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目盛の間隔を正確に図示する必要はなく、それぞれの端の大小と、黒丸白丸があっているかが重要です。(黒丸の場合はその点を含む、白丸の時はその点を含まないことを表します。不等号に=が入っているかどうかの違いとも言えます。) 例えば、 p: x>1 q:x≦2 のように与えられていた時、右向きの数直線上に左から1と2の点を書きます。 pについては、x>1(つまり「xは1より大きい」)であることから、先ほど書いた1の点に白丸を書き、そこから右上がりに少し直線を書き、そこから右向きに直線を伸ばします。新幹線のような形になります。この形は、1の点を含まないことを表すもので、白丸と同じ意味ですが、ぱっと見で分かるように両方使います。また、この線がpであることをどこかに書いておいてください。 qについては、x≦2(つまり「xは2以下」)であるので、2の点に黒丸を書き、そこから真下に少し直線を書き、左向きの直線を伸ばします。こちらは、電車のような形になります。この形は、2を含むことを表すもので、黒丸と同じ意味です。こちらの線にも、qであることを書いておいてください。 このように、範囲を一つ一つ図示していくと、次のようになります。 _______________ p / 2 ---------○-----●------->x 1 | q --------------- これを見れば、「pかつq」や、「pまたはq」「p⇒q は真か偽か」はすぐに分かるはずです。たとえば「pかつq」なら、pとqが重なっているところなので、10 軸>0 判別式≧0 で必要十分ですよね。これは大丈夫でしょうか? これの少しひねった問題が例えば二次方程式の解が00 f(1)>0 0<軸<1 判別式≧0 で必要十分です。これと先ほどの上の条件と比較すると同じような感じですよね?つまり端点のみに具体的な数字の条件があるときにこのような条件で進めていくのがセオリーです。 上の解法を知識ゼロから解けと言われたら厳しいものがあるかと思いますが、一通り通っていることなら問題を見たときに「あっ、この問題はこの解法かな?」と瞬時に判断できるはずです。その感覚が大事です。「あー、これどうすればいいんだっけ…?」みたいな感じになっているのは良くないです。 これは勉強する時は問題を解き始める前に一瞬立ち止まって考えください。これを意識するしないとでは雲泥の差です。これは私自身、現役の時には気づかなかったことですが、浪人してからはこのことを意識するだけで、解ける問題のレパートリーが増えました。 闇雲にただ問題をこなすだけなら、むしろその場しのぎになってしまいます。それなら、数学の問題とかは時間がないのなら問題をみてこのような解法でいけばいいかなと思えるなら解かなくていいです。 要は、解き方に“意識“して問題演習を行ってください。時間のかける方はこっちの方です。 模試の前とかは、全国模試であれば定期テストなどでできなかった問題の教科書レベルの類題を確認する感じでいいと思います。高校生は部活等で時間がないと思われますので。1f:T10ea,受験数学にひらめきは全く必要ありません。 実際、数学者と数学の得意な高校生が、受験数学で勝負すると高校生が圧勝します(実話です)。一体何が、高校生を勝たせるのだと思いますか? 受験数学には、確かに、「ひらめきのようなもの」を要求する場面があります。特に整数問題などで顕著ですが。しかし、ほとんどの問題は、今まで身につけてきた解法で対応できてしまうんですね。 例えばですが、多変数関数 f(x,y)の最大値、最小値を求めよという問題が出たとします。(f(x,y)の中身は、例えば、x^2 3xy y^2などですね。ここではそれは本質ではないのでスルーします。)その時、方針が何通りかあるんですが、それを列挙できますか? あるいは、図形問題に対して、どのようなアプローチを考えるべきか説明できますか? (答えはどちらも回答の最後に載せますね) もし1つも分からない場合や、何個かしか挙げられない時は、少し補充的な勉強をする必要があります。 問題ごとに、それを解くための最適な方針がありますね。それをメモ程度で十分なので、どんどんまとめていってください。すると、多種多様に見える問題も、スタートは必ず同じことをしていたり、何個かのパターンの方針しか使っていなかったりします。本当はこういうことを分かっていくのは、問題演習を通してだんだん培っていくべきものなんでしょうが、99%の人は出来ないでしょう。僕も全然出来ませんでしたし。 なんにせよ、こういう「解法の整理」をしていくと、全く手が付かない問題はほとんどなくなってきます。途中までは行けるようになるんですね。そして、「ひらめき」は大抵こういう場面で使うものですね。例えば最後の最後に有名不等式を使ったりなどでしょうか。しかし、これすらも、方針としてカテゴライズすることが可能です。いわゆる純粋なひらめきは、受験数学においてはあり得ないといって良いでしょう。大抵、「閃かない」時は、解法が浮かばない時です。かなり具体的な問題に帰着できましたね。 僕は、ノートの見開き1ページに、この問題が来たら、この方針がよく登場する!というフローチャートのようなものを作っていましたね。頭の中が整理されていく感じがして楽しいですよ。 ちなみに、基礎ができていないということは、多少あるにせよ直接的な原因ではなく、いくら固めたところで、成果が微々たるものしか出ないので、気をつけましょう。青チャート、フォーカスゴールド、どちらも持っている時点でフル装備なので、多少の復習はもちろん必要といえども、頑張る必要はありません。 さて、先ほどの問題、わからずじまいは良くないですから簡単に 多変数関数の最大最小問題: ・等式があればxかyに代入してそれを消去する(いわゆる文字消去) ・xかyのどちらかを定数とみなし、ただの1変数関数とみなして考える(いわゆる文字固定) ・有名不等式の利用(相加相乗平均の関係、コーシーシュワルツの不等式、三角不等式など) ・逆像法 ・線型計画法 ・グラフを書いて考える Etc. 図形問題のアプローチ ・まずは初等幾何で解けないか考える。 ・次に、位置ベクトルを導入することで、内積などを利用して解けないか考える。 ・もし対称性の高い図形だったら、座標平面を設定するのも考える。 僕がこの解法整理についての対策を編み出し、始めたのは12月の半ばです。今なら相当早いタイミングから対策できますから、ぜひ過去問での得点をぐんぐん挙げて、自信をつけていってほしいと思います。 では、有意義な秋をお過ごしください!20:T11fd,初めまして。 大学数学を理解していないと解けない問題とはどんな問題を指しているのでしょうか?私は受験生時代東北大の数学も13年分ほど解きましたが、そのように感じられる問題はなかった記憶です。 もちろん数3では高校範囲だと証明できないものがあります(中間値の定理等)。また、例えば「x>0のとき、不等式sinx>x-x^3/6を示せ」という問題であれば、不等式の右辺はどこから出てきたのだろうと思うことはあるかもしれません。 ですが総じて大学範囲を理解しなければならないことは問題にするほど多くないと思います。 私自身はこれまで数学にかなり没頭してきましたが、高校範囲で「受験勉強」として取り組んだ内容は無駄になったとは感じていません。少なくとも現在学んでいる微分積分学は数3をもっと厳密に、拡張したような話が多いですし、線形代数学は行列という新しい分野ですが、ところどころベクトルのときに学んだことが役立っているように感じます。 ですが、質問者様のように受験がそういったゲームのようになっているというのも分かります。昔中国では科挙という試験が実施されていましたが、その試験では漢詩を始めとした非常に多くのことを暗記して挑む試験でした。そして、その試験にさえ合格すれば将来は安泰どころか裕福に暮らせたようです。 現行の受験制度もある種そうなっているのでは無いでしょうか。ハッキリ言えば、高校生の勉強の受験ぐらいで上手くいかない人は「学問の学ぶ姿勢」を追求したところで自信で学び切ることは出来ないという根本的な考えがあるのだと思います。当然高校の勉強は出来なかったけど天才的な研究や発見をする人もいるのかもしれませんが、母数として多くないのは明白でしょう。 「たくさんの科目に手を出していることで、学問の本質を見失っている」という指摘については、私の意見としては「まだその指摘をするには早い」という感じですね。もちろん私がその指摘をするにもまだ早いです。大袈裟に言えば我々が寿命を迎えるときにはじめて「あの勉強はいらなかった」と思うことが出来ます。 結局この教育の根幹を作った人たちは、「幅広く学ばせて、どれかが当たればいいな」みたいな発想なのでしょう。 回答作成中に思ったことですが、数学より理科(物理、化学)の方が大学範囲を黙認することが多いですね。解く上で黙認すると言うよりは、その定義自体に黙認があったり。そもそも高校で学ぶ特に理系科目というのは、非常に限られた都合のいい場合のみ扱います。数学の条件付き極値問題であれば三角関数を使ったり線形計画法を使って上手く解けるような変数の次数になっていたり、物理の公式であればその厳密な証明が大学範囲の微分積分だったり、化学では条件が綺麗すぎたり。 学問というのは具体から始まると考えています。例えば物理学の、ニュートンの運動方程式は実験事実です。つまり実験(具体)を通して一般的に正しい(抽象)とされました。数学であれば、有名なフェルマーの最終定理がありますが、あの問題は初めからn一般について示された訳ではなく、まずはn=3,4,5と示されたのです。化学でいえば、周期表は具体→抽象の一例ではないでしょうか。 そういう観点で高校の学習を振り返ってみると、限られた場合にしか出来ないような解法、考え方であっても、今後「学問」を修めるにあたって抽象化するときに役に立つことでしょう。 以上。少々まとまりがない形になってしまい申し訳ありません。このような疑問をもてる質問者様は素晴らしいと思います。もしかしたら何かやりたい研究等があるのでしょうか。であれば私と同じですね。是非受験をパスしてやりたい学問に取り組み、貢献して欲しいですね。頑張ってください。応援しています。2:["$","main",null,{"className":"px-4 pt-4 pb-4","children":["$","div",null,{"className":"max-w-3xl mx-auto w-full","children":[["$","div",null,{"className":"mb-8","children":["$","$L7",null,{"href":"https://unilink-app.onelink.me/isbO/h6xeh63x?advice=OOt1ro1fLQES8lzT1KJA","target":"_blank","children":["$","$L8",null,{"src":"/images/web_to_app_banner.jpg","width":3660,"height":1500,"sizes":"100vw","style":{"width":"100%","height":"auto"},"alt":"UniLink WebToAppバナー画像","className":"mb-4 rounded"}]}]}],["$","h1",null,{"className":"text-xl font-semibold mb-2","children":"三角関数です"}],["$","div",null,{"className":"flex justify-between mb-4","children":[["$","div",null,{"className":"text-left text-xs text-caption","children":["クリップ(",1,") コメント(",0,")"]}],["$","div",null,{"className":"text-right text-xs text-caption","children":"5/29 20:25"}]]}],["$","div",null,{"className":"coach-mark mb-4","children":"UniLink利用者の80%以上は、難関大学を志望する受験生です。これまでのデータから、偏差値の高いユーザーほど毎日UniLinkアプリを起動することが分かっています。"}],["$","div",null,{"className":"mb-4","children":["$","$L13",null,{"clientImageUrl":null,"clientUserName":"大学生から見直す","infoString":"高3 新潟県 愛知東邦大学人間健康学部(38)志望","adviceId":"OOt1ro1fLQES8lzT1KJA"}]}],["$","div",null,{"className":"mb-8","children":[["$","div",null,{"className":"leading-loose whitespace-pre-wrap","children":[["$","div","consultation-part-0",{"children":[null,"0≦θ≦πで\nsin(θ-π/4)=-1/√2\nこれをとくとどうなりますか"]}]]}],["$","div",null,{"className":"pt-4","children":["$","$L14",null,{}]}],null]}],["$","h1",null,{"className":"text-xl font-semibold mb-2","children":"回答"}],["$","div",null,{"className":"mb-4","children":["$","$L15",null,{"adviserImageUrl":"https://firebasestorage.googleapis.com/v0/b/unilink-48e75.appspot.com/o/images%2Fs_mBLd8WqR2pOVKp8vWtIyDjP2qA03.jpg?alt=media&token=58f8baf6-452b-4be3-b749-a23ab3e01e88","adviserName":"たけなわ","adviserDepartment":"北海道大学法学部","adviceId":"OOt1ro1fLQES8lzT1KJA"}]}],["$","div",null,{"className":"coach-mark mb-4","children":"すべての回答者は、学生証などを使用してUniLinkによって審査された東大・京大・慶應・早稲田・一橋・東工大・旧帝大のいずれかに所属する現役難関大生です。加えて、実際の回答をUniLinkが確認して一定の水準をクリアした合格者だけが登録できる仕組みとなっています。"}],["$","div",null,{"className":"mb-8","children":[["$","div",null,{"className":"leading-loose whitespace-pre-wrap mb-4","children":[["$","div","advice-part-0",{"children":[null,"①一度、前提条件を考慮の外において、一般に\n  sin(x)=-1/√2\nとなる場合を考えてみましょう。これをxについて解くと、解はどうなるでしょうか。\n\n②では、一つ前提条件を追加して、xの定義域が0≦x≦πの場合、①の方程式の解はどうなるでしょうか。\n\n③x=θであれば、②で出した解がそのままθについても当てはまることになるでしょう。しかし、本問の場合は、x=θ-π/4です。どう工夫して解けば良いでしょうか。具体的にいうと、⑴θの定義域が0≦θ≦πだとしたら、xの定義域はどうなるでしょうか。⑵その定義域を考慮して、①の方程式をxについて解くと、解はどうなるでしょうか。⑶また、本問ではθについての解を求められていますが、⑵でxについて導いた解を、どのようにしたらθの解に持っていけるでしょうか。\n\n考えてみてください。"]}]]}],["$","div",null,{"className":"mb-4","children":["$","$L16",null,{"adviserImageUrl":"https://firebasestorage.googleapis.com/v0/b/unilink-48e75.appspot.com/o/images%2Fs_mBLd8WqR2pOVKp8vWtIyDjP2qA03.jpg?alt=media&token=58f8baf6-452b-4be3-b749-a23ab3e01e88","adviserName":"たけなわ","adviserDepartment":"北海道大学法学部","adviceId":"OOt1ro1fLQES8lzT1KJA","numberOfFan":43,"clipsAvg":7.826589595375722,"adviceRateAvg":4.770325203252033,"profile":"気が向いたときに、気が向いたご質問に回答しています。"}]}],["$","div",null,{"children":["$","$L7",null,{"href":"https://ck.jp.ap.valuecommerce.com/servlet/referral?sid=3364577&pid=884970531&vc_url=http%3A%2F%2Fshingakunet.com%2F%3Fvos%3Dnrmnvccp0000100","rel":"nofollow","target":"_blank","children":["$","$L8",null,{"src":"/images/document_request_banner.jpg","width":3660,"height":1500,"sizes":"100vw","style":{"width":"100%","height":"auto"},"alt":"UniLink パンフレットバナー画像","className":"mt-4 rounded"}]}]}],["$","div",null,{"className":"pt-4","children":["$","$L17",null,{"id":"adsbygoogle-init-under-advice"}]}]]}],["$","div",null,{"className":"flex justify-between","children":[["$","h1",null,{"className":"text-xl font-semibold","children":["コメント(",0,")"]}],["$","$L18",null,{"adviceId":"OOt1ro1fLQES8lzT1KJA"}]]}],["$","div",null,{"className":"mb-8","children":["$","div",null,{"className":"text-xs p-4","children":"コメントで回答者に感謝を伝えましょう!相談者以外も投稿できます。"}]}],["$","h1",null,{"className":"text-xl font-semibold","children":"よく一緒に読まれている人気の回答"}],["$","div",null,{"className":"mb-8","children":["$","div",null,{"className":"divide-y","children":[["$","div",null,{"children":["$","$L7",null,{"href":"/advice/XluOkWMBp00JfyF5pONv","children":["$","div",null,{"className":"flex items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"三角関数の変形の使い分けについて"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"質問者様は高2ということなので、数Ⅱまでの範囲で回答させていただきます。\n\n\n【三角関数を変形する目的】\n\nまず、三角関数を変形するのは必ず目的があります。\n①三角関数を含んだ方程式・不等式を解くため\n②三角関数を含んだ関数の最大値・最小値を求めるため\nなどがよくある目的ですね。\n\n《①について》\n方程式や不等式ははじめに因数分解で攻めます。\n(因数)(因数)=0\nといった形になれば、あとは簡単ですね。\n因数分解しない場合は②の考え方をそのまま借りましょう\n\n《②について》\nsinのみ、cosのみ、tanのみ、の式に帰着させます。そしたら見たことある関数(一次関数、二次関数など)になります。\nそのための手段として\n*三角関数の相互関係\n*加法定理を用いた公式\nなどが存在します。\n\n\n---------\n\n【質問主様の弱点と思われるところ】\n\n数Ⅱの三角関数に入ってからうまくいかなくなった高校生は加法定理を用いた公式につまづいている人が多いです。\n公式自体覚えていても、問題でうまく活用出来ないことがよくあります。\n\n先程の項目で書きました、変形のそもそもの目的を意識して演習してみてください。\n使い分けパターンは青チャートなどのテキストに詳しく記載されています。これを身につけることが大切です。\n\nパターンを繰り返しの演習で身につける際に、\n「因数分解を目指す!」\n「sinのみ、cosのみ、tanのみの式を目指す!」\nという意識を持って取り組むことで、何故その式変形を使うのかが体感出来ます。\n\n\n---------\n\n【最後に】\n\n問題のゴールから逆算して考えることが数学においては大切です。\n初めから逆算して考えることなんて出来ないから、パターンを演習によって身につけるわけですが、ゴールを意識してパターンを身につけなければ、何のためのパターンなのかがわかりません。\n必ず、式変形の目的を意識した演習を心掛けてください。"}],["$","div",null,{"className":"flex 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items-center py-4","children":[["$","div",null,{"className":"flex-1","children":[["$","div",null,{"className":"mb-1","children":"数学I(図形と計量)"}],["$","div",null,{"className":"text-xs text-caption line-clamp-2 mb-1","children":"回答させてもらいます!\n見た感じ計算はあってそうですね!\n\nセナさんの疑問としては\ncosAの時はb=-√2+√6(b>0)が答えとして出るのに\ncosCの場合はb=√6±√2がb>0の条件でどちらも有効で、cosAの時と同じにならないのではないかという疑問だと思って回答しますね!\n\nこの場合cosCで出てきたbの値に対して一つ有効な条件設定があります。それが「辺と角の関係」です。\nもしかしたらこの時点でピンと来たかもしれませんが、角度が大きい角の対面の辺が長くなるよって感じのやつですね(言葉がラフでごめんなさい。回答に書くときはしっかり教科書通りのやつ書いてね笑)\nその関係性を角Cと角Bに当てはめてみると角度が小さい方の対面の辺bは辺cより小さい必要があります。\n√6+√2と2√2の大小関係、√6-√2と2√2の大小関係はどういう風に考えるといいんでしたっけ?\n一回考えてみてください🙆‍♂️(逆にいうとその考えがめんどくさくて回答はcosAを採用したのかもしれませんね…)\n\nまた、他にも考え方があると思うのでこういう考え方もあるよ!ってのを思いついたら是非教えてくださいね🥸\n頑張って!"}],["$","div",null,{"className":"flex mb-1","children":[["$","svg",null,{"stroke":"currentColor","fill":"currentColor","strokeWidth":"0","viewBox":"0 0 24 24","className":"text-subPrimary mr-1","children":["$undefined",[["$","path","0",{"fill":"none","d":"M0 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mb-1","children":"こんにちは!\nこうしんと申します!\n\n指数関数…というと範囲が難しいので、\n最大最小問題の解き方→指数関数の処理方法\nという形で話を進めていきますね!\n\nまず最大最小問題ですが、これは方程式・関数を扱う分野で出てきます。\nこの分野の攻略方法は以下の通りです\n・文字を見分ける\n・解答法を知る\n(方程式として解く、関数として解く、不等式として解く)\n\n一つずつ説明していきますね。\n\n・文字を見分ける\n文字は、定数と変数があります。物理ではこれがはっきり決まってますが、数学では全く別の性質で、定数でさえ値を動かすことがあります。\nなので\n定数…中心にはない文字\n変数…中心に扱っていく文字(〜と解く、微分する、といった文字の中心となります)\nこれをまず見分ける必要があります。\n見分け方は、定数が「分布(どういう値をとるのか?)を知りたい文字」であるという性質がある点です。他には、定数の方が次元が高い、扱いづらいという特徴がありますね。\nこうして、変数を絞り込んでおきます。\n変数は1個にしてください。\n\n・解答法を知る\n解答法は3つに分かれます。\n\n方程式としてみる\n→解の配置(0より大小となる点を探す)・座標・対称式\n関数としてみる\n→微分してグラフを描く\n不等式としてみる\n→実数の2乗は0以上を使う、コーシーシュワルツ、相加相乗平均\n(不等式は難しいので、関数としてみた方が早いです)\n\nこれらの解答法を調べてみてください!完璧にすると対応ができます!\n\n\n最大値というのは、\n・関数がそれ以上に増えない値\n・それを満たすxが一つ定義域に存在する値\nであるという性質を持ちます。\n最小値は、反転した性質ですね。\n\nそのため最大値の候補は絞られます\n→①極大値 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