Comments on Chapter 4

Page 99

The last sentence in the first paragraph should be singular: “The second derivative \(f^{(2)}\) (respectively, \(y^{(2)}\)) is usually written as \(f''\) (respectively, \(y''\)), and when it exists at some point \(a\), we shall say that \(f\) is twice differentiable at \(a\).”

Page 104

The first sentence (part c of Exercise 4.1.0) is missing a terminal period.

In Exercise 4.1.1b, the unstated domain of \(f\) is, of course, the nonnegative real numbers. In other words, you can replace the restriction that \(a\ne 0\) by the restriction that \(a\gt 0\).

Page 108

A terminal period is missing in Exercise 4.2.7a.

In Exercise 4.2.8, notice that the two parts of the question ask very different things. The first part asks about differentiability on an interval in the sense of Definition 4.6 on page 102, so you have to study a one-sided limit at 0. The second part asks about differentiability at a point in the sense of Definition 4.1 on page 98, so you have to study a two-sided limit at 0.

Page 110

In the second paragraph, instead of invoking “a similar proof”, one could reduce the case of a minimum to the case of a maximum by passing to the function \(-f\).

Page 111

In the paragraph following the proof of Theorem 4.15, the reference to “Remark 14.33” should be to Remark 14.32.

Page 112

In the proof of Theorem 4.17, the claim that part ii follows “by part i” is wrong, for part i is silent about what happens when the derivative equals zero. But part ii follows by the proof of part i. Namely, if \(f'\) is identically zero, then \[f(x_2)-f(x_1)=f'(c)(x_2-x_1)=0,\] so \(f(x_2)=f(x_1)\), and since the points \(x_2\) and \(x_1\) are arbitrary, the function \(f\) is indeed constant.

In the proof of Theorem 4.18, the statement “By symmetry” should come at the beginning, before \(c\) is fixed, and it should really say, “By symmetry, it suffices to prove part ii”.

In the second line from the bottom of the page, the “i.e.” is inappropriate, for this abbreviation stands for “that is”, but the phrases “nowhere continuous” and “uncountably many points of discontinuity” are not equivalent. Instead of “i.e.”, either “thus” or “in particular” would serve.

Page 113, proof of Theorem 4.19

The exposition is not consistent about whether the interval is closed \([a,b]\) or open \((a,b)\). The argument is cleaner if the open interval is used everywhere.

In the second paragraph, the restriction in the definition of the set \(A_j\) should be “\(x\in (a,b)\)” rather than “\(x\in\mathbf{R}\)”.

In the third paragraph, “in \([a,b]\)” should be “in \((a,b)\)” to be compatible with the second paragraph. Moreover, it is not necessarily true that the supposed infinite set \(A_{j_0}\) can be exhibited as an increasing sequence \(x_1\lt x_2\lt \ldots\). (It might be a decreasing sequence.) What is true (which suffices for the proof) is that for every choice of a natural number \(n\), there are points \(x_1\), \(\ldots\), \(x_n\) in the set such that \(x_1\lt x_2\lt \ldots \lt x_n\).

Page 113, proof of Example 4.20

In the second line, “strictly increasing on \((0,\infty)\)” should say “strictly increasing on \([0,\infty)\)”.

Page 114, proof of Theorem 4.21

In the second line of the proof, “\(t\in[-1,\infty)\)” should be “\(t\in[0,\infty)\)”.

Page 115, Strategy for 4.23

In the third sentence, “extrema” is plural, so the matching verb should be “occur” (not “occurs”). Moreover, it would be more precise to say “occur only where” rather than “occur where”.

Page 116

In Exercise 4.3.1, it is reasonable to assume known the derivatives of the sine, exponential, and logarithm functions. The derivative of the sine function is officially known from Exercise 4.2.9 on pages 108–109, but the derivatives of the logarithm and exponential are not officially known until later (Exercise 4.5.5 on pages 128–129).

In the statement of Exercise 4.3.10, the cited Exercise 4.1.8 defines local maximum but not proper local maximum. The latter term is defined on page 178 in the paragraph preceding Theorem 5.63.

The hint (on page 650) for Exercise 4.3.11 states that the exercise is the only one in the section having no relation to the mean-value theorem, but Exercise 4.3.7 is a second exercise in the same category.

Section 4.4

If you look at other sources, you may see “l'Hôpital” spelled “l'Hospital”. The letter “s” is silent.

Page 118

At the end of the proof, the property that \(x\ne c\) is needed, but not in the clause where it appears. The property is used only in the subsequent clause to guarantee that one can divide by \(F'(c)\).

Four lines from the bottom of the page, it is not “the denominator” that “gets smaller”; rather, the fraction \(1/(n+1)!\) gets small because the denominator gets large.

Three lines from the bottom of the page, it is not the case “in general” that the approximation improves when \(n\) gets large; rather, the improvement happens in special cases when there is control on the size of the numerator \(f^{(n+1)}(c)\). See Remark 7.41 on page 250 for an indication of what can go wrong in general.

Page 119

In equation (18), the letter \(x\) is overloaded in \(P_n^{e^x,0}(x)\). Using the notation that has been introduced, one could write simply \(P_n(x)\) on the left-hand side. An analogous comment applies to equation (19).

Notice that there is no “right” answer to Example 4.25c, for if a value of \(n\) works, then so does a larger \(n\). Actually \( |e^x - P_n(x)|\lt 0.00005\) when \(x\in[-1,1]\) if \(n\ge 7\), although to make the crude upper bound \(3/(n+1)!\) less than \(0.00005\) requires taking \(n\) one unit larger.

In Example 4.26, if you notice that \(P_{2n+1} = P_{2n+2}\), then you can improve the upper bound in part b to \(1/(2n+3)!\). Hence in part c (at the top of page 120), taking \(n\) equal to 2 suffices.

Page 120, Theorem 4.27

The statement of the theorem is supposed to allow additionally the possibility that \(A=-\infty\).

In the proof, two lines above the start of Case 1, the conditions \(x,y\ge a\) and \(x,y\le a\) should be \(x,y\gt a\) and \(x,y\lt a\) with strict inequalities, for the hypotheses of the theorem give no information about the value of the function \(g\) precisely at the point \(a\).

Page 121

In line 5, the symbol \(t\) should be \(y\).

Actually, there is a gap in the proof at lines 4–5. It could happen that there are distinct values of \(k\) and \(n\) for which \(x_k=x_n\). In that case, the modified equation (20) breaks down, for there is a zero in the denominator. One needs to go back to the proof of the Sequential Characterization of Limits and check that it suffices to consider sequences of distinct points.

Five lines after equation (22), the inequality \(n\ge N_0\) should be \(n\gt N_0\) (with strict inequality), since \(c_{N_0,N_0}\) is undefined.

It is interesting to note that the proof of Case 2 does not use the hypothesis that \( \lim_{x \to a} f(x) =\infty\).

Page 122

Example 4.29 is actually of the form \( (-\infty)/\infty\), not \(\infty/\infty\), so formally it is not covered by the statement of Theorem 4.27. But of course one can factor out a minus sign.

Page 123

As written, the solution of Example 4.30 shows only that if the original limit exists, then the value is \(e^3\). A rearrangement of the argument will show that the limit does exist.

Example 4.31 is misstated. The limit can only be a one-sided limit as \(x\to 1{+}\), for the functions \( (\log x)^{1-x}\) and \(\log \log x\) are not real functions when \(x\lt 1\).

The answer to Exercise 4.4.0a in the back of the book says “false”, and this answer is correct, but only because \(x/\log x\) is not a real function when \(x\lt 0\).

In Exercise 4.4.1b, the upper bound can be improved to \(1/(2n+2)!\) since \(P_{2n}=P_{2n+1}\).

For Exercise 4.4.2c (on the next page), the answer in the back of the book has a misleading number of significant digits, since the method gives a valid but usually unnecessarily large value of \(n\). A more reasonable choice for \(n\) would be the round figure of 2000. A more sophisticated analysis shows that the sharp cutoff value for \(n\) is actually 1000.

Page 124

The inequality in Exercise 4.4.4 actually holds when \(x\) is an arbitrary positive number, not just when \(0\lt x \lt\pi\), but the proof of this generalization requires extra work.

In Exercise 4.4.5d, notice that the limit exists as a two-sided limit.

Exercise 4.4.5f is nonsense, for \( (\log x)^x\) is not a real function when \(x\) is in the interval \( (0,1)\). A sensible problem would be \( \lim_{x\to 0{+}} |\log x|^x\).

In Exercise 4.4.6b, notice that the roles of the two functions get interchanged.

Page 125

In Exercise 4.4.8, the conclusion remains true when \(n\) is even, but the proof requires an extra step.

In Exercise 4.4.9a, the constraint that \(0\lt\delta\le 1\) is unnecessarily restrictive. The inequality holds when \(\delta \ge 0\), or even for all \(\delta\) if you replace \(\delta^3\) with \(|\delta|^3\). Similarly, in part b, the upper bound on \(\delta\) is not needed.

In Exercise 4.4.10, the hint probably should be a statement about derivatives: “\(g'(x)/f'(x) \to 0\) when \(f(x)/g(x) \to \pm \infty\)”.

Page 126

In the first sentence, it is not that we may suppose the existence of two points in the interval; rather, we have supposed in the hypothesis of the theorem that the interval is nondegenerate.

In the last paragraph on the page, line 3, for “\(f\) is not continuous” read “\(f^{-1}\) is not continuous”.

In the last paragraph on the page, there is of course a special case if \(y_0\) happens to be an endpoint of the interval \(J\); one of the directions of continuity is then vacuous.

Page 127

In the proof of Theorem 4.33, the two references to Theorem 4.33 (at lines 1 and 5) should be to Theorem 4.32.

Harold P. Boas