Download Section 2.1 Rates of Change and Limits Calculus

Section 2.1 Rates of Change and Limits DISCUSSION
Calculus
A rate is a ratio of two quantities of different units (examples, m/s, $/kg,). In
general, if we are graphing two units on a 2 dimensional (xy) plane, the rate
consists of the y units divided by the x units.
A unit rate is the numerical value of the rate where the bottom value (x axis
value) is 1.
Example 1. Finding an average speed.
A rock breaks loose from the top of a tall cliff. What is its average speed during
the first 2 seconds of fall?
For short distances near the earth’s surface, the distance an object falls, y (in
meters), over a time, t (in seconds), is given by the relationship:
y = 4.9 t2
The independent variable, _____, is graphed on the horizontal axis and the
dependent variable, _______, is graphed on the vertical axis. The units of
speed,
in this case, are
______
. The average speed during the first two seconds, then,
______
is computed as:
______ _____________
_____
.

 ______
______ _____________
____
Example 2: Finding the instantaneous speed (speed at an instant of time)
For the problem of example 1, what is the instantaneous speed at t = 2 seconds?
Numerically, to do this we calculate the average speed from t = 2 seconds to
some slightly later time t = 2 + h. This is expressed as:
_____ ___________________

_____ ___________________

_________________________
_________________________

______________________
______________________
 ___________________________
Therefore, the average speed at t = 2 has a limiting value as the difference of
time approaches 0, which is:
____________________________________________________
Limits describe how the outputs of a function behave as the inputs approach
some particular value.
The formal definition of a limit is somewhat confusing, but makes sense if you
reason through it slowly. The AP Calculus exam usually does not have
questions relating to the formal definition of a limit, but in some college
calculus courses it is required to be known and understood.
The formal definition of a limit is:
Let c and L be real numbers. The function f has limit L as x approaches
c if, given any positive number ε, there is a positive number δ such that
for all x:
0  x  c    f x   L  
lim f  x   L
We write:
x c
The diagrams below illustrate that the existence of a limit as x → c does not
depend on how the function may or may not be defined at c.
y
y
y
2
2
2
1
1
1
1
-1
f (x ) =
x
1
-1
x2 - 1
x-1
x
x2 - 1
,x≠1
x-1
h(x) = x + 1
g(x) =
1,
x=1
In all cases in this example, lim f  x   lim g  x   lim h x   2
x 1
x 1
Properties of Limits
Limit of a function that is equal to a constant
Limit of the function y = x as x approaches c
1
-1
x 1
x
Theorem 1 Properties of Limits
If L, M, c, and k are real numbers and lim f  x   L and lim g  x   M , then:
x c
Sum Rule:
Difference Rule:
Product Rule:
Constant Multiple Rule:
Quotient Rule:
Power Rule: If r and s are integers, s ≠ 0, then
Examples using Properties of Limits:


x4  x2  1
Find lim x  4 x  3 and lim
x c
x c
x2  5
3
2
x c
One-sided and Two-sided Limits
Right-hand limit: lim f x 
x  c
Left-hand limit:
lim f  x 
x  c
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. In symbols,
lim f x   L  lim f x   L and lim f x   L
x  c
x c
x  c
Example: Discuss the limits that exist at all points for the graph shown below:
y
2
y = f(x)
1
-1
1
2
3
4
x
If a limit cannot be found directly, it can sometimes be found indirectly by the
sandwich theorem.
The Sandwich Theorem
If g x   f  x   h x  for all x ≠ c in some interval about c, and
lim g  x   lim h x   L ,
x c
x c
then
lim f  x   L
x c
Example
sin  x 
1
x 0
x
Application of the Sandwich Theorem to show that lim
y
1
x
1 x
Section 2.1 Rates of Change and Limits
Calculus
For problems 1 – 10, tell whether or not the function has a limit as x approaches
c; if so, tell what the limit equals.
1.)
2.)
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
x c
xc
xc
3.)
x c
xc
xc
4.)
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
x c
xc
xc
x c
xc
xc
5.)
6.)
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
x c
xc
xc
7.)
x c
xc
xc
8.)
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
x c
xc
xc
x c
xc
xc
9.)
10.)
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
Does lim exist? __________
x c
xc
xc
x c
xc
xc
Part II. Determine the limit by substitution. Support graphically.
11.) lim  x3  3x 2  2 x  17 
x1
12.) lim e x cos x
x0
Part III. Determine the limit graphically. Confirm algebraically.
13.) lim
x 1
x 1
x2  1
1
1

14.) lim 2  x 2
x 0
x
sin x
x 0 2 x 2  x
15.) lim
3sin 4 x
x 0 sin 3 x
16.) lim
Match the function with the table.
x2  x  2
17.) y1 
___________
x 1
x2  x  2
18.) y1 
__________
x 1
x2  2 x  1
19.) y1 
__________
x 1
x2  x  2
20.) y1 
__________
x 1
X
.7
.8
.9
1
1.1
1.2
1.3
X = .7
Y1
-.4754
-.3111
-.1526
0
.14762
.29091
.43043
X
.7
.8
.9
1
1.1
1.2
1.3
X = .7
(a)
X
.7
.8
.9
1
1.1
1.2
1.3
X = .7
Y1
2.7
2.8
2.9
ERROR
3.1
3.2
3.3
(c)
Y1
7.3667
10.8
20.9
ERROR
-18.9
-8.8
-5.367
(b)
X
.7
.8
.9
1
1.1
1.2
1.3
X = .7
-.3
-.2
-.1
ERROR
.1
.2
.3
(d)