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 = fw(s)dx + D. y If we divide this equation by (11) and use the fact that T /T is y', we y X obtain (2) y' = .le_ fw(s)ds + D' To where D' is D/T0• If the function w(s) is given, we can calculate 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 it hangs in the form of a semicircle. Suggestion: Take the semicircle to be the lower half of x2+y2 = 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 = T0 and (1) becomes 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) = ax2+b, find the equation of the cable. Ans. y = (ax4+6bx2)/12T . 0 3. A heavy chain is suspended at its two extremities and forms an arc of the parabola y = x2/4p. 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-/a2-x2• Then y1 = x(a2-x2)-l/2 and y" = a2(a2-x2)-312, ds/dx = /l+y'2::-a(a2-x2)-l/2. Then from (3), w(s) = aT0/(a2-x"). 2, Carry out the obvious integrations and use the facts that y1 and y are 0 at X = 0. 3. We can think of w(s)ds/dx as a function w(x) of x since sis. Now use (3). Since y = x2/4p, y" = ½P, and w(x) is a constant. B. Projectile Motion in a Resisting Medium. After taking up projectile motion in a vacuum (Chap. one can take up the case of motion in a resisting medium. 18, Sect. 4) Since the
