5 Steps To A 5 Ap Calculus Ab 2023 William Ma Full Chapter
5 Steps To A 5 Ap Calculus Ab 2023 William Ma Full Chapter
5 Steps To A 5 Ap Calculus Ab 2023 William Ma Full Chapter
William Ma
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Preface
Acknowledgments
About the Author
Introduction: The Five-Step Program
Appendix
Bibliography
PREFACE
As an old martial artist once said, “First you must understand. Then you
must practice.” Have fun and good luck!
ACKNOWLEDGMENTS
I could not have written this book without the help of the following people:
WILLIAM MA has taught calculus for many years. He received his BA and
MA from Columbia University. He was the chairman of the Math
Department at the Herricks School District on Long Island, New York, for
many years before retiring. He also taught as adjunct instructor at Baruch
College, Fordham University, and Columbia University. He is the author of
several books, including test preparation books for the SAT, ACT, GMAT,
and AP Calculus AB and BC. He is currently a math consultant.
INTRODUCTION: THE FIVE-
STEP PROGRAM
The book concludes with a summary of math formulas and theorems related
to the AP Calculus exam. (Please note that the exercises in this book are
done with the TI-89 Graphing Calculator.)
This icon points out a very important concept or fact that you should not
pass over.
This icon calls your attention to a strategy that you may want to try.
This icon indicates a tip that you might find useful.
STEP 1
IN THIS CHAPTER
Summary: Learn what topics are tested in the exam, what the format is,
which calculators are allowed, and how the exam is graded.
Key Ideas
The AP Calculus AB exam has 45 multiple-choice questions and 6 free-
response questions. There are two types of questions, and each makes up
50% of the grade.
Many graphing calculators are permitted on the exam, including the TI-
98.
You may bring up to two approved calculators for the exam.
You may store programs in your calculator, and you are not required to
clear the memories in your calculator for the exam.
A ND Cooksey took me to tea, that same day, with his little old friend Mr.
Fitch. I was greatly charmed by Mr. Fitch, who was small and frail and
wore a dust-coloured beard; and his first suspicion of me (he was afraid
of the young) was allayed when he found that I knew and adored a particular
Roman church or two, remote and neglected, which he didn’t suppose that a
casual intruder like myself would have discovered. I remember how
Cooksey threw an arm of patronage around me and explained that he had
been my guide to the holy places of the city; but Mr. Fitch caught my eye
with a twinkle of intelligence, quickly withdrawn, which set up a happy
understanding between us on the spot. He did the honours of his apartment
with pleasant chirps and fidgets, hospitably bustling about the tea-tray,
beaming and fussing and apologizing, with bird-like cries to the stout maid-
servant who was energetically seconding his welcome.
Mr. Fitch was a scholar, a student, who worked daily in the library of the
Vatican. I believe he was a hundred years old, and indeed he looked it; but
he didn’t appear to have grown old, only to have suffered a slow deposit of
time to accumulate upon his person. Time was deep upon his hair and face
and clothes; but a few score years more or less could have made no
difference to the cheerful little bird-spirit in his breast, and it was because he
was shy and defenceless, not because he was old, that he feared the
onslaught of the young. A young person, however, who was found to have
made his way unaided to the church of San Cesareo, far away among the
vineyards on the verge of the city, was one towards whom Mr. Fitch could
hop and twitter in kindly confidence, and he did so. Before we parted he
invited me to lunch with him a day or two later, and I fully understood that
this was for him a remarkable demonstration. “Gina!” he called, and Gina,
the voluble maid-servant, came from the kitchen with a run, to receive his
command concerning the festival. She was delighted, she swept me into the
happy plan, she seemed to be immediately arranging a treat for two merry
little children, for me and Mr. Fitch. We were like children between her
broad palms, all but hugged to her bosom; and with dancing eyes she told us
to leave it all to her—she would do something splendid. “Gina will see to it,”
said Mr. Fitch; and he asked her whether he shouldn’t invite some other
young thing to join the party—what about the giovanotto who had called the
other day? “Quel poverino?” said Gina—yes, the very thing. So we should
be a party of three; and Gina clapped her hands and ran back to the kitchen,
as though to set about her preparations there and then.
Mr. Fitch lived in the Via Giulia, deep in the depth of Rome, not far from
the great mass of the Farnese palace. He had the craziest little apartment, a
tangle of rooms with bare tiled floors, in which his funny frumpy English
furniture, which might have come straight (and no doubt it had) from his
mother’s parlour at Cheltenham, looked strangely shocked and ill at ease.
Forty years of the Via Giulia (it can hardly have been less) had not
reconciled the mahogany overmantel and the plush-topped tea-table to the
ramshackle ways of foreign life; mutely they protested, keeping themselves
to themselves, wrapped in their respectability. Mr. Fitch, I think, had never
so much as noticed their plight; he sat on a chair, he made tea on a table, and
one chair or table was as good as another for the purpose. He himself looked
homely and frumpy enough, to be sure, lodged there under the wing, so to
speak, of Julius the Pope; but he didn’t feel at a loss, and he tripped along
the proud-memoried street of his abode, with his decent English beard and
his little mud-gaiters on his boots, as brisk as a sparrow. He accompanied us
down the street and left us to go and invite the “poverino” to meet me at
lunch; I see him waving us good-bye at some grand dark street-corner, where
he turned and pattered off on his errand. Cooksey treated him with large
protective kindness and contempt, out of which the old man seemed to slip
with a duck of his head and a gleam of fright and amusement in his two
bright eyes.
The luncheon-party, a day or two later, was a great success. I climbed to
the apartment on the stroke of the hour, but the other young man was already
there before me, and Mr. Fitch ceremoniously performed an introduction.
The name of the youth was Maundy, and he proved to be one of those
aspiring priests, novices, seminarists—I don’t know what their rightful name
may be, but you know them well, you remember how they converge in long
lines upon the Pincian Hill towards evening, how they pick up their skirts
and romp with the gaiety of the laity upon the greensward of the Villa
Borghese. Maundy was his name, and he didn’t look, for his part, as though
he had had much romping; he was pale and meagre, he reclined in a
contorted cat’s-cradle of thin arms and legs on one of Mr. Fitch’s fringed and
brass-nailed arm-chairs. If Gina’s word for him meant a poor young
specimen of chilly lankness she was right; his limp black soutane (is it a
soutane?) couldn’t disguise his sharp-set knees or the lean little sticks of his
arms. He jumped up, however, quite alert and spritely for our introduction,
and he greeted me with a friendly high-piping composure that made it
unnecessary to pity him. I had begun to pity him, as I always do feel
compassionate, so gratuitously, at the sight of his kind—at the sight of the
young novices, caught and caged and black-skirted in their innocence,
renouncing the world before they have had the chance to taste it; but
Maundy turned the tables upon me in a moment, and he revealed himself as
a perfectly assured young son of the world, with whom I had no call to be
sympathetically considerate. He shook hands with me, using a gesture which
at that time, so long ago, was reputed a mark of distinction—I forget how it
went exactly, but I think the pair of clasped hands was held high and waved
negligently from side to side. Maundy achieved it with an air, not failing to
observe that I had stepped forward to meet him with the ordinary pump-
handle of the vulgar.
And so we sat down to Gina’s admirable meal, and Mr. Fitch was in a
flutter of pleasure and excitement, and Maundy talked and talked—he led
the conversation, he led it almost beyond our reach, he led it so masterfully
that it hardly escaped him at all. Mr. Fitch lost his hold on it at once; he sat
with his head on one side, making small clucking noises of assent and
question now and then, while Maundy piped and swept away from us in his
monologue. But no, I oughtn’t to say that he left us both behind, for he kept
turning and waiting for me to catch him up, he flatteringly showed me that
he wished for my company. “Such a blessing,” he said, “to get away from
piety”—and he intimated with a smile that it was I who represented the
impious. He desired my company, not my talk; and he might have been
breaking out with the relief of unwonted freedom, soaring forth into topics
that were discouraged in the congregation of the poor caged lambs; and I
dare say he enjoyed the spread of his wings among the tinted and perfumed
vapours of his fancy. It was all beyond Mr. Fitch, who clearly couldn’t
explain him with my ready mixture of metaphor; Mr. Fitch was bewildered.
But to me the fancies of Maundy were sufficiently familiar; I knew the like
of them from of old, and I fear we both took a certain pleasure in noting the
bedazzlement of our host. The good soul, he sat and plied us with food and
wine, while Maundy rattled away in his emancipation and I assumed the
most impious look (I had small opportunity for more than looks) that I could
accomplish.
Maundy threw off a light word or two about his place of residence and
instruction in Rome—the seminary, the college, I forget how he referred to
it. He seemed disdainful of all its other inmates; he couldn’t regard them as
companions for a person of intelligence and fine feeling. How he came to
have placed himself among them, submitting to their rule, he didn’t explain
at the time, but I afterwards made out a little of his history. He had written a
great deal of poetry at Oxford, and he had kept an old silver oil-lamp
burning night and day before a Greek statuette, and he had had his favourite
books bound in apricot linen, and he had collected thirty-five different kinds
of scented soap—and I know it sounds odd, but he appeared to consider
these achievements as natural stages on the path to Rome. He didn’t go quite
so far as to say that he repented of having made the journey and embraced
the Roman discipline; but after a year in the college or the seminary his
mind, I think, was in a state of more painful confusion than he allowed me to
see. Somehow the argument at one end, the Oxford end, where he had
draped his dressing-table with an embroidered rochet (he told me so),
seemed to have so little in common with the argument at the other, the
Roman end, where he walked out with his young associates for exercise in
the Villa Borghese and not one of them had heard of the poetry of Lionel
Johnson; and somehow he had perceived the discrepancy without
discovering where the chain of his reasoning had failed, and in the privacy of
his discontent he was still floundering backwards and forwards, trying to
persuade himself of the soundness of all the links—and perhaps seeking with
a part of his mind (a growing part) to be convinced that he had reasoned
wrong. Something of this kind, I believe, was fretting his life in Rome, and
how it may have ended I never knew; he didn’t confide his troubles to me—
he simply hailed me as one who would possibly understand what it meant to
him to have once, in an eating-house of Soho, been introduced to Aubrey
Beardsley.
“The passion of his line,” he said, referring to that artist; and again, “The
passion of his line!”—and he described the scene in Soho, mentioning that
the impression had wrought upon him so potently that afterwards he had sat
up all night, with some golden Tokay beside him in a blue Venetian glass
(not drinking it, only refreshed by the sight of it), and had written a poem, a
sonnet of strange perfumes and fantastic gems, which he had dedicated in
Latin to the hero of the evening. And then he had gone out into the dawn,
and had wandered through Leicester Square to Covent Garden, and had
bought a bunch of mauve carnations; and he had thought of sending them,
with the sonnet, to the master who had inspired him—but then he had
returned to his lodging and had burnt the sonnet, heaping the carnations for a
pyre, having resolved to guard the experience, whole and rounded and
complete, in the secrecy of a faithful memory. He pointed out that to share
these things is to lose them; as soon as you turn them into words for
another’s eye they cease to be perfectly yours, they are dissipated into the
common air; which was why a friend of his, at Oxford, had insisted that one
should write no words, paint or carve no colour or line, but only make one’s
images and pictures and poems out of the rainbow-tinted substance of
memory, that exquisite material always awaiting and inviting the hand of an
artist. So one avoids, you see, the sick disillusion of the writer who flings
forth his maiden fancy to the ribaldry of the crowd; and Maundy himself had
tried to rise to this height of disinterested passion, and in the dying perfume
of the mauve carnations he had sacrificed what he saw to be a vulgar
ambition. Oh yes, depend upon it, the greatest works of art have never been
seen of any but their maker; and to Maundy it was a beautiful thought, the
thought of the white secret statues locked away by the thousand in their
secluded shrines, safe from the world, visited now and again by the one and
only adorer who possessed the key. “But stay,” said Mr. Fitch, “have you
considered—” oh yes, Maundy had felt the weight of that objection, and
Dickson after all (Dickson was the friend at Oxford) had written and printed
his volume, but that was because he had found no other way to rid himself of
an obsession; the white statue in his case had become more real than life,
and he had cast it forth to retain—to retain, you might say, his sanity.
Well, we must publish or go mad; that is the melancholy conclusion. Mr.
Fitch stared doubtfully, and I shook my head like one whose hold upon his
senses is precarious indeed. Maundy was quick to interpret my movement,
and it encouraged him to yet giddier flights. He was hovering upon the
climax of one of these when Gina happened to come clattering in with a
dish; and she paused, sinking back upon her heels, the dish held high before
her, and she threw up her head and she flashed out such an amusing
challenging bantering look at Maundy, where he flourished his thin fingers
in the zest of his eloquence, that I have never forgotten the picture of her
mirth and her plumpness as it was framed at that moment in the doorway.
“Ah, the poor little fellow,” she said to herself, “he loves to talk!” And she
too began to talk, breaking into his monologue with unabashed and ringing
frankness; she set down her dish on the table with a dancing gesture,
whipping her hands away from it like an actress in a play, and she stood by
his side, patting him on the shoulder, approving him, scolding him, bidding
him eat, eat!—and Maundy turned round to her with a peal of sudden light
laughter, a burst of naturalness that changed his whole appearance; so that
Gina had transformed the temper of the party and had raised it at once to a
breezier level of gaiety than it would ever have touched without her. It was
delightful; I couldn’t understand a word she said, for her words flew shining
and streeling over our heads as quick as thought, and I dare say Maundy
answered their spirit rather than their meaning; but he responded well, he
had some good neat conversational turns of idiom that he shot back at her
with a knowing accent, and she chuckled, she threatened him, she bustled
out of the room with a smile for me and Mr. Fitch and a last fling of
playfulness over her shoulder for Maundy. Mr. Fitch had said that Gina
would “see to it,” and he was quite right; we started afresh in a much better
vein, all three of us, after her incursion.
Mr. Fitch produced a bottle of “vino santo” at the end of the meal and
charged our glasses. The sacred liquor was exceedingly good, and he took
heart from it to talk more freely. Gina had relaxed the strain of Maundy’s
preciosity, and he had begun to cross-question our host about his occupation,
his early life, his establishment in Rome, with an inquisitive and youthful
familiarity under which the old man shyly and prettily expanded. He told us
how in the dim ages he had received a commission to do a little historical
research among the manuscripts of the Vatican, and how he had taken his
seat in the library, with a pile of volumes around him, and had never left it
again from that moment to this. His first commission was long ago fulfilled,
but it had revealed a point of singular interest, some debatable matter in
connexion with a certain correspondence about a question raised in a
contemporary version of an unofficial report of a papal election in the
seventeenth century—yes, a matter which had chanced to be overlooked by
previous investigators; and Mr. Fitch, sitting fast in his chair at the library,
day after day, year after year, had been enabled to throw a little light upon
the obscurity, and had even published a small pamphlet—“not, I must admit,
for the very cogent reason that prompted your friend at Oxford, but from a
motive that I justify as a desire for historical accuracy, and that I condemn as
vanity”; and Mr. Fitch, so saying, beamed upon us with a diminutive
roguishness, more sparrow-like than ever, which he immediately covered by
plying us anew with the sacred bottle.
And then he told us of the long evenings he had spent, year after year, in
wandering among the ancient byways of the city—every day, when he was
turned out of the library at the closing hour, he had set forth to explore the
grand shabby old city that had now perished, he said, bequeathing little but
its memory to the smart new capital of to-day. Rome had changed around
him, he only had remained the same; but he could truthfully claim that he
knew nothing, save by report, of Rome’s rejuvenation—say rather of its
horrible pretentious bedizenment in the latest fashion; for he had long
abandoned his old pious pilgrimages, he now went no farther than his
lodging here and the library over there, and he was proud to declare that he
had never set eyes on a quarter of the monstrosities of which he heard tell.
There was a break of indignation in his voice as he spoke of them; he had
loved that Rome of the far-away golden evenings, it was all he ever had
loved except his work, and he had been robbed of it, bit by bit, till nothing
was left him but his well-worn seat among the state-papers and the pontifical
dust that nobody had taken the trouble to clear away. I don’t mean that he
said all this, but it was all in his gentle regretful tone; he seemed to stand
solitary and disregarded among the riot of modernity, and to utter a little tiny
dismal reproach, barely audible in the din—the plaintive “how can you, how
can you?” of a small bird whose nest has been trampled down by a pack of
stupid louts on a holiday. It was hard on him; the louts might just as well
have stamped and scuffled somewhere else; but so it was, they had violated
his wonderful Rome, and nobody noticed the sad small squeak of protest that
arose here and there from a scholar, a student, a lover.
What did Maundy think of it all? Mr. Fitch brightened in hospitable care
for our amusement; he didn’t often have two young things to lunch with him,
and he mustn’t blight the occasion with his griefs; and so he recovered his
spirit and tried to set Maundy off again in one of his droll tirades. What did
Maundy think of it? Oddly enough the question of Rome, in the light in
which it appeared to Mr. Fitch, hadn’t seemingly occurred to him; Maundy’s
Rome had been predominantly a matter of Spanish altar-lace and rose-tinted
chasubles, and a year by the Tiber had brought him to think that Oxford is
now more purely, more daintily Roman than the city of the Popes; and that
was really his only conclusion on the subject, and I don’t believe he had
given a thought to the Roman romance, vanished or vanishing, that had
inspired the tenderness of Mr. Fitch. Maundy knew nothing of San Cesareo,
nothing of the enchanted evenings among the ruins and the cypresses that
were still to be recaptured, I could give Mr. Fitch my word for it, even in the
desolation of to-day. “Ah yes, no doubt of it,” said Mr. Fitch, “if one
happens to be twenty years old to-day!”—but this he threw out in passing,
and he returned to the strange case of Maundy, which perplexed and troubled
him. It seemed that Maundy, whenever he went wandering through Rome,
had only one interest in view; I forget what it was, but it had something to do
with a point of ritual that Maundy excessively cherished; and he used to go
hunting round the city to discover the churches in which it was properly
observed, keeping a black-list of those which failed to make good. It was the
only aspect in which San Cesareo could engage him, and Mr. Fitch and I had
both neglected it.
With Rome ancient or modern Maundy was otherwise little concerned.
He listened blankly to Mr. Fitch’s melancholy regrets; for him they were the
mild ravings that you naturally expect from the very old. He was ignorant of
the past, so ignorant that it couldn’t raise the least stir in his imagination; he
had lived upon flimsiness, upon a little sentiment and a little second-hand
art, and he hadn’t the stomach, I suppose, for Rome. It was curious to see
how his insensibility puzzled Mr. Fitch. Maundy’s glibness about unknown
artists, about poems that hadn’t been written and statues that drove you mad,
had certainly surprised and impressed him; but the gulf of vacuity that
yawned beneath Maundy’s culture was a shock. Of course it only showed
what a featherweight of a tatter it was, that culture; if you are thus artistic in
the void, with the empty inane below you, it proves that your art hasn’t
substance enough to make it drop. But Mr. Fitch was too humble and kindly
for that harsh judgment, and he seemed to be beating about in his courtesy to
find an explanation more honourable to Maundy. Surely the young man was
very able, very original and brilliant; if he spurned the treasures of the past
he must have some clever new reason for doing so. I think I could have told
Mr. Fitch that Maundy’s reason was no newer than simple ignorance; and
perhaps I began to parade my own slender stock of learning to mark the
contrast. But Mr. Fitch was unconvinced, and I still see him eyeing young
Maundy with a sort of hesitating admiration, hovering on the edge of a
question that he couldn’t formulate. As for Maundy, he was thoroughly at
ease; Mr. Fitch had confessed that the name of Aubrey Beardsley was
unknown to him.
Anyhow the party had been most successful, and Mr. Fitch might go
trotting back to his afternoon’s work with the pleased sense that two very
young people had made friends under his and Gina’s auspices. He liked to
observe that Maundy and I were making a plan to meet next day, and he
blessed our alliance, taking credit for the good thought of acquainting
Maundy’s brilliance with my—my what?—my honest and old-fashioned
enthusiasm. Gina too was satisfied; she stood at her kitchen-door as we went
out, and she cordially invited us to come again. She pointed out that Maundy
set me an example with his soutane and his aspiration to the priesthood, and
she assured me that I couldn’t do better than to place myself under his
guidance; but at the same time she allowed that it wasn’t for all of us to aim
so loftily, and perhaps I was wise to be content with a lower standard. She
cheerily dismissed us; she had developed these reflections in twenty seconds
of farewell. We descended to the street, the three of us, and Mr. Fitch waved
his hat as he sped off to his happy labours, and Maundy and I turned away in
the direction of his seminary, where it was now time for him to rejoin his
black-skirted brethren. I was rather proud to be seen walking beside his
sweeping robe and clerical hat; it seemed so intimately Roman. But I found
to my surprise that Maundy was quite uneasy and apologetic about it; he
hated his uniform, he well understood that a man should feel shy of its
company. “If I were you,” he said, twitching his skirt disdainfully, “I should
hate to appear in public along with this.” He was an odd jumble of cross-
purposes, poor Maundy, and here was another glimpse of his natural mind.
He was more of a self-conscious school-boy than ever he was of a musk-
scented sonnetteer; but in either character I am afraid, or I hope, that he
didn’t fit comfortably into his Roman retreat. I can’t think that the cage was
to hold him much longer.
VI. VILLA BORGHESE