite Function
If f and g are functions such that
lim f (g(x)) =
x→c
lim g(x) = L
x→c
and
lim f (x) = f (L)
x→L
, then
______________ = _______________ = ___________
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INTERMEDIATE VALUE THEOREM FOR CONTINUOUS
FUNCTIONS:
Functions that are continuous on intervals have properties that make them
particularly useful in mathematics and its applications. One of these is the
intermediate value property.
THE INTERMEDIATE VALUE THEOREM
If f is continuous on the closed interval _____________ and k is any
number between f(a) and f(b), then there is at least one number c in [a,
b] such that
______________________
Note: The IVT Theorem tells you at least one c exists, but it does not
give a method for finding c. These are called
_______________________________________.
*The IVT guarantees the existence of at least one number c in the closed
interval [a, b]. There may be ______________________________ number
c such that f(c) = k.
Real Life Example:
Why IVT useful?
*If f is continuous on [a, b] and f(a) and f(b) differ ____________________
the IVT guarantees the existence of at least ______________________ of f
in the closed interval [a, b].
Example 1: An application of the Intermediate Value Theorem.
Use the Intermediate Value Theorem to show the polynomial function
has a zero in the interval [0, 1].
Then find the zero to four decimal places by using your calculator.
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2.4 RATES OF CHANGE AND TANGENT LINES
The average rate of change of a quantity over a period of time is the
amount of ____________ divided by ___________________.
Average rate of change =
_______________
It can be thought of as the slope of a ___________________ to a curve.
Example 1 Finding Average Rate of Change:
3
Find the average rate of change of f (x) = x − x over the interval [1, 3].
Note: Finding the average rate of change of a function over an interval is
simply finding the slope of the line containing the endpoints of the interval.
Average rate of change over the interval [a, b] = ___________________
Try It Average Rate of Change (MML #1-3)
Find the average rate of change of the function over each interval.
1)
f (x) = 4x + 1
2)
f (x) = cot x
a) [0, 2]
a) [π 4, 3π 4]
b) [10, 12]
b) [π 6, π 2]
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Example 2 Growing Drosophila in a Laboratory:
Use the points P (23, 150) and Q (45, 340) in Figure 2.27 to compute the
average rate of change and the slope of the secant line PQ.
Note: We can always think of an average rate of change as the
slope of a secant line.
Example 3 Finding Slope and Tangent Line:
Find the slope of the parabola y = x2 at the point P(2,4). Write an equation
for the tangent to the parabola at this point.
Slope of a Curve:
DEFINITION SLOPE OF A CURVE AT A POINT
The slope of the curve y = f(x) at the point P(a, f(a)) is the number
m = ______________________________
provided the limit exists.
The tangent line to the curve at P is the line through P with this slope.
14
Example Tangent to a Curve (MML #5-7)
2
Given y = x + 2 at x = -1 find:
a) The slope of the curve
b) An equation of the tangent
c) An equation of the normal
d) Then draw a graph of the curve, tangent line, and normal line in the
same square viewing window.
DEFINITION NORMAL TO A CURVE
The normal line to a curve at point is the line ___________________
To the tangent line at the point.
The slope of the normal line is the _____________________________
of the slope of the tangent line.
Slope of a Curve
All of the following mean the same:
1) The slope of y = f(x) at x = a
2) The slope of the tangent line to y = f(x) at x = a
3) The _______________________________ rate of change of f(x) with
respect to x at x = a
4) __________________________________
5) The expression ______________________ is the difference quotient
15
of f at a.
Example 4 Exploring Slope and Tangent (MML #9, 10)
Let f(x) = 1/x
a)Find the slope of the curve at x = a.
b)Where does the slope equal -1/4?
c)What happens to the tangent to the curve at the point (a, 1/a) for different
values of a?
Example 5 Finding a Normal Line:
2
Write an equation for the normal to the curve f (x) = 4 − x at x = 1
16
Speed Revisited:
The function y = 16t2 is an object’s position function.
A body’s average speed along a coordinate axis for a given period of time is
the average rate of change of its position y = f(t).
Its instantaneous speed at any time t is the instantaneous rate of change
of position with respect to time t, or
___________________________________
Example Finding Instantaneous Rate of Change (MML #11 & #12)
Find the instantaneous rate of change of the position function y = f(t) in feet
at the given time t in seconds.
f (t) = 2t 2 − 1 at t = 2
Example 7 Investigating Free Fall:
A rock breaks loose from the top of a tall cliff with position given from the
equation y = 16t 2.
Find the speed of the falling rock at t = 1 sec.
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