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will compare which result is bigger: sqrt(1)+sqrt(2)+...+sqrt(100) or 2000/3? Make sure you don't use a calculator! Prefer a quick read? Here's the solution: / 134845326...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3418842/e3526ddf890a8d39b2a1b38c3f12eab2/564x318_1"},"target":"_self","position":"2","reqid":"1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL","summary":{"fullTextUrl":"/int_search_summary?data=http%3A%2F%2Fwww.youtube.com%2Fwatch%3Fv%3DwMwiRzd8GiU","linkTemplate":"/video/preview/914143672597820554?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Use calculus, NOT calculators!","related_orig_text":"Not Calculus","related_porno":false,"related_less_3m_off":true,"client":"d2d","no_cnt":1,"related_src":"serp","related":"{\"porno\":false,\"vfp\":1,\"orig_text\":\"Not Calculus\",\"url\":\"http:\\/\\/www.youtube.com\\/watch?v=wMwiRzd8GiU\",\"src\":\"serp\",\"rvb\":\"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_E60CggQkAYAEKyqLARABGniB__n8-_wFAAP2-wgGCPwCDf36AQn-_wD9BQMAAgX-APYE__cBAAAA_RL69AQAAADxC_4GAwAAAAz3A_EDAAAADPr5_P4AAAALDfAC_wAAAP4A-AYD_wAADAEKBQAAAAD2BP8A__8AAP_6-wUAAAAAAvgFBgAAAAAgAC1bBuE7OBNACUhOUAIqcxAAGmAMJwA2SxLcuxNe2Bn6HijNNwAOns8W_xzt_xQx1hoOPITMADD_EesX6psAAAATzssvCwDyf7f5zuEN8BWf_606T2sSCdYdGBT9A84uNsnOAAnA9Q8AmgoRA_250iXxNSogAC06Ng47OBNACUhvUAIqrwYQDBqgBgAAiEEAACzCAACoQgAAMMEAABDBAABAwQAASEIAAABBAABcwgAAAMEAAIA_AABAQgAAsMEAAKBAAAAsQgAA8EEAADDBAADAwQAAqEEAACDBAACAQQAA4MAAAMDBAAAAQAAA-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_AACYQQAAmEEAAAAAAAAIwgAAlkIAAKjBAAA4QgAAuMEAALDBIAA4E0AJSHVQASqPAhAAGoACAAAMvgAAMD0AALg9AACgPAAAoLwAAHw-AAAQPQAAB78AADA9AABwvQAADL4AAFC9AAAQPQAAVD4AABy-AADYvQAAbD4AAIg9AAAwPQAAET8AAH8_AACyvgAAUL0AAHw-AACAOwAAHD4AAIC7AACIPQAADD4AAEQ-AACIPQAA6r4AAEC8AABQPQAA-D0AAAy-AADIvQAAvr4AAFy-AAAcvgAA-L0AAIg9AABMPgAALL4AAIi9AADgvAAAFD4AANi9AACWvgAAFD4AACy-AAAsPgAA6D0AAOA8AACSvgAAUD0AADs_AADYPQAAdD4AAIo-AABQvQAAkj4AAHA9AAAMviAAOBNACUh8UAEqjwIQARqAAgAAqL0AAFw-AABQvQAAH78AANi9AABkPgAArj4AABA9AABQPQAAcD0AAOA8AAAUvgAAFD4AAOi9AAAkPgAAQLwAAFA9AADqPgAAJL4AAJY-AAC4vQAAUD0AAJg9AADgvAAA4LwAAOi9AACYvQAA4DwAAIi9AAA0vgAAoDwAACQ-AACGvgAAyL0AANi9AADIvQAABD4AACw-AAB0vgAA-L0AADy-AAAwPQAAED0AAFA9AAAsPgAAQLwAAH-_AABUPgAAPD4AAGw-AAAkPgAAUD0AAJi9AACqPgAAQLwAAAQ-AACAuwAAcL0AAEA8AADoPQAALD4AAAS-AAB0PgAAEL0gADgTQAlIfFABMAk4AUoAYABoAA,,\"}","related_url":"http://www.youtube.com/watch?v=wMwiRzd8GiU","parent-reqid":"1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL","related_vfp":1,"relatedVideo":"yes"},"cwidth":1280,"cheight":720,"cratio":1.77777,"dups":["914143672597820554"],"episode":0,"season":0,"isEmbedOnly":false,"greenHost":"YouTube","hasTranslation":true,"contentTypeId":null,"censored":false},"16786679170945077717":{"videoId":"16786679170945077717","docid":"34-3-15-ZDB22F7417C23A49B","description":"Calculus 1 - Genel Matematik Vize-Final / Genel Tekrar.","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3936459/c1194eec1d2f4732bde0c133b17fa1c3/564x318_1"},"target":"_self","position":"3","reqid":"1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=videoid:16786679170945077717","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitmeni tarafından sunulan, üç saat aralıksız devam eden kapsamlı bir ders anlatımıdır. Eğitmen, Instagram'da paylaştığı bir paylaşım sonrası gelen katkılar üzerine bu videoyu hazırladığını belirtmektedir.","Video, kalkülüs ve genel matematik konularını kapsamlı şekilde ele almaktadır. İçerik, fonksiyonların tanım kümeleri, birebir ve örten fonksiyonlar, ters fonksiyonlar, bileşke fonksiyonlar, tek-çift fonksiyonlar, tam değer fonksiyonu, işaret fonksiyonu, trigonometrik fonksiyonlar, hiperbolik fonksiyonlar, limitler, fonksiyonların sürekliliği, türev, ara değer teoremi, fonksiyon analizi ve asimptotlar gibi konuları kapsamaktadır.","Eğitmen, 19 soru içeren bir dosya kullanarak konuları anlatmakta ve her bir konuyu örnek sorular üzerinden pekiştirmektedir. Video, özellikle AYT sınavına hazırlanan öğrenciler için hazırlanmış olup, özel üniversitelerdeki vize sınavlarından seçilen sorular da çözülmektedir. Eğitmen, öğrencilerin kendi uygulamalarını yapmaları için MATLAB veya Desmos gibi araçları kullanmalarını önermektedir."]},"endTime":7845,"title":"Kalkülüs ve Genel Matematik Vize Hazırlık Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Giriş ve Paylaşım Hakkında Bilgilendirme","list":{"type":"unordered","items":["Konuşmacı, iki-üç gün önce Instagram'da kalkülüs ve genel matematik için vize öncesi faydalı bir paylaşım yapmış.","İzleyicilerden dosyalar, eski vize soruları ve örnek çözümlü sorular paylaşmaları istenmiş.","Konuşmacı, bu paylaşımın analiz dersi için değil, genel matematik dersi için olduğunu vurguluyor."]},"beginTime":1,"endTime":54,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1&ask_summarization=1"},{"index":1,"title":"Analiz Dersi Hakkında Bilgiler","list":{"type":"unordered","items":["Konuşmacı, analiz dersinde doğrudan türev soruları yerine ispat yaparak ilerlediklerini belirtiyor.","Altı-yedi soru sorulduğunda beş-altı tanesi ispatlı, bir tanesi uygulama sorusu oluyor.","Limit sorularına bile iki-üç teorem kullanılıyor ve sandviç teoremi gibi kavramlar kullanılıyor."]},"beginTime":54,"endTime":78,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=54&ask_summarization=1"},{"index":2,"title":"Dosya İçeriği ve Kullanım Önerileri","list":{"type":"unordered","items":["Dosya içerisinde teorem kullanabileceğimiz bazı sorular bulunuyor.","Konuşmacı, hafta içinde sadece delta sadece de soruları içeren bir dosya da hazırlamayı düşünüyor.","Dosya on dokuz soru içeriyor ve izleyicilerin eksik hissettiği yerleri görebilecekleri belirtiliyor."]},"beginTime":78,"endTime":139,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=78&ask_summarization=1"},{"index":3,"title":"Müfredat Farklılıkları ve Dosya Paylaşımı","list":{"type":"unordered","items":["Her okulda müfredat farklı olabiliyor; bazılarında fonksiyon tanımları ve grafikler üzerinde çok durulmuş, bazılarında limitte daha çok durulmuş.","Konuşmacı, tüm soru tiplerini koymaya çalıştığını belirtiyor.","Dosya herkese açık paylaşılacak ancak iki-üç gündür çok emek harcanmış olduğu için sadece katılanlar için değil, tüm izleyiciler için paylaşılacak."]},"beginTime":139,"endTime":172,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=139&ask_summarization=1"},{"index":4,"title":"Giriş ve Üyelik Bilgileri","list":{"type":"unordered","items":["İzleyiciler videoyu beğenirse \"katıl\" butonundan 5-10 lira arasında bir üyelik ücreti ödeyebilirler.","Üyelik, video sahibi için teşekkür etmek için bir yol olarak sunulmaktadır.","Videoda 19 adet sorunun çözümü paylaşılacaktır."]},"beginTime":177,"endTime":219,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=177&ask_summarization=1"},{"index":5,"title":"Fonksiyonların Tanım Kümeleri","list":{"type":"unordered","items":["Fonksiyonların tanım kümelerini bulmak için fonksiyon bilgisine sahip olmak gerekir.","Kök fonksiyonunun tanım kümesi, kök içindeki ifadenin sıfırdan büyük veya eşit olması koşuluyla belirlenir.","Logaritma fonksiyonunun tanım kümesi, logaritmanın içi sıfırdan büyük olması koşuluyla belirlenir."]},"beginTime":219,"endTime":256,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=219&ask_summarization=1"},{"index":6,"title":"Örnek Soruların Çözümü","list":{"type":"unordered","items":["A şıkkında, iki farklı fonksiyonun tanım kümelerinin kesişimi alınarak çözüm kümesi bulunur.","B şıkkında, sinüs fonksiyonunun sıfır olduğu tam sayı değerleri çıkarılarak tanım kümesi belirlenir.","C şıkkında, arccos ve arcsin fonksiyonlarının tanım kümeleri incelenerek çözüm kümesi bulunur."]},"beginTime":256,"endTime":575,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=256&ask_summarization=1"},{"index":7,"title":"Daha Karmaşık Bir Soru","list":{"type":"unordered","items":["D şıkkında, kosinüs fonksiyonunun belirli aralıklarda değerleri incelenerek tanım kümesi bulunur.","Tablo metodu kullanılarak fonksiyonun işaret tablosu oluşturulur.","Çözüm kümesi, kosinüs fonksiyonunun belirli aralıklarda 1/2'den büyük veya eşit olması koşuluyla belirlenir."]},"beginTime":575,"endTime":927,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=575&ask_summarization=1"},{"index":8,"title":"Birebir ve Örten Fonksiyonlar","list":{"type":"unordered","items":["Matematik bölümünde birebir ve örten fonksiyonlar konusunda ciddi sorular sorulabilir.","Birebir fonksiyon için, tanım kümesinden alınan tüm x₁ ve x₂ elemanları için, eğer bunların görüntüleri birbirlerine eşit iken tanımları da eşit ise f birebirdir.","Örten fonksiyon için, B kümesi üzerinden her y elemanı için en az bir x elemanı vardır ki f(x) = y olsun."]},"beginTime":934,"endTime":1122,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=934&ask_summarization=1"},{"index":9,"title":"Örnek Soruların Çözümü","list":{"type":"unordered","items":["f(x) = 1 - x² fonksiyonu birebir değildir çünkü iki kökü vardır (x = -1 ve x = 1), ancak örten bir fonksiyondur.","|x| fonksiyonu birebir değildir çünkü farklı x değerleri aynı görüntüye sahip olabilir, ancak örten bir fonksiyondur.","f(x) = 2^(x-1) fonksiyonu birebirdir ancak örten değildir çünkü [0,1] aralığını örtemez.","f(x) = -2 - 1/2(x+1) fonksiyonu birebirdir ve örten bir fonksiyondur."]},"beginTime":1122,"endTime":1443,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1122&ask_summarization=1"},{"index":10,"title":"Fonksiyonların Tersi","list":{"type":"unordered","items":["Üçüncü soruda bir fonksiyonun tanım kümesi belirleniyor ve tersi alınarak bir aralığı yapılıyor.","Fonksiyonun tanım kümesinde paydasını sıfır yapan değerler (örneğin 3) bulunmamalıdır.","Ters fonksiyon bulmak için x gördüğünüz yerlere f⁻¹(x) dönüşümü uygulanır ve f⁻¹(x) = (4+3x)/(x+2) olarak bulunur."]},"beginTime":1447,"endTime":1591,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1447&ask_summarization=1"},{"index":11,"title":"Fonksiyonların Bileşkesi","list":{"type":"unordered","items":["Fonksiyonların bileşkesi (fog) tanımlanırken, f fonksiyonu A'dan B'ye, g fonksiyonu B'den C'ye gider.","A'dan C'ye gitmek için önce f fonksiyonu, sonra g fonksiyonu uygulanır.","f(g(x)) = (1-x³)/(1-1/x³) şeklinde hesaplanır."]},"beginTime":1591,"endTime":1782,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1591&ask_summarization=1"},{"index":12,"title":"Tek ve Çift Fonksiyonlar","list":{"type":"unordered","items":["Bir fonksiyon f tek fonksiyon ise, f(-x) = -f(x) olmalıdır.","f(x) = x³ - x³ + x² + 1 = x/x+1 olduğundan f tek fonksiyondur.","f(x) = log₂(x+√(x²+1)) fonksiyonunun tek olup olmadığı için, ifadenin eşleniği ile çarpılıp bölünerek log₂(1/(√(x²+1)+x)) = -log₂(√(x²+1)+x) olarak yazılır ve f tek fonksiyondur."]},"beginTime":1782,"endTime":2068,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1782&ask_summarization=1"},{"index":13,"title":"Tek ve Çift Fonksiyonlar","list":{"type":"unordered","items":["Fonksiyonların tek veya çift olup olmadığını belirlemek için mutlak değer ve fonksiyonların özellikleri kullanılır.","Reel sürekli iki tek fonksiyonun çarpımı çift fonksiyon, bölümü de tek fonksiyon verir.","Arctan(2x) fonksiyonu tek fonksiyondur çünkü tanjant x'in orijine göre simetrik olduğu için arctan(2x) de tek fonksiyondur."]},"beginTime":2072,"endTime":2249,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2072&ask_summarization=1"},{"index":14,"title":"Tam Değer Fonksiyonu","list":{"type":"unordered","items":["Tam değer fonksiyonu (flor fonksiyonu) tam sayı değerleri için değişmez, ancak reel kısım varsa en küçük reel kısmına götürür.","Negatif değerler için tam değer fonksiyonu, kendisinden küçük en büyük tam sayıya gider.","Tam değer fonksiyonlarının grafiği merdiven şeklinde olup, tam sayılar grafiklerde harici bir şekilde gösterilir."]},"beginTime":2249,"endTime":2533,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2249&ask_summarization=1"},{"index":15,"title":"İşaret Fonksiyonu","list":{"type":"unordered","items":["İşaret fonksiyonu (signum fonksiyonu) için, fonksiyonun içi pozitifse görüntü 1, negatifse görüntü -1'dir.","İşaret fonksiyonunun grafiği, x=1'in sağında 1, solunda -1, x=0'da 0'dır.","İşaret fonksiyonunun grafiği, fonksiyonun kökleri ve işaret değişim noktalarına göre belirlenir."]},"beginTime":2533,"endTime":2658,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2533&ask_summarization=1"},{"index":16,"title":"Signum Fonksiyonu","list":{"type":"unordered","items":["Eksi bir ve altı aralığında görüntüler negatif gelecektir.","Signum fonksiyonu, negatif değerler için belirli bir değer alır.","Signum fonksiyonu ile ilgili soru çözümü isteyenler yorumlarda belirtebilir."]},"beginTime":2661,"endTime":2735,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2661&ask_summarization=1"},{"index":17,"title":"Periyot Bulma","list":{"type":"unordered","items":["Periyot bulma soruları zordur ve ispatları istenebilir.","Sinüs fonksiyonunun grafiğini çizerken, katsayı a değiştirildiğinde sadece frekans değişir, periyota etki etmez.","Periyot, f(x) = f(x+T) sağlayan en küçük pozitif T sayısıdır."]},"beginTime":2735,"endTime":2813,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2735&ask_summarization=1"},{"index":18,"title":"Periyot Etkileyen Faktörler","list":{"type":"unordered","items":["İçerideki ifadeyi x/2 şeklinde yazmak, grafiği P/2 kadar sağa veya sola oynatmak demektir ve periyotta değişikliğe sebep olmaz.","Fonksiyona k eklemek, fonksiyonu yukarı veya aşağı oynatmak dışında hiçbir etkiye sebep olmaz.","Periyodu etkileyecek tek şey, içerideki m katsayısıdır."]},"beginTime":2813,"endTime":2850,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2813&ask_summarization=1"},{"index":19,"title":"Sinüs ve Kosinüs Periyotları","list":{"type":"unordered","items":["Sinüs 2x fonksiyonunun periyodu P/2'dir, sinüs x'in periyodu P'dir.","Sinüs 3x fonksiyonunun periyodu 2P/3'tür.","Negatif periyot olmaması için periyodu bulurken ifadenin önüne mutlak değer koyulur."]},"beginTime":2850,"endTime":2921,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2850&ask_summarization=1"},{"index":20,"title":"Trigonometrik Fonksiyonların Periyotları","list":{"type":"unordered","items":["Kosinüs ve sinüs fonksiyonlarının periyotları aynıdır.","Tanjant fonksiyonunun periyodu P/2'dir.","Sin²x + cos²x = 1 ve sin²x = (1 - cos2x)/2 dönüşümleri periyot hesaplamalarında kullanılır."]},"beginTime":2921,"endTime":3040,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2921&ask_summarization=1"},{"index":21,"title":"Karmaşık Fonksiyonların Periyotları","list":{"type":"unordered","items":["Sinüs x çarpı kosinüs x ifadesi, sinüs 2x/2 şeklinde yazılabilir.","İki veya daha fazla trigonometrik fonksiyonun toplamının periyodu, bu fonksiyonların periyotlarının en küçük ortak katıdır.","Örneğin, cos(3x) + sin(4x) fonksiyonunun periyodu 2π'dir."]},"beginTime":3040,"endTime":3142,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=3040&ask_summarization=1"},{"index":22,"title":"Mutlak Değerli Trigonometrik Fonksiyonlar","list":{"type":"unordered","items":["Mutlak değerli trigonometrik fonksiyonlar (sinüs mutlak değer x, kosinüs mutlak değer x) için değişiklikler söz konusudur.","Kosinüs x artı t = mutlak değer kosinüs x eşitliğini sağlayan en küçük t değeri aranmaktadır.","Mutlak değerli fonksiyonların periyotları, orijinal fonksiyonların periyotlarının yarısıdır (örneğin sinüs x'in periyodu π ise, sinüs mutlak değer x'in periyodu π/2'dir)."]},"beginTime":3150,"endTime":3312,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=3150&ask_summarization=1"},{"index":23,"title":"Periyodik Fonksiyonların Özellikleri","list":{"type":"unordered","items":["Sinüs x kare fonksiyonunun periyodik olup olmadığı incelenmektedir.","Periyodik bir fonksiyon için f(x) = f(x+T) eşitliği sağlanmalıdır.","Sinüs x kare fonksiyonunun periyodik olmadığı, bu varsayımla çelişki yaratıldığı gösterilmiştir."]},"beginTime":3312,"endTime":3616,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=3312&ask_summarization=1"},{"index":24,"title":"Hiperbolik Fonksiyonlar","list":{"type":"unordered","items":["Hiperbolik fonksiyonların grafiklerini bilmek önemlidir.","Cosinüs hiperbolik x = (e^x + e^(-x))/2 formülü ve grafiği incelenmektedir.","Sinüs hiperbolik x = (e^x - e^(-x))/2 formülü ve grafiği de açıklanmıştır."]},"beginTime":3616,"endTime":3694,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=3616&ask_summarization=1"},{"index":25,"title":"Matematik Problemleri Çözümü","list":{"type":"unordered","items":["E üzeri x = 1 ise x = ln 2 olarak bulunur.","Arksinüs x = sinüs hiperbolik y ise, her tarafı e üzeri y ile çarparak ve e üzeri y - 1'in karesi ile işlem yaparak y = ln(x + √(x² + 1)) sonucuna ulaşılır.","Tanjant hiperbolik x = e üzeri x / (e üzeri x + e üzeri -x) ise, her tarafı e üzeri y ile çarparak ve e üzeri 2y - 1'in karesi ile işlem yaparak y = 1/2 ln(1 + x / 1 - x) sonucuna ulaşılır."]},"beginTime":3699,"endTime":4049,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=3699&ask_summarization=1"},{"index":26,"title":"Limit Problemleri","list":{"type":"unordered","items":["Limit x giderken sıfıra sinüs ax/b ifadesinde, a'yı çarpan olarak alarak ve u dönüşümü yaparak limitin b'ye eşit olduğu gösterilir.","Limit x giderken a sinüs mx/nx ifadesinde, m/n şeklinde yazılabilir ve aynı sonuç elde edilir.","Limit x giderken a nx/sinüs mx ifadesinde de n/m şeklinde yazılabilir ve aynı sonuç elde edilir.","Limit x giderken a tanjant mx/tanjant nx ifadesinde, paydasını sinüsle vererek aynı sonuç elde edilir."]},"beginTime":4049,"endTime":4210,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4049&ask_summarization=1"},{"index":27,"title":"Limit Problemleri Çözümü","list":{"type":"unordered","items":["Limit x giderken sıfıra sinüs 5x/x⁵ ifadesinde, sinüs 2x/x = 2 olduğundan, 5 tane 2'nin çarpımı olan 32 sonucuna ulaşılır.","Limit x giderken eksi 2'ye (5x+10)√(x+3)+1 / (√(x+3)-1) ifadesinde, eşlilikle çarpma yerine sadeleştirme yaparak limitin 10 olduğuna ulaşılır."]},"beginTime":4210,"endTime":4329,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4210&ask_summarization=1"},{"index":28,"title":"Limit Problemleri Çözümü","list":{"type":"unordered","items":["Cos 2x = 1 - 2sin²x dönüşümü kullanılarak limit problemi çözülüyor.","Sinüs x bölü x ifadesi limit x sonsuza giderken sıfıra gider çünkü sıkıştırma teoremi gereği -1 ≤ sinx/x ≤ 1 olur.","Limit problemlerinde sinüs ve tanjant fonksiyonlarının limit değerleri kullanılarak çözüm bulunuyor."]},"beginTime":4330,"endTime":4507,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4330&ask_summarization=1"},{"index":29,"title":"Türev ve Limit İlişkisi","list":{"type":"unordered","items":["Tanjant kök x fonksiyonunun türevi, limit h sıfıra giderken (f(x+h) - f(x))/h formülüyle hesaplanıyor.","Tanjant kök x'in türevi 1/2x sekant x olarak bulunuyor.","Limit problemlerinde trigonometrik fonksiyonların limit değerleri kullanılarak çözüm yapılıyor."]},"beginTime":4507,"endTime":4649,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4507&ask_summarization=1"},{"index":30,"title":"Trigonometrik Dönüşümler","list":{"type":"unordered","items":["Cos 6x - 1 ifadesi cos 2x = 1 - 2sin²x dönüşümü kullanılarak 1 - 2sin²(3x) şeklinde yazılıyor.","Sinüs ve tanjant fonksiyonlarının limit değerleri kullanılarak limit problemi çözülüyor.","Sinüs 3x/x, tanjant 2x/x ve sinüs x/x ifadelerinin limit değerleri 3, 2 ve 1 olarak bulunuyor."]},"beginTime":4649,"endTime":4777,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4649&ask_summarization=1"},{"index":31,"title":"Kosinüs Fark Formülü","list":{"type":"unordered","items":["Kosinüs farkı formülü cos(x+y) - cos(x-y) = -2sin(x)sin(y) olarak açıklanıyor.","Kosinüs fonksiyonlarının toplam formülü cos(x+y) = cosx.cosy + sinx.siny olarak hatırlatılıyor.","Limit probleminde sinüs ve tanjant fonksiyonlarının limit değerleri kullanılarak çözüm bulunuyor."]},"beginTime":4777,"endTime":4949,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4777&ask_summarization=1"},{"index":32,"title":"Limit Problemleri Çözümü","list":{"type":"unordered","items":["Eğitmen, limit problemlerini çözerek öğrencilerin vizelerinin güzel geçmesini diliyor.","İlk limit probleminde, kosinüs fonksiyonunun sürekli olduğu için limiti içeriye alarak çözümü gösteriyor.","İkinci limit probleminde, mutlak değer ve signum fonksiyonlarını kullanarak x=2 noktasının sağ ve sol limitlerini hesaplıyor."]},"beginTime":4952,"endTime":5093,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=4952&ask_summarization=1"},{"index":33,"title":"Sonsuzluk Problemleri","list":{"type":"unordered","items":["Sonsuzluk sorularında en büyük kuvveti alarak çözümü gösteriyor.","Üçüncü limit probleminde, kosinüs ve sinüs fonksiyonlarının -1 ile 1 aralığında olduğunu kullanarak sıkıştırma yöntemi uyguluyor.","Dördüncü limit probleminde, mutlak değer ve kök fonksiyonlarını kullanarak çözümü gösteriyor."]},"beginTime":5093,"endTime":5320,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5093&ask_summarization=1"},{"index":34,"title":"Tam Değer Fonksiyonu ve Sonsuzluk","list":{"type":"unordered","items":["Beşinci limit probleminde, tam değer fonksiyonunu kullanarak x=2 noktasının sağ ve sol limitlerini hesaplıyor.","Altıncı limit probleminde, kök fonksiyonunun eşleniğini kullanarak sonsuzluk problemini çözüyor.","Sonuç olarak, limitin y=1 doğrusu olan yatay asimptotu olduğunu belirtiyor."]},"beginTime":5320,"endTime":5566,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5320&ask_summarization=1"},{"index":35,"title":"Sürekli Fonksiyonlar","list":{"type":"unordered","items":["Sürekli fonksiyonlar için limitli olmak ve o noktanın görüntüsünün limite eşit olması gerekiyor.","Bir fonksiyonun sürekli olması için x'in solu ve sağının limitinin eşit olması ve o noktanın görüntüsünün limite eşit olması gerekiyor.","Verilen örnekte a=1 ve b=4/3 olarak bulunuyor."]},"beginTime":5576,"endTime":5661,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5576&ask_summarization=1"},{"index":36,"title":"Limit Problemi Çözümü","list":{"type":"unordered","items":["Limit x giderken 101'e soldan veya sağdan giderken, sinüslü ifadelerde dönüşüm yaparak limit hesaplanıyor.","x'in 101'e soldan gittiği durumda, sinüs ifadesi eksi sonsuza gidiyor ve limit 2 olarak bulunuyor.","x'in 101'e sağdan gittiği durumda, c-1 ifadesi 2 olarak bulunuyor ve c=203 olarak hesaplanıyor."]},"beginTime":5661,"endTime":5950,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5661&ask_summarization=1"},{"index":37,"title":"Son Konular","list":{"type":"unordered","items":["Dosyada curve streching ve ikinci derece türevin yorumları konuları da var.","İkinci derece türevin yorumları dosyadaki en güzel soru olarak belirtiliyor."]},"beginTime":5950,"endTime":5960,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5950&ask_summarization=1"},{"index":38,"title":"Ara Değer Teoremi ve Polinomlar","list":{"type":"unordered","items":["Ara değer teoremi, kapalı [a,b] aralığında tanımlı ve sürekli bir fonksiyon için, f(a) \u003c 0 ve f(b) > 0, o zaman [a,b] aralığında en az bir c elemanı vardır ki f(c) = 0, der.","Tek dereceli bir polinom fonksiyonu için, limit x → -∞ p(x) = -∞ ve limit x → ∞ p(x) = ∞ olduğunda, ara değer teoremi gereği en az bir c değeri vardır ki f(c) = 0, der.","Beşinci dereceden bir polinomun en fazla beş reel kökü vardır."]},"beginTime":5965,"endTime":6107,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=5965&ask_summarization=1"},{"index":39,"title":"Ters Fonksiyonun Türevi","list":{"type":"unordered","items":["Ters fonksiyonun türevi, f⁻¹(f(x)) = x eşitliğinden türetilir ve f⁻¹'(x) = 1 / f'(f(x)) formülüyle hesaplanır.","Verilen fonksiyonun tersi doğrudan alınamadığı için, türev formülü kullanılarak f⁻¹'(-1) = 1/f'(-1) hesaplanır.","f(x) = x³ - 3x² - 1 fonksiyonunun x = 3 noktasında türevi f'(3) = 9 olduğundan, f⁻¹'(-1) = 1/9 olarak bulunur."]},"beginTime":6107,"endTime":6304,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6107&ask_summarization=1"},{"index":40,"title":"Türevin Limit Tanımı ile Hesaplanması","list":{"type":"unordered","items":["Türevin limit tanımı kullanılarak, f'(x) = lim[h→0] (f(x+h) - f(x)) / h formülü uygulanır.","Verilen fonksiyonun türevi, limit tanımı kullanılarak -1/(2(x+2)³) olarak hesaplanır.","Bu sonuç, f(x) = (x+2)⁻² fonksiyonunun türevinin f'(-1) = -1/2 olarak hesaplandığı sonucuna eşdeğerdir."]},"beginTime":6304,"endTime":6495,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6304&ask_summarization=1"},{"index":41,"title":"Teğet ve Normal Line'lar","list":{"type":"unordered","items":["Bir eğrinin üzerindeki bir noktadan çizilen teğetin eğimi, o noktadaki türev değerine eşittir.","Verilen denklem x²y - xy² = x²y² + 2y + 2 için kısmi türev alınarak f'(x,y) = (2xy + x² - y² - 2xy) / (2xy² + 2y + 2) bulunur.","x = 0, y = 2 noktasında türev değeri hesaplanarak teğet line'ın eğimi bulunur."]},"beginTime":6495,"endTime":6597,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6495&ask_summarization=1"},{"index":42,"title":"Teğet ve Normal Line Problemi","list":{"type":"unordered","items":["dy/dx türevi hesaplanarak, x=2 noktasında dy/dx=4 olarak bulunuyor.","Teğet denklemi, (y-2)=4(x-2) şeklinde yazılabilir ve sonucu y=4x+2 olarak elde edilir.","Normal line, teğetin dik olduğu doğrudur ve eğimleri çarpımı -1 olmalıdır."]},"beginTime":6603,"endTime":6695,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6603&ask_summarization=1"},{"index":43,"title":"Logaritmik Fonksiyonun Türevi","list":{"type":"unordered","items":["y=x^(sinx) fonksiyonunun türevi için her iki tarafın da logaritması alınır: ln(y)=sinx·ln(y).","Türev alınarak y' = y·(sinx·cosx + 1/x) olarak bulunur.","x=2 noktasında y' = 1 olarak hesaplanır ve normal line denklemi y=x-1 olarak bulunur."]},"beginTime":6695,"endTime":6911,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6695&ask_summarization=1"},{"index":44,"title":"Balon Hacmi ve Yüzey Alanı Problemi","list":{"type":"unordered","items":["Balon hacmi V=4/3πr³ formülüyle hesaplanır ve r=5 metre olduğunda hacim 1000 metreküp/dakika olarak verilir.","Zincir kuralı kullanılarak dr/dt = 1 olarak bulunur.","Yüzey alanı A=8πr formülüyle hesaplanır ve r=5 metre olduğunda yüzey alanı 40π birim² olarak bulunur."]},"beginTime":6911,"endTime":7113,"href":"/video/preview/16786679170945077717?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=6911&ask_summarization=1"},{"index":45,"title":"Fonksiyonun Grafiği ve Özellikleri","list":{"type":"unordered","items":["Birinci türev, fonksiyonun artanlığını ve azalanlığını gösterir, işaret değiştirdiği noktalara ekstremum nokta denir.","İkinci türev, fonksiyonun konkavitesini (dışbükey veya içbükey) gösterir.","Konkavitenin değiştiği noktaya büküm noktası denir, ancak her ikinci türevin olduğu nokta büküm noktası 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O yüzden yavaş yavaş konuları hem pdf haline getirmeye hem de anlatmaya çalışıyorum.","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3952324/14c3b3c95e3c9e2e6de3172165c000b3/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/uzwzLwIAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"12","reqid":"1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=videoid:8219283975777781596","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitmeni tarafından sunulan integral konusunu anlatan kapsamlı bir eğitim içeriğidir. Eğitmen, kendi hazırladığı notları kullanarak konuyu detaylı bir şekilde açıklamaktadır.","Video, belirsiz integrallerin temel kavramlarını ele almaktadır. İlk olarak ilkel fonksiyon ve belirsiz integral arasındaki ilişki açıklanmakta, ardından integral sabiti (C) kavramı, integral ve türev arasındaki ilişki, diferansiyel kavramı ve integral özellikleri anlatılmaktadır. Daha sonra temel integral formülleri (x üzeri alfa, 1/x, a üzeri x, e üzeri x, sinüs, kosinüs, tanjant, sekant, kosekant fonksiyonlarının integralleri) örneklerle açıklanmakta ve son olarak hiperbolik fonksiyonlar ile trigonometrik fonksiyonların türevleri ele alınmaktadır.","Eğitmen, ilerleyen zamanlarda belirsiz integralleri, integrasyon metotlarını ve altı farklı başlıkta inceleyeceğini belirtmektedir. Video, matematiksel kavramları grafikler üzerinden görselleştirerek anlatmakta ve bir sonraki videoda değişken değiştirme metodu ve rasyonel metot gibi konuların işleneceğini belirterek sona ermektedir."]},"endTime":1962,"title":"İntegral Dersi: Belirsiz İntegraller ve İlke Fonksiyonlar","beginTime":0}],"fullResult":[{"index":0,"title":"İntegral Serisi ve Belirsiz İntegral","list":{"type":"unordered","items":["Calculus serisi için integral konusunu anlatacak ve notları kendisi hazırlayarak hızlı bir şekilde ilerleyecek.","Belirsiz integralleri ve integrasyon metotlarını anlatacak, altı başlıkta inceleyecek ve ispatlı gidecek.","Videoda belirsiz integraller, ilkel fonksiyonlar ve türevle ilgili temel kavramlar ele alınacak."]},"beginTime":0,"endTime":52,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=0&ask_summarization=1"},{"index":1,"title":"İlke Fonksiyon Tanımı","list":{"type":"unordered","items":["İlke fonksiyon, bir fonksiyonun türevi olarak verilen başka bir fonksiyon olarak tanımlanır ve anti-türev olarak da adlandırılır.","Bir fonksiyonun ilkel fonksiyonu, o fonksiyonun türevi olarak verilen fonksiyonu üretir.","İlke fonksiyon, türevlenebilir olduğu aralıkta sürekli olmak zorundadır."]},"beginTime":52,"endTime":220,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=52&ask_summarization=1"},{"index":2,"title":"İlke Fonksiyon Örnekleri","list":{"type":"unordered","items":["x⁴/4 fonksiyonu x³ fonksiyonunun ilkelidir çünkü türevi x³'e eşittir.","Sinüs fonksiyonu kosinüs fonksiyonunun ilkelidir çünkü türevi kosinüse eşittir.","1/√(1-x²) fonksiyonu, (-1,1) aralığında ilkelidir çünkü bu aralıkta sürekli ve 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tüm ilkel fonksiyonların kümesini ifade eder.","Belirsiz integral, bir ifadenin türevi verilen fonksiyonu bulmayı amaçlar.","İntegral sembolü kullanılırken, integrand (entegre edilen ifade) ve integrasyon değişkeni belirtilir."]},"beginTime":551,"endTime":659,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=551&ask_summarization=1"},{"index":5,"title":"İntegrasyon Sabiti","list":{"type":"unordered","items":["İntegral hesaplarında C olarak gösterilen sabit, integrasyon sabiti olarak adlandırılır.","İntegrasyon sabiti, belirsiz integralin tüm ilkel fonksiyonları kapsamasını sağlar.","Her fonksiyonun belirsiz integralinin var olduğu anlamına gelmez, bu konu ileride incelenecektir."]},"beginTime":659,"endTime":692,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=659&ask_summarization=1"},{"index":6,"title":"Signum Fonksiyonunun İntegrali","list":{"type":"unordered","items":["Signum fonksiyonu, x>0 için 1, x=0 için 0, x\u003c0 için -1 değerlerini verir.","Signum fonksiyonunun eksi sonsuz ile sonsuz aralığındaki ilkel fonksiyonu yoktur çünkü sıfır noktasında türevi yoktur.","Sıfır noktasında sivri nokta olduğu için fonksiyon süreksiz olur ve bu nedenle ilkel fonksiyonu bulunamaz."]},"beginTime":692,"endTime":907,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=692&ask_summarization=1"},{"index":7,"title":"İntegral Özellikleri ve Diferansiyel","list":{"type":"unordered","items":["İntegralin türevi, orijinal fonksiyona eşittir.","Diferansiyel operatörü (d/dx) türevi temsil ederken, d(f(x)) ifadesi diferansiyeli temsil eder.","İntegral dışındaki diferansiyel, d(f(x)) = f'(x)dx şeklinde ifade edilir."]},"beginTime":907,"endTime":985,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=907&ask_summarization=1"},{"index":8,"title":"İntegral Kavramı ve Özellikleri","list":{"type":"unordered","items":["Diferansiyel integrali alınabilir ve diferansiyel efix'in integrali büyük fix artı c olarak bulunur.","İntegral alma sırasında alfa sıfırdan farklıysa, alfa integral dışına çıkabilir.","İntegral alma sırasında alfa artı beta çarpı gx gibi bir ifade varsa, alfa ve beta dışarı çıkarılabilir."]},"beginTime":993,"endTime":1117,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=993&ask_summarization=1"},{"index":9,"title":"İntegral Özellikleri ve Örnekler","list":{"type":"unordered","items":["a ile b arasında büyük eşit küçük ise, integral alınırken a katsayısı varsa bir bölü a katsayısı eklenmelidir.","Sinüs ve kosinüs fonksiyonlarının integrali, türevleri belli olan fonksiyonlarla ilişkilendirilerek bulunabilir.","İntegral sabiti olan c her zaman hesaplanırken dikkate alınmalıdır."]},"beginTime":1117,"endTime":1318,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1117&ask_summarization=1"},{"index":10,"title":"Temel İntegral Formülleri","list":{"type":"unordered","items":["x üzeri alfa fonksiyonunun integrali, alfa artı bir kuvveti alıp alfa artı bir'e bölmekle bulunur, alfa eksi bir'den farklı olmak zorundadır.","1 bölü x dx integrali ln|x| artı c olarak bulunur.","a üzeri x fonksiyonunun integrali a üzeri x bölü ln(a) artı c olarak bulunur, a bir'den farklı olmak zorundadır."]},"beginTime":1318,"endTime":1471,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1318&ask_summarization=1"},{"index":11,"title":"Trigonometrik Fonksiyonların İntegrali","list":{"type":"unordered","items":["Sinüs fonksiyonunun integrali eksi kosinüs x artı c olarak bulunur.","Kosinüs fonksiyonunun integrali sinüs x artı c olarak bulunur.","Tanjant fonksiyonunun integrali ln|sekant x| artı c olarak bulunur, kotanjant fonksiyonunun integrali eksi ln|sinüs x| artı c olarak bulunur."]},"beginTime":1471,"endTime":1615,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1471&ask_summarization=1"},{"index":12,"title":"Hiperbolik Fonksiyonlar","list":{"type":"unordered","items":["Hiperbolik kosinüs fonksiyonu cosh(x) = (e^x + e^-x) / 2 olarak tanımlanır.","Hiperbolik sinüs fonksiyonu sinh(x) = (e^x - e^-x) / 2 olarak tanımlanır.","Tanjant ve kotanjant formülleri trigonometrik formüllerle aynı şekilde geçerlidir."]},"beginTime":1617,"endTime":1653,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1617&ask_summarization=1"},{"index":13,"title":"Türevler ve İntegral Formülleri","list":{"type":"unordered","items":["Kosinüs x'in türevi sinüs x'tir, sinüs x'in türevi ise kosinüsteir.","Tanjant x'in türevi 1/cos²x veya sekant²x'tir.","Tanjant x'in integrali arctan(x) + C, arctan(x)'in türevi 1/(1+x²)'dir."]},"beginTime":1653,"endTime":1850,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1653&ask_summarization=1"},{"index":14,"title":"Trigonometrik Fonksiyonların İntegrali","list":{"type":"unordered","items":["Arksinüs x'in türevi 1/√(1-x²)'dir.","Arksinüs x'in integrali x + C olarak bulunur.","Bu videoda belirsiz integralin tanımı ve temel integralleri ele alınmıştır."]},"beginTime":1850,"endTime":1953,"href":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1850&ask_summarization=1"}],"linkTemplate":"/video/preview/8219283975777781596?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=%%timestamp%%&ask_summarization=1"},"isAdultDoc":false,"relatedParams":{"text":"Calculus-1 : Belirsiz İntegral ve İlkel Fonksiyon Tanımı -1","related_orig_text":"Not Calculus","related_porno":false,"related_less_3m_off":true,"client":"d2d","no_cnt":1,"related_src":"serp","related":"{\"porno\":false,\"vfp\":1,\"orig_text\":\"Not 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the channel to take advantage of the benefits and support the channel: / tunç kurt matematik Hello, in response to your messages, I'm working on Calculus 1 homework solutions and preparatory...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/1033459/4f8ff6446b207ca50f9b253bcc44adc8/564x318_1"},"target":"_self","position":"13","reqid":"1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=videoid:6538981202598713059","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik eğitmeni tarafından sunulan kapsamlı bir türev dersidir. Eğitmen, tahtada veya yazılım arayüzünde matematiksel ifadeleri göstererek konuyu adım adım açıklamaktadır.","Video, türev konusunun temel kavramlarından ileri seviye konularına kadar uzanan bir yapıya sahiptir. İçerikte fonksiyonların tanım kümeleri, e üzeri fonksiyonların türevi, ters fonksiyonların türevi, ark tanjant, arksinüs fonksiyonlarının türevi, kısmi türevler, ln fonksiyonlarının türevi, normal line bulma, teğet doğruları, paralel doğrular ve Intermediate Value Theorem gibi konular ele alınmaktadır.","Videoda özellikle vize ve final sınavlarına hazırlanan öğrenciler için önemli bilgiler verilmekte, türev alma yöntemleri detaylı şekilde gösterilmektedir. Ayrıca, kübik fonksiyonların reel köklerinin varlığı ve bir fonksiyonun sürekli ve türevlenebilir olması için gerekli koşullar da açıklanmaktadır. Video, bir serinin parçası olup, ikinci dönemde devam edeceği belirtilmektedir."]},"endTime":2733,"title":"Kalkülüs Dersi: Türev Konusu","beginTime":0}],"fullResult":[{"index":0,"title":"Giriş ve Ders İçeriği","list":{"type":"unordered","items":["Önceki kayıtta kalkülüs tekrar testi yüklenmiş, ancak türev kısmı eksik kalmıştır.","Matematik bölümü öğrencileri için ilk vize genellikle fonksiyonların tanım kümeleri ve grafiği konularını kapsar, türev sadece tanım kısımları içerir.","Mühendislikte veya servis dersleri olan matematikte (103, 104, 109 kodları) türev konusu daha erken gelebilir."]},"beginTime":8,"endTime":65,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=8&ask_summarization=1"},{"index":1,"title":"Video İçeriği ve Destek İsteği","list":{"type":"unordered","items":["Videoda eski üniversitelerden çıkan vize soruları çözülecek ve bu pratikler gelecek sorulara hazırlık sağlayacaktır.","İzleyicilerden videoyu beğenip paylaşmaları ve \"katıl\" butonuna tıklamaları isteniyor.","Katıl kısmına katılanlar PDF'lere erişebilir ve ilerleyen dönemlerde çözülen soruların dijital notlarına da ulaşabilirler."]},"beginTime":65,"endTime":128,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=65&ask_summarization=1"},{"index":2,"title":"Türev Soruları","list":{"type":"unordered","items":["Bu dosyada sadece türev soruları olacak, limit ve fonksiyon tanım kümesi ile ilgili sorular için açıklama kısmında linkler bırakılmıştır.","Fonksiyonun tanım kümesi sorusunda, üslü fonksiyonların tanım kümesi reel sayılar olup, ark tanjant fonksiyonunun tanım kümesi de reel sayılar olarak belirlenmiştir.","Kök içerisindeki ifadenin tanımlı olması için 1+ln x ≥ 0 şeklinde bir eşitsizlik kurulmuş ve x > 1 olarak 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özelliğinden türetilir.","g⁻¹(g(x)) = x eşitliğinin her iki tarafının türevi alınarak g⁻¹'(g(x)) × g'(x) = 1 ilişkisi elde edilir.","Ark tanjant fonksiyonunun türevi, tanjant fonksiyonunun türevi kullanılarak g⁻¹'(tan(x)) = 1 / (1 + tan²(x)) olarak bulunur."]},"beginTime":446,"endTime":550,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=446&ask_summarization=1"},{"index":5,"title":"Ark Tanjant Fonksiyonunun Türevi","list":{"type":"unordered","items":["Ark tanjant x'in türevi 1/(1+tan²x) olarak hesaplanır.","Eğer ark tanjant u veya ark tanjant f(x) gibi bir ifadenin türevine bakıyorsak, 1/(1+x²) çarpı f'(x) şeklinde hesaplanır.","Geçen derste arksinüs ve arksinüs'ün gösterimleri yapılmıştır."]},"beginTime":552,"endTime":618,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=552&ask_summarization=1"},{"index":6,"title":"Fonksiyonun Türevi Hesaplama","list":{"type":"unordered","items":["e üzeri ark tanjant ifadesinin türevi, e üzeri ark tanjant kök içerisinde 1+lnx çarpı (1/(1+lnx)²) olarak hesaplanır.","Kök fonksiyonunun türevi, icerisinin türevi bölü 2 kök içerisinde ifadenin aynısı veya tepesinde 1/2 şeklinde rasyonel formatta yazılır.","Fonksiyonun kendisiyle tersinin bileşkesinin türevi, fonksiyonun kendisiyle tersinin çarpımı olarak hesaplanır."]},"beginTime":618,"endTime":759,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=618&ask_summarization=1"},{"index":7,"title":"Ters Fonksiyonların Türevi","list":{"type":"unordered","items":["Ters fonksiyonların 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hesaplanır."]},"beginTime":898,"endTime":1038,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=898&ask_summarization=1"},{"index":9,"title":"Logaritmik Fonksiyonların Türevi","list":{"type":"unordered","items":["f(x) = ln(cosx + 3√2 + sinx) fonksiyonunun türevi, f'(x) = (cosx + 3√2 + sinx) / (ln(cosx + 3√2 + sinx)) olarak bulunur.","f(x) = 3ˣ fonksiyonunun türevi, f'(x) = 3ˣ × ln3 olarak hesaplanır.","g(x) = log₂x fonksiyonunun türevi, g'(x) = 1 / (x × ln2) olarak bulunur."]},"beginTime":1038,"endTime":1284,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1038&ask_summarization=1"},{"index":10,"title":"Arksinüs Fonksiyonunun Türevi","list":{"type":"unordered","items":["f(x) = arksinüs(sinüs x) fonksiyonunun tersini bulmak için türev alma yöntemi kullanılır.","Arksinüs fonksiyonunun türevi, içerideki ifadenin türevi bölü kök(1 - içerideki ifadenin karesi) şeklinde hesaplanır.","Arksinüs(x²) fonksiyonunun türevi ise 2x/(1 - x⁴) olarak bulunur."]},"beginTime":1295,"endTime":1491,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1295&ask_summarization=1"},{"index":11,"title":"Tanjant Doğrusu Bulma","list":{"type":"unordered","items":["Bir eğrinin belirli bir noktadaki tanjant doğrusunun eğimi, o noktadaki fonksiyonun türevine eşittir.","f(x) = eˣ + 2 + 4x³ fonksiyonunun 0,1 noktasındaki türevi 3'tür.","Eğimi 3 olan ve (0,1) noktasından geçen doğru denklemi y = 3x + 1 olarak bulunur."]},"beginTime":1491,"endTime":1630,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1491&ask_summarization=1"},{"index":12,"title":"Paralel Doğrunun Üzerindeki Nokta","list":{"type":"unordered","items":["Bir eğrinin belirli bir doğrusuna paralel olan bir noktasının varlığını göstermek için, o noktadaki fonksiyonun türevinin paralel doğrunun eğimine eşit olması gerekir.","f(x) = eˣ + 2 + 4x³ fonksiyonunun türevi f'(x) = eˣ + 2 + 4x² + 4x³'tür.","f'(x) = 2 eşitliğini sağlayan bir x değeri için, intermit value teoremi kullanılarak bu noktanın varlığı gösterilir."]},"beginTime":1630,"endTime":1830,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1630&ask_summarization=1"},{"index":13,"title":"İç İçe Fonksiyonların Türevi","list":{"type":"unordered","items":["f(x) = sin(sec x) fonksiyonunun türevi hesaplanırken, sec x = 1/cos x olarak tanımlanır.","Bölüm türevi formülü kullanılarak f'(x) = tan x / cos sec x bulunur.","Bu sonuç, sec x'in türevine eşittir."]},"beginTime":1830,"endTime":1866,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1830&ask_summarization=1"},{"index":14,"title":"Türev Hesaplama Yöntemleri","list":{"type":"unordered","items":["İç içe geçmiş fonksiyonların türevi hesaplanırken en dışarıdan başlayıp türevliye türevliye gidilmelidir.","Sekant fonksiyonunun türevi sekant x çarpı tanjant x'tir.","Türev hesaplamasında f'(x) = f'(g(x)) × g'(x) formülü kullanılır."]},"beginTime":1869,"endTime":2006,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=1869&ask_summarization=1"},{"index":15,"title":"Logaritma Fonksiyonlarının Türevi","list":{"type":"unordered","items":["Logaritma fonksiyonlarının türevi hesaplanırken, logaritma fonksiyonu g(x) = 5^f(x) şeklinde yazılabilir.","g'(x) = (g'(x) / 5^f(x)) × ln(5) formülü kullanılır.","Logaritma fonksiyonlarının türevi hesaplanırken, logaritmanın tabanı ve fonksiyonun değeri dikkate alınmalıdır."]},"beginTime":2006,"endTime":2101,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2006&ask_summarization=1"},{"index":16,"title":"Teğet ve Normal Line","list":{"type":"unordered","items":["Teğet line ile normal line birbirleriyle dik kesişir ve eğimlerinin çarpımı -1'e eşittir.","Eğrinin türevi alınarak teğet line'ın eğimi (m) bulunabilir.","Normal line'ın eğimi, teğet line'ın eğiminin tersi ve işaretinin tersi olarak hesaplanır."]},"beginTime":2101,"endTime":2270,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2101&ask_summarization=1"},{"index":17,"title":"Fonksiyonların Sürekliliği ve Türevlenebilirliği","list":{"type":"unordered","items":["Polinom tipteki fonksiyonlar tüm reel sayılarda süreklidir ve türevlenebilirdir.","Intermediate Value Theorem kullanılarak fonksiyonların belirli aralıklarda değerleri karşılaştırılabilir.","Fonksiyonun belirli aralıklarda değerleri karşılaştırıldığında, en az bir noktada fonksiyonun belirli bir değere eşit olduğu sonucuna varılabilir."]},"beginTime":2270,"endTime":2366,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2270&ask_summarization=1"},{"index":18,"title":"Kübik Fonksiyonun Kökleri","list":{"type":"unordered","items":["Kübik fonksiyonun kökleri hesaplanarak P₁ = -3, P₂ = 8 ve P₃ = 12 olarak bulunuyor.","Fonksiyonun üç reel kökü olduğu gösteriliyor.","Türev yöntemi ile de köklerin varlığı gösterilebilir: f'(x) = 3x² - 5 = 0 denkleminin kökleri -√5/3 ve +√5/3 olup, bu noktaların görüntülerinin zıt işaretli olması üç reel kökün varlığını garanti eder."]},"beginTime":2367,"endTime":2543,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2367&ask_summarization=1"},{"index":19,"title":"Süreklilik ve Türevlenebilirlik","list":{"type":"unordered","items":["Sabit c ve d değerleri için hem sürekli hem diferansiyellenebilir bir fonksiyon bulunması isteniyor.","Süreklilik için x=3 noktasındaki soldan ve sağdan limitlerin eşit olması gerekiyor: 3c+4 = 8d.","Türevlenebilirlik için x=3 noktasındaki sağ ve sol türevlerin eşit olması gerekiyor: 6d = c.","Bu denklemlerden d = -2/5 ve c = -12/5 değerleri bulunuyor."]},"beginTime":2543,"endTime":2685,"href":"/video/preview/6538981202598713059?parent-reqid=1769842687513388-6984151502383330951-balancer-l7leveler-kubr-yp-vla-140-BAL&text=Not+Calculus&t=2543&ask_summarization=1"},{"index":20,"title":"Dersin Kapanışı","list":{"type":"unordered","items":["Eğitmen, dersi hazırlarken ve çözerken keyif 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