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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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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 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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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+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 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İlk bölümde x üzeri n ifadelerinin integrali, rasyonel ifadelerde integral alma ve trigonometrik fonksiyonların integrali ele alınırken, ikinci bölümde değişken değiştirme yöntemi ve özellikle kosinüs fonksiyonlarının integrali adım adım gösterilmektedir.","Eğitmen, integral hesaplamalarında kullanılan temel formülleri ve trigonometrik eşitlikleri kullanarak çeşitli soruları çözmekte ve bir sonraki videoda değişken değiştirme yöntemlerinin detaylı olarak anlatılacağını belirtmektedir."]},"endTime":859,"title":"Temel İntegral Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Temel İntegral Kavramı","list":{"type":"unordered","items":["Videoda temel integral kavramları anlatılacak ve değişken değiştirme yöntemi de ele alınacak.","İntegral, bir ifadenin türevini alarak üzerine olan ifadeyi bulma işlemidir.","Polinom tipteki fonksiyonların integrali, derecenin üstünü bir arttırıp arttırdığımız dereceye bölerek 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şeklinde yazılabilir.","İntegral hesaplamalarında trigonometrik dönüşümler ve temel trigonometrik eşitlikler önemlidir."]},"beginTime":393,"endTime":543,"href":"/video/preview/2791149328929414095?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=393&ask_summarization=1"},{"index":3,"title":"Üstel İntegral Örnekleri","list":{"type":"unordered","items":["Üstel fonksiyonların integrali, türevi alınarak bulunur.","950x² dx integrali, 950x×9/9 + 150x/ln15 + 250x/ln25 + C şeklinde çözülür.","İntegral hesaplamalarında değişken değiştirme yöntemi de kullanılabilir."]},"beginTime":543,"endTime":600,"href":"/video/preview/2791149328929414095?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=543&ask_summarization=1"},{"index":4,"title":"İntegral Hesaplama ve Değişken Değiştirme","list":{"type":"unordered","items":["İntegral hesaplamasında değişken değiştirme yöntemi kullanılabilir, 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(Analiz) ile Grafik Çizimi konu anlatımı ders videosu. 10 binden fazla ücretsiz ders videosu için: https://www.khanacademy.org.tr Matematikten sanat tarihine, ekonomiden fen bilimlerine...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/1667319/14719cea1a8b7fe99915307b0f86ebe5/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/9jwAyAAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"9","reqid":"1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=videoid:5290259787722281402","teaser":[{"list":{"type":"unordered","items":["Bu video, bir eğitmen tarafından sunulan matematik dersi formatında hazırlanmıştır. Eğitmen, öğrencilere adım adım bir fonksiyonun grafiğini çizme sürecini göstermektedir.","Videoda, f(x) = 3x⁴ - 4x³ + 2 fonksiyonunun grafiğini çizmek için gerekli analizler yapılmaktadır. Önce kritik noktalar (x=0 ve x=1) bulunmakta, ardından ikinci türev kullanılarak çukurluk yönü incelenmekte ve potansiyel büküm noktaları (x=0 ve x=2/3) belirlenmektedir. Bu bilgiler kullanılarak fonksiyonun grafiği çizilmektedir.","Video, türev kavramını kullanarak fonksiyonların analizi ve grafik çizimi konusunu ele almaktadır. Eğitmen, analiz becerilerini kullanarak karmaşık bir fonksiyonun grafiğini çizme sürecini detaylı bir şekilde göstermektedir."]},"endTime":979,"title":"Fonksiyonun Grafiğini Çizme: Türev ve Analiz Yöntemleri","beginTime":0}],"fullResult":[{"index":0,"title":"Fonksiyonun Grafiğini Çizme","list":{"type":"unordered","items":["Türev, çukurluk yönü, maksimum, minimum ve büküm noktaları bilgisi kullanılarak hesap makinesi olmadan bir fonksiyonun grafiği çizilebilir.","Fonksiyon f(x) = 3x⁴ - 4x³ + 2 olarak tanımlanmıştır.","Fonksiyonun grafiğini çizmek için öncelikle kritik noktalar belirlenmelidir."]},"beginTime":0,"endTime":32,"href":"/video/preview/5290259787722281402?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=0&ask_summarization=1"},{"index":1,"title":"Kritik Noktaların Bulunması","list":{"type":"unordered","items":["Kritik noktalar, türevin sıfır veya tanımsız olduğu noktalardır.","Verilen fonksiyon her yerde türevlidir, 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Eğitmen, fonksiyonların türevlerini hesaplayarak grafiklerini çizme sürecini adım adım göstermektedir.","Videoda, f(x) = x⁴ + 27'nin doğal logaritması fonksiyonunun birinci ve ikinci türevlerinin hesaplanması, kritik noktaların bulunması ve büküm noktası adaylarının incelenmesi anlatılmaktadır. Eğitmen önce türevleri hesaplayıp, ardından kritik noktaları ve büküm noktası adaylarını belirlemek için ikinci türevin işaret değişimini incelemektedir. Son olarak, bu bilgileri kullanarak fonksiyonun grafiğini çizer ve hesap makinesi ile çizimini doğrular.","Videoda ayrıca, hesap makinesiz grafik çizme teknikleri gösterilmekte ve grafikli hesap makinesiyle kontrol edilebileceği belirtilmektedir. Sonuç olarak, x = 3 ve x = -3'ün büküm noktaları olduğu, x = 0'un ise minimum nokta olduğu belirlenmektedir."]},"endTime":1187,"title":"Matematik Dersinde Fonksiyonun Türevleri ve Grafiği Çizimi","beginTime":0}],"fullResult":[{"index":0,"title":"Fonksiyonun Türevleri","list":{"type":"unordered","items":["Fonksiyonumuz f(x) = x⁴ + 27'nin doğal logaritması olarak tanımlanmıştır.","Fonksiyonun birinci ve ikinci türevleri alınıp, hesap makinesiz grafiği çizilecektir.","Birinci türev f'(x) = 4x³ / (x⁴ + 27) olarak hesaplanmıştır."]},"beginTime":1,"endTime":76,"href":"/video/preview/846790392502462576?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=1&ask_summarization=1"},{"index":1,"title":"İkinci Türev Hesaplama","list":{"type":"unordered","items":["İkinci türev f''(x) hesaplanırken çarpım kuralı kullanılmıştır.","İkinci türev f''(x) = (12x²(x⁴ + 27) - 16x⁶) / (x⁴ + 27)² olarak sadeleştirilmiştir.","İkinci türevin paydası x⁴ + 27 kare olarak ifade edilmiştir."]},"beginTime":76,"endTime":275,"href":"/video/preview/846790392502462576?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=76&ask_summarization=1"},{"index":2,"title":"Kritik Noktalar ve Büküm Noktaları","list":{"type":"unordered","items":["Birinci türevin sıfır olduğu nokta x = 0'dır ve bu kritik noktadır.","Kritik noktada y koordinatı ln(27) ≈ 3,30'dur.","İkinci türevin sıfır olduğu noktalar büküm noktası adaylarıdır, ancak ikinci türevin sıfır olması kesinlikle büküm noktası anlamına gelmez.","Büküm noktası için ikinci türevin işaretinin değişmesi gerekir."]},"beginTime":275,"endTime":469,"href":"/video/preview/846790392502462576?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=275&ask_summarization=1"},{"index":3,"title":"Büküm Noktalarının Bulunması","list":{"type":"unordered","items":["Büküm noktaları için aday noktalar bulunmalı ve ikinci türevin sıfır olduğu noktalar incelenmelidir.","İkinci türevin payı 27×12x²-4x⁶=0 denklemiyle hesaplanır ve x=±3 değerleri büküm noktası adayları olarak bulunur.","x=0 noktası ikinci türevin sıfır olduğu bir noktadır ancak büküm noktası değildir çünkü ikinci türevin işareti bu noktada değişmez."]},"beginTime":470,"endTime":750,"href":"/video/preview/846790392502462576?parent-reqid=1773837033928665-3936108505706160201-balancer-l7leveler-kubr-yp-vla-77-BAL&text=New+Calculus&t=470&ask_summarization=1"},{"index":4,"title":"Büküm Noktalarının Doğrulanması","list":{"type":"unordered","items":["x=3 civarında ikinci türevin işareti 3'e yaklaşırken pozitif, 3'ten büyük olduğunda ise negatif olur.","x=-3 civarında da benzer şekilde, -3'e yaklaşırken ikinci türev pozitif, -3'ten küçük olduğunda ise negatif olur.","Bu işaret değişiklikleri, x=3 ve x=-3'nin büküm noktaları olduğunu 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