Transcript Slide 1

Homework
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Homework Assignment #1
Review Section 2.1/Read Section 2.2
Page 66, Exercises: 1 – 9 (Odd), 13 – 29
(EOO)
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
1. A ball is dropped from a state of rest at t = 0. The
distance traveled after t sec is s(t) = 16t2.
(a) How far does the ball travel during the time
interval [2, 2.5]?
s  2.5  16  2.5   16  6.25   100 ft
2
s  2   16  2   64 ft  s  2.5   s  2   36 ft
2
(b) Compute the average velocity over [2, 2.5].
36 ft
avg. velocity 
 72 ft s
0.5s
Rogawski Calculus
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Homework, Page 66
1. (c) Compute the average velocity over the time
intervals [2, 2.01], [2, 2.005], [2, 2.00001]. Use this to
estimate the object’s instantaneous velocity at t = 2.
2
s  2.01  s  2  64.6416  64
ROC 

 64.16
2.01  2
0.01
s  2.005   s  2 
2
ROC 
2.005  2
s  2.00001  s  2 
2
ROC 
64.6416  64

 64.08
0.005
2.00001  2
64.00064  64

 64.0
0.00001
Instantaneous velocity at t  2 is  64 ft s
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
3. Let v  20 T . Estimate the instantaneous ROC of
v with respect to T when T = 300.
20 300.01  300
0.00577
ROC 

 0.577 m
s
300.01  300
0.01

ROC 
ROC 
20
20



300.0001  300
300.0001  300
  0.0000577  0.577 m
300.00001  300
300.00001  300
0.0001
s
  0.00000577  0.577 m
0.00001
Instantaneous ROC at T  300 is  0.577 m
s
s
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
A stone is tossed in the air from ground level with an initial
velocity of 15 m/s. Its height at time t is h(t) = 15t – 4.9t2 m.
5. Compute the stones average velocity over the time interval
[0.5, 2.5] and indicate corresponding the corresponding secant
line on a sketch of the graph of h(t).
y



ROC 
h  2.5   h  0.5 


(0.5, 6.275)
2.5  0.5
(2.5, 6.875)



0.6

 0.3m / s
2



x








Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
7. With an initial deposit of $100, the balance in a bank
account after t years is f(t) = 100(1.08)t dollars.
(a) What are the units of the ROC of f(t)?
The units of the ROC of f (t) are dollars per year.
(b) Find the average ROC over [0, 0.5] and [0, 1].
f  0.5   f  0 
103.92  100
ROC 

 7.85$ / yr
0.5  0
0.5
f 1  f  0  108  100
ROC 

 8$ / yr
1 0
1
Rogawski Calculus
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Homework, Page 66
(c) Estimate the instantaneous ROC at t = 0.5 by
computing the average ROC over intervals to the right
and left of t = 0.5.
f  0.5   f  0.4999  103.9230  103.9222
ROC 

0.5  0.4999
0.0001
 8.00$ / yr
ROC 
f  0.5001  f  0.5 
0.5  0.0001
 8.00$ / yr
103.9238  1009230

1
Instantaneous ROC  t  0.5   0.08
Rogawski Calculus
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Homework, Page 66
Estimate the instantaneous ROC at the indicated point.
9. P  x   4 x 2  3; t  2
P  2   P 1.9999 
13  12.9984
ROC 

 15.9996
2  1.9999
0.0001
P  2.0001  P  2  13.0016  13
ROC 

 16.0004
0.5  0.0001
0.0001
Instantaneous ROC  x  2   16
Rogawski Calculus
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Homework, Page 66
Estimate the instantaneous ROC at the indicated point.
13. f  x   e x ; x  0
f  0.0001  f  0 
1.0001  1
ROC 

1
0.0001  0
0.0001
f  0   f  0.0001 1  0.9999
ROC 

1
0   0.0001
0.0001
Instantaneous ROC  x  0   1
Rogawski Calculus
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Homework, Page 66
17. The atmospheric temperature T (in ºF) above a
certain point is T = 59 – 0.00356h, where h is the
altitude in feet (h ≤ 37,000 ft). What are the average
and instantaneous ROC of T with respect to h? Why
are they the same? Sketch the graph of T.
The secant line and
the graph of T are the
same line, so the average
and instantaneous ROC
are both equal to – 0.00356.
y
Temperature
50
40,000
ft
x
-50
Rogawski Calculus
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Homework, Page 66
21. Assume that the period T (in sec) of a pendulum is
3
T
L where L is the length (in m).
2
(a) What are the units for ROC of the T with respect to
L? Explain what this rate measures.
The units of the ROC of T are sec/m, giving the rate at
which the period changes when length changes.
(b) What quantities are represented by the slopes of
lines A and B in Figure 10?
Lines A and B represent instantaneous and average
ROC.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
25. The fraction of a city’s population infected by a flu
virus is plotted as a function of time in weeks in Figure 14.
(a) Which quantities are represented by the slopes of
lines A and B? Estimate these slopes.
The slope of line A represents the average ROC over
weeks 4 through 6. The slope of line B represents the
instantaneous ROC in week 6. The slopes appear to be
about 0.09  0.045 and 0.12  0.02 , respectively.
2
6
Rogawski Calculus
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Homework, Page 66
25. (b) Is the flu spreading most rapidly at t = 1, 2, or 3?
The flu is spreading most rapidly at t = 3 , as the slope
is greatest at that point.
(c) Is the flu spreading most rapidly t = 4, 5, or 6?
The flu is spreading most rapidly at t = 4, as the slope
is greatest at that point.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework, Page 66
29. Sketch the graph of f (x) = x (1 – x) over [0, 1]. Refer
to the graph and find:
(a) The average ROC over [0, 1].
average ROC = 0
(b) The instantaneous ROC at x = 0.5.
instantaneous ROC = 0
(c) The values of x for which ROC is positive.
ROC is positive for x < 0.5.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Jon Rogawski
Calculus, ET
First Edition
Chapter 2: Limits
Section 2.2: Limits: A Numerical and
Graphical Approach
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
In previous courses, we have considered limits when
looking at the end behavior of a function. For instance,
the function y = 2 + e–x asymptotically approaches y = 2
as x approaches infinity. Mathematically, we could
write this as:
x
lim  2  e
x
Similarly:
lim  2  e x
x
2

In this section we will look at the behavior of the values
of f (x) as x approaches some number c, whether or not f
(c) is defined.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
sin x
Consider the function f  x   x as x approaches 0.
The function is not defined at x = 0, but it is defined for
values of x very close to 0. By examining these values
in the table below, we see that f (x) approaches 1 as x
approaches 0 or lim sin x  1
x0
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 76
Fill in the table and guess the value of the limit.
4. lim f   ,where f   
sin   
 0
θ
f (+θ)
f (–θ)
±0.002
±0.0001

3
±0.00005
±0.00001
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Recalling that the distance between points a and b is |b – a|,
If the values of f (x) do not converge to any finite value as x
approaches c, we say that lim f  x  does not exist (DNE)
x c
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 76
Verify each limit using the limit definition.
14. lim  5 x  7   8
x 3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Theorem 1 states two simple, yet important concepts,
which are illustrated in the graph below.
y
y=k
c
x
The value of constant k is independent of the value of x,
as is the value of c.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Graphical Investigation
Use the graphing calculator to graph the function in the
vicinity of the value of x in question.
If the graph appears to approach the same point from
either side of the value in question, then we may say
that the limit exists, and the limit may frequently be
estimated directly from the graph.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 76
Verify each limit using the limit definition.
14. lim  5 x  7   8
x 3
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Numerical Investigation
1. Construct a table of values of f (x) for x close to, but
less than c.
2. Construct a second table of values of f (x) for x close
to, but greater than c.
3. If both tables indicate convergence to the same
number L, we take L to be an estimate for the limit.
lim f  x   L
x c
Rogawski Calculus
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The graph and table below illustrate graphical and
numerical investigation of:
x9
lim
x 9
x 3
Rogawski Calculus
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The graph and table below illustrate graphical and
numerical investigation of:
lim x 2
x4
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
The graph and table below illustrate graphical and
numerical investigation of:
eh  1
lim
h 0
h
Rogawski Calculus
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The graph below doesn’t really answer the question of the
existence of a limit, but the table shows that it does not
exist as the values of sin (1/x) have opposite signs when
the same distance from zero in the positive and negative
directions is considered. Thus:
1
limsin D.N.E.
x0
x
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
In Figure 6, we see that f  x  does
not approach the same value of y
as x approaches 0 from the negative
and positive directions. In this case, lim f  x  D.N.E.,
x 0
but we may state: lim f  x   1 and lim f  x   1.
x0
x0
lim f  x  and lim f  x  are referred to as one-sided
x0
x0
limits and read as the limit as x approaches 0 from the
positive and negative directions, respectively.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Figure 7 shows the graph of a
piecewise function. The limits
as x approaches 0 and 2 do not
exist, but the limit as x
approaches 4 appears to exist,
as the graph on both sides of
x = 4 seems to converge on the
same value of y. The following one-sided limits also
appear to exist: lim f  x  , lim f  x  , and lim f  x  ,
x0
x 2
x 2
but lim f  x  DNE as the values of f (x) continue to
x 0
oscillate between ±1 as x gets ever closer to 0.
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Definition
The limit of a function at a given point exists if, and
only if, the one-sided limits at the point are equal.
Mathematically, we state:
lim f  x  exists  lim f  x   lim f  x 
xc 
xc
x c 
or
lim f  x   lim f  x   lim f  x 
xc
xc 
x c 
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 76
38. Determine the one-sided limits at c = 1, 2, 4 of the
function g(t) shown in the figure and state whether the
limit exists at these points.
y



x





Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Graphs (A) and (B) in Figure 8 illustrate functions with
infinite discontinuities. Graph (C) illustrates a function
with the interesting property: lim f  x    and
lim f  x   .
x 
x0
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Example, Page 76
Draw the graph of a function with the given limits.
48. lim f  x   , lim f  x   0, lim f  x   
x3
x1

x3
y



x









Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
Homework
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Homework Assignment #2
Read Section 2.3
Page 76, Exercises: 1 – 53 (EOO)
Quiz next time
Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company