Transcript Real Numbers - University of Houston

Another look at D=RT
If you travel 240 miles in a car in 4 hours,
your average velocity during this time is
distance traveled 240

 60 m i / hr
elapsedtim e
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This does not mean that the car’s
speedometer was on 60 mph at all times;
this is only your average velocity during
this time interval.
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2.6 Constant velocity
If a car’s cruise control was set at 60 mph for 4 hours of
travel, what would the shape of the graph of distance
traveled to elapsed time be?
The graph is a straight line and the slope of the line is 60.
So in general, if the graph of distance to time is a
straight line, at every instant the velocity is constantly
the same, that is, at every instant the velocity is the
slope of the line.
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2.6 Varying velocity
Now let’s consider the case when the velocity
of a car is varying over time.
What is the velocity of
the car at time t=3?
If the graph were a straight
line, the answer would be
the slope of the line as before.
Let’s zoom-in on the graph
near the point t=3.
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2.6 Varying velocity continued
We see that after magnifying the graph near t=3,
the curve looks like a straight line between the
inputs
t=2.992 and t=3.004.
It seems reasonable
to assume that the curve behaves like the straight
line and that the velocity is constant during this
time interval. So at each instant during this time,
which includes t=3, the velocity is the slope of this
line.
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2.6 Varying velocity continued
Two points on the graph
are approximately
(2.992, 26.85) and
(3.004, 27.05).
Therefore the
velocity at t=3 is
velocityat t  3
 slope of line
27.05  26.85
0.2


 16.7 ft / sec
3.004 2.992 0.012
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2.6 Varying velocity continued
For the functions we are studying it can be
proven that the more you zoom-in on the
graph of the function at a specified input,
the curve will look more and more like a
particular straight line – a tangent line.
Putting this all together, we define the
instantaneous rate of change of a function at
a specified input x=a to be the slope of the
tangent line at the point ( a, f(a) ).
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2.6 Instantaneous rate of change
The instantaneous rate of change of a
function f at the input x=a
= slope of tangent line at x=a
= derivative of f at x=a
'
= f (a)
This is read “f prime of a”
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2.6 Example of Instantaneous Rate of Change
The distance in feet traveled by a car moving along a
straight road x seconds after starting from rest is given by
f(x) = 2x2, 0< x <30
Use a tangent line to approximate the (instantaneous)
velocity of the car at x=22.
Solution: On a graph
of the function, draw a tangent
line at x=22. Then find its slope
from any 2 points on the line.
1 4 9 0 4 7 0
f ( 2 2) 
2816
1020

 85
12
'
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2.6 Limit Definition of
Instantaneous Rate of Change
The instantaneous rate of change of a
function f at the input x=a is defined by
instaneous rate of changeof f at x  a
 f ' (a)
 lim
h0
f (a  h)  f (a)
h
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2.6 When is a function not differentiable, that is, f `
(a) does not exist?
• You have learned that if the graph of a function is broken
at a point, then the function is not continuous at the point.
That is, the graph of a continuous function is unbroken.
• It can be shown that a differentiable function is continuous.
This means the graph of a differentiable function must be
unbroken too. But there is another requirement. A function
is not differentiable wherever the graph has a sharp turning
point, a cusp or a vertical tangent line at the point.
The function whose graph
is shown here is not
differentiable at the
points x= -2, x=0 and
x=1.
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2.6 When, What, How and By How Much
• A function has output “the weight of an infant w(t) in lb” and input
“age t in mo”.
Write a sentence to interpret (explain the meaning) of the instantaneous
rate of change w`(3)=1.5.
Answer:
• A function has output “body temperature of a patient F(t) in oF” and
input “t, hours after taking a fever-reducing drug”. In a sentence,
interpret F`(3) = -0.25.
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2.6 Average Rate of Change
The average rate of change of a function f
from the input x to the input x+h, or over
the interval [x, x+h], is given by
f[ x , x  h ] ( x ) 
f ( x  h)  f ( x ) f ( x  h)  f ( x )

( x  h)  x
h
In words, this is the change in the outputs
divided by the change in the inputs.
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2.6 Geometric Interpretation of ARC
• If a straight line goes thru the two points (x1,y1)
and (x0, y0), then the slope of the line is given by
y1  y0
x1  x0
• So the average rate of change of a function from
input x0 to input x1, is the same as the slope of the
straight line going thru the points x0 and x1.
This line is called a secant line.
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2.6 Average rate of change of f between 2
inputs equals slope of a secant line
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2.6 Example of Average Rate of Change
The distance in feet traveled by a car
moving along a straight road x seconds after
starting from rest is given by
f(x) = 2x2, 0< x <30
For each of the following three time
intervals, calculate the average velocity of
the car.
[22, 23], [22, 22.1], [22,22.01]
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