SOLUTIONS MANUAL
TO
CALCULUS
AN INTUITIVE AND PHYSICAL APPROACH
2ND ED
BY
MORR Is Ku NE
, Introduction
I
1. The Solutions In This Manual.
The solutions of all the exercises in the text are given in full.
The primary reason is to save professors' time. Choosing exercises for
homework assignments can be a laborious matter if one must solve fifteen,
twenty or more to determine which are most suitable for his class. A
glance at the solutions will expedite the choices.
The second reason is that in many institutions calculus is taught
by teaching assistants who have yet to acquire both the training and ex-
perience in handling many of the mathematical and physical problems. The
availability of the solutions should help these teachers.
2. Suggestions For The Use Of The Text.
The one-volume format of this second edition should give profes-
sors more latitude in the choice of topics which might be suitable to the
interests of the students or to the length of the course.
Several types of choices might be noted. Because precalculus
courses have become more common since the publication of the first edi-
tion, some of the analytic geometry topics may no longer have to be
taught in the calculus course. The most elementary topics of analytics
have been put in an appendix to Chapter 3, Section 4 of Chapter 4, Sec-
tion 5 of Chapter 7, and the Appendix to Chapter 7. If familiar to the
students, all or some can be omitted.
Though I believe strongly in the importance of physical and, more
generally, real applications to supply rnotiVation and meaning to the
calculus, again class interests and available time must enter into de-
termining how many of these applications can be taken up. I have there-
fore starred all Lhose sections and chapters which can be omitted without
disrupting the continuity.
The last chapter, which is intended as an introduction to the
theory or rigor, can be taken up at almost any point after Chapter 10.
However, I personally believe that the intuitive approach should be
maintained throughout and that this chapter should be left for the last
and then taken up only if time permits.
The complete text is intended for a three semester, three hours a
week course. However, in view of the number of sections and chapters
that are not essential to the continuity the text can be used for shorter
courses including those offered in the fourth high school year.
3. Some Additional Topics.
Some physical applications which were included in the first edi-
tion were omitted in.the second one and replaced in the text proper by
applications to economics and to other social science areas. A few of
those omitted are reproduced here. They may be useful as suggestions
, 2
for additional work which bright or somewhat advanced students can under-
take, as fill-ins for periods which for one reason or another cannot be
used for regular work, or as material for a mathematics club talk. Ex-
ercises and solutions relevant to these additional topics are also in-
cluded here.
A. The Hanging Chain.
In the text proper we derived the equation of the chain or cable
suspended from two points (Chap. 16, Sect. 4) on the assumption that the
weight per unit length of the cable is the same all along the cable.
However, the theory developed there can be used to solve more general
problems. One is to determine the shape of the cable if the weight per
unit length or, one can say, the density per unit length is specified.
The second is, given the desired shape of the cable, how can we fix the
distribution of the mass along the cable so that it assumes the desired
shape? Both of these problems are readily solved with the theory at hand.
The derivation of (21), the equation of the cable, in the text
proper, presupposed that the weight of the cable per unit foot is con-
stant all along the cable. Let us now see what we can do when we let the
weight of the cable vary from point to point. Let us denote by w(s) the
function that gives the weight per unit foot at points. Then (11) and
(13) still hold, but (14) must be changed to read
(1) T
y
= fw(s)dx + D.
If we divide this equation by (11) and use the fact that T /T is y', we
obtain y X
(2) y' = .le_ fw(s)ds + D'
To
where D' is D/T • If the function w(s) is given, we can calculate
0
Jw(s)ds, The quantity D' can now be fixed by lettings be Oat y' = 0.
We now have y' as a function of s. Next we may proceed as we did in the
case where w(s) is a constant and seek to obtains as a function of x
through
ds
= ✓ l+y''
dx
but y' is now given by (2). If the integration can be performed ands is
obtained as a function of x, we can substitute this value of sin (2) and
attempt to obtain y as a function of x.
We can also solve the second problem. Suppose that we wish
to distribute weight along the cable so that the cable hangs in a given
shape; that is, we presume that we know the equation of the cable and we
wish to find w(s). To solve this problem, we differentiate (2) with
respect to x. On the left side differentiation with respect to x pro-
duces y", On the right side to differentiate with respect to x we use
the chain rule and differentiate with respect to sand multiply by ds/dx.
The derivative of fw(s)ds with respect to s must be w(s) because the
integral is that function whose derivative is w(s). Thus our result is
, 3
( 3)
Because we presume that we know the equation of the curve, we can calcu-
late y" and ds/dx. Hence we can find w(s), that is, the variation of
weight along the curve that produces the particular'shape of the hanging
cable, Of course, the shape of the cable need no longer be a catenary,
It is often called a non-uniform catenary.
The theory presented in this section is useful under more general
conditions than those so far described. In the derivations of the text
and of (2), we attributed the weight to the cable, However, the weight
w(s) might be the load on the cable, that is, the load of the bridge
itself, if the cable's weight is negligible, or the combined weight of
cable and load. In the case of the theory in the text this load would
have to be proportional to the arc length of the cable; that is, the load
would have to be the same for each unit of length of the cable, In the
case of (2), the load could vary along the cable or the combined weight
of load and cable could vary along the cable, and the function w(s) would
have to represent the variation of the total weight with arc length.
Exercises:
1, Find the law of variation of the mass of a string suspended from two
points at the same level and acted upon by gravity so that i t hangs
in the form of a semicircle. Suggestion: Take the semicircle to be
the lower half of x 2 +y 2 = 2ay and use (3).
2, The derivation given in (2) for a cabJe whose load varies with arc
length applies also to a cable whose load varies with horizontal dis-
tance from, say, the lowest point. Thus Tx = T and (1) becomes
0
T = Jw(x)dx+D. Then (2) is y' = (1/T Jfw(x)+D'. Given that the load
y 0
per horizontal foot is w(x) = ax 2 +b, find the equation of the cable.
Ans. y = (ax 4 +6bx 2 )/12T .
0
3. A heavy chain is suspended at its two extremities and forms an arc of
the parabola y = x 2 /4p. Show that the weight per horizontal foot is
constant. Suggestion: Use (3).
Solutions:
1. The lower half of the semicircle is given by y = a-/a 2 -x 2 • Then
y 1 = x(a 2 -x 2 )-l/ 2 and y" = a 2 (a 2 -x 2 ) - 312 , ds/dx = /l+y' 2 ::-a(a 2 -x 2 )-l/ 2 .
Then from (3), w(s) = aT /(a 2 -x").
0
1
2, Carry out the obvious integrations and use the facts that y and y are
0 at X = 0.
3. We can think of w(s)ds/dx as a function w(x) of x since s i s . Now use
(3). Since y = x 2 /4p, y" = ½P, and w(x) is a constant.
B. Projectile Motion in a Resisting Medium.
After taking up projectile motion in a vacuum (Chap. 18, Sect. 4)
one can take up the case of motion in a resisting medium. Since the
TO
CALCULUS
AN INTUITIVE AND PHYSICAL APPROACH
2ND ED
BY
MORR Is Ku NE
, Introduction
I
1. The Solutions In This Manual.
The solutions of all the exercises in the text are given in full.
The primary reason is to save professors' time. Choosing exercises for
homework assignments can be a laborious matter if one must solve fifteen,
twenty or more to determine which are most suitable for his class. A
glance at the solutions will expedite the choices.
The second reason is that in many institutions calculus is taught
by teaching assistants who have yet to acquire both the training and ex-
perience in handling many of the mathematical and physical problems. The
availability of the solutions should help these teachers.
2. Suggestions For The Use Of The Text.
The one-volume format of this second edition should give profes-
sors more latitude in the choice of topics which might be suitable to the
interests of the students or to the length of the course.
Several types of choices might be noted. Because precalculus
courses have become more common since the publication of the first edi-
tion, some of the analytic geometry topics may no longer have to be
taught in the calculus course. The most elementary topics of analytics
have been put in an appendix to Chapter 3, Section 4 of Chapter 4, Sec-
tion 5 of Chapter 7, and the Appendix to Chapter 7. If familiar to the
students, all or some can be omitted.
Though I believe strongly in the importance of physical and, more
generally, real applications to supply rnotiVation and meaning to the
calculus, again class interests and available time must enter into de-
termining how many of these applications can be taken up. I have there-
fore starred all Lhose sections and chapters which can be omitted without
disrupting the continuity.
The last chapter, which is intended as an introduction to the
theory or rigor, can be taken up at almost any point after Chapter 10.
However, I personally believe that the intuitive approach should be
maintained throughout and that this chapter should be left for the last
and then taken up only if time permits.
The complete text is intended for a three semester, three hours a
week course. However, in view of the number of sections and chapters
that are not essential to the continuity the text can be used for shorter
courses including those offered in the fourth high school year.
3. Some Additional Topics.
Some physical applications which were included in the first edi-
tion were omitted in.the second one and replaced in the text proper by
applications to economics and to other social science areas. A few of
those omitted are reproduced here. They may be useful as suggestions
, 2
for additional work which bright or somewhat advanced students can under-
take, as fill-ins for periods which for one reason or another cannot be
used for regular work, or as material for a mathematics club talk. Ex-
ercises and solutions relevant to these additional topics are also in-
cluded here.
A. The Hanging Chain.
In the text proper we derived the equation of the chain or cable
suspended from two points (Chap. 16, Sect. 4) on the assumption that the
weight per unit length of the cable is the same all along the cable.
However, the theory developed there can be used to solve more general
problems. One is to determine the shape of the cable if the weight per
unit length or, one can say, the density per unit length is specified.
The second is, given the desired shape of the cable, how can we fix the
distribution of the mass along the cable so that it assumes the desired
shape? Both of these problems are readily solved with the theory at hand.
The derivation of (21), the equation of the cable, in the text
proper, presupposed that the weight of the cable per unit foot is con-
stant all along the cable. Let us now see what we can do when we let the
weight of the cable vary from point to point. Let us denote by w(s) the
function that gives the weight per unit foot at points. Then (11) and
(13) still hold, but (14) must be changed to read
(1) T
y
= fw(s)dx + D.
If we divide this equation by (11) and use the fact that T /T is y', we
obtain y X
(2) y' = .le_ fw(s)ds + D'
To
where D' is D/T • If the function w(s) is given, we can calculate
0
Jw(s)ds, The quantity D' can now be fixed by lettings be Oat y' = 0.
We now have y' as a function of s. Next we may proceed as we did in the
case where w(s) is a constant and seek to obtains as a function of x
through
ds
= ✓ l+y''
dx
but y' is now given by (2). If the integration can be performed ands is
obtained as a function of x, we can substitute this value of sin (2) and
attempt to obtain y as a function of x.
We can also solve the second problem. Suppose that we wish
to distribute weight along the cable so that the cable hangs in a given
shape; that is, we presume that we know the equation of the cable and we
wish to find w(s). To solve this problem, we differentiate (2) with
respect to x. On the left side differentiation with respect to x pro-
duces y", On the right side to differentiate with respect to x we use
the chain rule and differentiate with respect to sand multiply by ds/dx.
The derivative of fw(s)ds with respect to s must be w(s) because the
integral is that function whose derivative is w(s). Thus our result is
, 3
( 3)
Because we presume that we know the equation of the curve, we can calcu-
late y" and ds/dx. Hence we can find w(s), that is, the variation of
weight along the curve that produces the particular'shape of the hanging
cable, Of course, the shape of the cable need no longer be a catenary,
It is often called a non-uniform catenary.
The theory presented in this section is useful under more general
conditions than those so far described. In the derivations of the text
and of (2), we attributed the weight to the cable, However, the weight
w(s) might be the load on the cable, that is, the load of the bridge
itself, if the cable's weight is negligible, or the combined weight of
cable and load. In the case of the theory in the text this load would
have to be proportional to the arc length of the cable; that is, the load
would have to be the same for each unit of length of the cable, In the
case of (2), the load could vary along the cable or the combined weight
of load and cable could vary along the cable, and the function w(s) would
have to represent the variation of the total weight with arc length.
Exercises:
1, Find the law of variation of the mass of a string suspended from two
points at the same level and acted upon by gravity so that i t hangs
in the form of a semicircle. Suggestion: Take the semicircle to be
the lower half of x 2 +y 2 = 2ay and use (3).
2, The derivation given in (2) for a cabJe whose load varies with arc
length applies also to a cable whose load varies with horizontal dis-
tance from, say, the lowest point. Thus Tx = T and (1) becomes
0
T = Jw(x)dx+D. Then (2) is y' = (1/T Jfw(x)+D'. Given that the load
y 0
per horizontal foot is w(x) = ax 2 +b, find the equation of the cable.
Ans. y = (ax 4 +6bx 2 )/12T .
0
3. A heavy chain is suspended at its two extremities and forms an arc of
the parabola y = x 2 /4p. Show that the weight per horizontal foot is
constant. Suggestion: Use (3).
Solutions:
1. The lower half of the semicircle is given by y = a-/a 2 -x 2 • Then
y 1 = x(a 2 -x 2 )-l/ 2 and y" = a 2 (a 2 -x 2 ) - 312 , ds/dx = /l+y' 2 ::-a(a 2 -x 2 )-l/ 2 .
Then from (3), w(s) = aT /(a 2 -x").
0
1
2, Carry out the obvious integrations and use the facts that y and y are
0 at X = 0.
3. We can think of w(s)ds/dx as a function w(x) of x since s i s . Now use
(3). Since y = x 2 /4p, y" = ½P, and w(x) is a constant.
B. Projectile Motion in a Resisting Medium.
After taking up projectile motion in a vacuum (Chap. 18, Sect. 4)
one can take up the case of motion in a resisting medium. Since the
