Homework
Homework Assignment #1 Review Section 2.1/Read Section 2.2 Page 66, Exercises: 1 – 9 (Odd), 13 – 29
(EOO)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 661. 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]?
(b) Compute the average velocity over [2, 2.5].
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
2
2.5 16 2.5 16 6.25 100
2 16 2 64 2.5 2 36
s ft
s ft s s ft
36avg. velocity 72
0.5
ftft s
s
Homework, Page 661. (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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2
2
2
2.01 2 64.6416 6464.16
2.01 2 0.01
2.005 2 64.6416 6464.08
2.005 2 0.005
2.00001 2 64.00064 6464.0
2.00001 2 0.00001
Instantaneous velocity at 2 is 64
s sROC
s sROC
s sROC
t ft s
Homework, Page 663. Let . Estimate the instantaneous ROC of v with respect to T when T = 300.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
20 300.01 300 0.005770.577
300.01 300 0.01
20 300.0001 300 0.00005770.577
300.0001 300 0.0001
20 300.00001 300 0.000005770.577
300.00001 300 0.00001
Instantaneous ROC at 300 is 0.577
mROC s
mROC s
mROC s
mT s
20v T
Homework, Page 66A 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).
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
2.5 0.5
2.5 0.50.6
0.3 /2
h hROC
m s
x
y
(2.5, 6.875)(0.5, 6.275)
Homework, Page 667. With an initial deposit of $100, the balance in a bank account after t years is f(t) = 100(1.08)t dollars.
(a) What are the units of the ROC of f(t)?
The units of the ROC of f (t) are dollars per year.
(b) Find the average ROC over [0, 0.5] and [0, 1].
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
0.5 0 103.92 1007.85$ /
0.5 0 0.51 0 108 100
8$ /1 0 1
f fROC yr
f fROC yr
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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
0.5 0.4999 103.9230 103.9222
0.5 0.4999 0.00018.00$ /
0.5001 0.5 103.9238 1009230
0.5 0.0001 18.00$ /
Instantaneous ROC 0.5 0.08
f fROC
yr
f fROC
yr
t
Homework, Page 66
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
29. 4 3; 2P x x t
2 1.9999 13 12.998415.9996
2 1.9999 0.00012.0001 2 13.0016 13
16.00040.5 0.0001 0.0001
Instantaneous ROC 2 16
P PROC
P PROC
x
Estimate the instantaneous ROC at the indicated point.
Homework, Page 66
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
13. ; 0xf x e x
0.0001 0 1.0001 11
0.0001 0 0.00010 0.0001 1 0.9999
10 0.0001 0.0001
Instantaneous ROC 0 1
f fROC
f fROC
x
Estimate the instantaneous ROC at the indicated point.
Homework, Page 6617. 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. Rogawski Calculus
Copyright © 2008 W. H. Freeman and Company
x
y
50
40,000 ft
Tem
pera
ture
-50
Homework, Page 6621. Assume that the period T (in sec) of a pendulum is
where L is the length (in m).
(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 CalculusCopyright © 2008 W. H. Freeman and Company
3
2T L
Homework, Page 6625. 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 , respectively.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
0.09 0.120.045 and 0.02
2 6
Homework, Page 6625. (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 CalculusCopyright © 2008 W. H. Freeman and Company
Homework, Page 6629. 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 CalculusCopyright © 2008 W. H. Freeman and Company
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Chapter 2: LimitsSection 2.2: Limits: A Numerical and
Graphical Approach
Jon Rogawski
Calculus, ET
First Edition
Rogawski CalculusCopyright © 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.
lim 2 2x
xe
lim 2 x
xe
Mathematically, we could write this as:
Similarly:
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 CalculusCopyright © 2008 W. H. Freeman and Company
Consider the function 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
sin xf x
x
0
sinlim 1x
x
x
Example, Page 76Fill in the table and guess the value of the limit.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
30
sin4. lim ,where f f
θ ±0.002 ±0.0001 ±0.00005 ±0.00001
f (+θ)
f (–θ)
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Recalling that the distance between points a and b is |b – a|,
lim (DNE)x c
f x does not exist
If the values of f (x) do not converge to any finite value as x approaches c, we say that
Example, Page 76Verify each limit using the limit definition.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3
14. lim 5 7 8x
x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Theorem 1 states two simple, yet important concepts, which are illustrated in the graph below.
x
y
y = k
c
The value of constant k is independent of the value of x, as is the value of c.
Graphical InvestigationUse the graphing calculator to graph the function in the vicinity of the value of x in question.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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.
Example, Page 76Verify each limit using the limit definition.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
3
14. lim 5 7 8x
x
Numerical Investigation1. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
limx c
f x L
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The graph and table below illustrate graphical and numerical investigation of:
9
9lim
3x
x
x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The graph and table below illustrate graphical and numerical investigation of:
2
4limx
x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
The graph and table below illustrate graphical and numerical investigation of:
0
1lim
h
h
e
h
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
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:
0
1limsin D.N.E.x x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
In Figure 6, we see that does
not approach the same value of
as approaches 0 from the negative
and positive directions.
f x
y
x
0
In this case, lim D.N.E.,x
f x
0 0
but we may state: lim 1 and lim 1.x x
f x f x
0 0
lim and lim are referred to as one-sided
limits and read as the limit as approaches 0 from the
positive and negative directions, respectively.
x xf x f x
x
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
Figure 7 shows the graph of a piecewise function. The limitsas x approaches 0 and 2 do notexist, 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:
0 2 2
lim , lim , and lim , x x x
f x f x f x
0
limx
f x
but DNE as the values of f (x) continue to
oscillate between ±1 as x gets ever closer to 0.
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:
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
lim exists lim limx c x c x c
f x f x f x
or
lim lim limx c x c x c
f x f x f x
Example, Page 7638. 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.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
x
y
Rogawski CalculusCopyright © 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: 0
lim andx
f x
lim .
xf x
Example, Page 76Draw the graph of a function with the given limits.
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company
1 3 3
48. lim , lim 0, limx x x
f x f x f x
x
y
Homework
Homework Assignment #2 Read Section 2.3 Page 76, Exercises: 1 – 53 (EOO) Quiz next time
Rogawski CalculusCopyright © 2008 W. H. Freeman and Company