Download Lesson 2.1 Rates of Change & Limits

Lesson 2.1
Rates of Change & Limits
What you’ll learn about
Average & instantaneous speed
Definition of Limit
Properties of Limits
One-sided and two-sided limits
Sandwich Theorem (tomorrow)
Example 1
Finding Average Speed
A moving body’s average speed during an interval of time is
found by dividing the distance covered by the elapsed
time.
A rock breaks loose from the top of a tall
cliff. What is its average speed during the
first 2 seconds of fall?
Distance covered is d = 16t2 feet in the first t seconds.
y
t
Example 2
Finding an Instantaneous Speed
Find the speed of the rock in Example 1 at
the instant t = 2 seconds.
Algebraically: We can calculate the average
speed of the rock over the interval from time
t=2 to any slightly later time t = 2+h.
y
t
Definition of Limit
Assume f is defined in a neighborhood of c and let c and L
be real numbers. The function f has limit L as x
approaches c if, given any positive number m, there is
a positive number n such that for all x,
0 | x  c | m | f ( x)  L | n
We write:
lim
f ( x)  L
xc
Read: “The limit of f of x as x approaches c equals L.”
Theorem 1:Properties of Limits
lim
f ( x)  L
If L, M, c, and k are real numbers and x  c
and, then
1) Sum Rule
lim
f ( x)  g ( x)  L  M
xc
3) Product Rule
Rule
lim
f ( x)  g ( x)  L  M
xc
5) Quotient Rule
lim f ( x) L

,M  0
x  c g ( x) M
lim
g ( x)  M
xc
2) Difference Rule
lim
f ( x)  g ( x)  L  M
xc
4) Constant Multiple
lim
k  f ( x)  k  L
xc
6) Power Rule
lim
( f ( x)) r / s  Lr / s
xc
Example 3: Using Properties of Limits
lim
lim
k k
Find the following limits given that x  c
and x  c x  c
a)
lim
( x 3  4 x 2  3)
xc
b)
lim x 4  x 2  1
x  c x2  5
Theorem 2
(Page 62)
Polynomial and Rational Functions
1) If f(x) is any polynomial function and c
is a real number, then the limit as x
approaches c of f(x) = f(c).
2) If f(x) and g(x) are polynomials and c is
any real number, then the limit of
f(x)/g(x) = f(c) / g(c), provided g(c)≠0.
Example 4
Using Theorem 2 & the Substitution Method
Find each limit.
a)
b)
lim
[ x 2 ( 2  x )]
x3
lim x 2  2 x  4
x2 x2
Example 5 Using the Product Rule
Determine
lim tan x
x0 x
Solve Graphically:
Graph the function and find the y value at x = 0.
Confirm Analytically: Use the fact that tan x = sin x / cos x
Example 6
Exploring a Nonexistent Limit
Use a graph to show that
does not exist.
lim x 3  1
x 2 x2
 Can we find the limit using substitution?
 What is the behavior of f(x) at x=2 if we look
from the left?
If we look from the right?
 What y-value does the function INTEND to
have at x = 2?
Theorem 3
One-sided and Two-sided Limits
A function f(x) has a limit as x
approaches c if and only if the
right-hand and left-hand limits
at c exist and are equal!
Example 7
Function Values Approach Two Numbers
Find the right and left hand limits of
f(x)=int x as x->3.
lim
a) x  3 int x
lim
b)
int x

x3
c)
lim
int x
x3
Example 8
Exploring Right & Left-Hand Limits
Find the following limits given the piecewise function f(x).
At x=0
At x=1
At x=2
At x=3
At x=4
f(x) = -x+1,
1,
2,
x-1,
-x+5,
0≤x<1
1≤x<2
x=2
2<x≤3
3<x≤4
Homework
Page 66
1-35 odds
Theorem 4
Sandwich Theorem
The Theorem refers to a function f whose values are
sandwiched between the values of two other functions,
g and h. If g and h have the same limit as x->c, then f
has that limit too.
If g(x)≤f(x)≤h(x) for all x ≠ c in some interval about c, and
lim
lim
g ( x) 
h( x )  L
xc
xc
Then
lim
f ( x)  L
xc
Example 9
Using the Sandwich Theorem
Show that
1)
2)
3)
lim
[ x 2 sin( 1 / x)]  0
x0
Graph the function. From the graph define g(x) and
h(x). f(x) must be between them! The parent functions
will be part of your given function every time.
From the graph we see that g(x) = x2 and h(x) = -x2.
Since
then by the Sandwich
lim 2
lim
2
x 
x 0
x0
x0
Theorem
lim
[ x 2 sin( 1 / x)]  0
x0
Homework
Page 66
37-43 odds,
45-49, 51, 54, 59, 60, 63
Let’s spend ONE MORE DAY
practicing LIMITS!!!
 Warm up
Pages 67-68: #50, 65-70
 Correct Homework: Q & A
 Partner work on assignment, finish as
homework
Page 66 #55, 57, 61, 64
Page 75 Quick Review 1-10