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KONU PDF https://drive.google.com/file/d/1GIXc... Platin Kitap İnternet Satiş LİNK: Miray Yayınları...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3779974/66906eb3de231fce1b206ad9b452e110/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/rWM9KAIAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"7","reqid":"1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=videoid:9951096995897381327","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik öğretmeninin türev konusunu anlattığı eğitim içeriğidir. Öğretmen, tahtada çizimler yaparak konuyu görsel olarak açıklamaktadır.","Video, fonksiyonların artan ve azalanlık özelliklerini türevle ilişkisini detaylı şekilde ele almaktadır. Öğretmen önce teorik bilgileri vererek artan fonksiyonların türevinin sıfırdan büyük, azalan fonksiyonların türevinin sıfırdan küçük ve sabit fonksiyonların türevinin sıfır olduğunu açıklar, ardından bu bilgileri kullanarak örnek problemler çözer ve grafiksel olarak fonksiyonların artan-azalanlık özelliklerini gösterir.","Videoda ayrıca fonksiyonların artan ve azalan olduğu aralıkların belirlenmesi için türev alma yöntemi, işaret tablosu oluşturma, parabolün tepe noktasının bulunması ve ekstremum noktalarının (yerel maksimum ve minimum) belirlenmesi gibi konular da anlatılmaktadır. Öğretmen, ÖSYM ve MEB sınavlarında kullanılan grafik okuma tekniklerini de detaylı şekilde açıklamaktadır."]},"endTime":2494,"title":"Türev Dersi: Fonksiyonların Artan ve Azalanlık Özellikleri","beginTime":0}],"fullResult":[{"index":0,"title":"Türev Dersinin Önemi","list":{"type":"unordered","items":["Öğretmen, türev konusunu ayrıntılı anlattığını ve bu videoları dinleyenlerin türevden soru kaçırma olasılığının yüzde alacağını belirtiyor.","Altın kitaptan bir fasikül kadar soru çözdüğünü ve tüm soru tiplerini vermeye çalıştığını söylüyor.","Türevin önemli kısımlarından biri olan artanlık-azalanlık konusuna geçiliyor."]},"beginTime":7,"endTime":99,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=7&ask_summarization=1"},{"index":1,"title":"Artan Fonksiyonlar","list":{"type":"unordered","items":["Bir f fonksiyonu, AB aralığında x₁ \u003c x₂ olduğunda f(x₁) \u003c f(x₂) oluyorsa artan fonksiyondur.","Artan fonksiyonların grafiğinde soldan sağa doğru yokuş tırmanılır.","Artan fonksiyonların çizilen teğetlerinin eğim açıları dar açıdır, bu nedenle eğimleri sıfırdan büyüktür ve birinci türevleri sıfırdan büyüktür."]},"beginTime":99,"endTime":356,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=99&ask_summarization=1"},{"index":2,"title":"Azalan Fonksiyonlar","list":{"type":"unordered","items":["Bir f fonksiyonu, AB aralığında x₁ > x₂ olduğunda f(x₁) > f(x₂) oluyorsa azalan fonksiyondur.","Azalan fonksiyonların grafiğinde soldan sağa doğru yokuş aşağı inilir.","Azalan fonksiyonların çizilen teğetlerinin eğim açıları geniş açıdır, bu nedenle eğimleri negatiftir ve birinci türevleri sıfırdan 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Özellikleri","list":{"type":"unordered","items":["Artan bir fonksiyonda, teğetlerin eğim açısı dar açı olduğundan türev değeri sıfırdan büyüktür.","Azalan bir fonksiyonda, teğetlerin eğim açısı geniş açı olduğundan türev değeri sıfırdan küçüktür.","Sabit giden bir fonksiyonda, teğetlerin eğim açısı sıfır olduğundan türev değeri de sıfırdır."]},"beginTime":1225,"endTime":1306,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1225&ask_summarization=1"},{"index":7,"title":"Türev ve Fonksiyon Arasındaki İlişki","list":{"type":"unordered","items":["Fonksiyonun artan olduğu aralıkta birinci türev sıfırdan büyük, azalan olduğu aralıkta birinci türev sıfırdan küçük, sabit olduğu aralıkta birinci türev sıfıra eşittir.","Birinci türev sıfırdan büyükse fonksiyon artandır, sıfırdan küçükse azalandır, sıfıra eşitse sabit değerlidir.","Fonksiyonun artan veya azalan olduğunu belirlemek için fonksiyonun 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sorularında fonksiyonun türevini alıp, türevi sıfıra eşitleyerek ekstremum noktalarını buluruz.","Türevin işaretini inceleyerek fonksiyonun artan ve azalan olduğu aralıkları belirleriz.","Türevin işaret değiştirdiği yer, yerel minimum veya maksimum noktası olarak adlandırılır."]},"beginTime":1512,"endTime":1671,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1512&ask_summarization=1"},{"index":10,"title":"Parabol Örneği","list":{"type":"unordered","items":["f(x) = ax² + bx + c şeklindeki bir parabolün tepe noktasının x koordinatı, birinci türevin sıfıra eşitlenmesiyle bulunur: x = -b/2a.","Parabolün tepe noktasına kadar azalır, tepe noktasından sonra artar.","Artanlık ve azalanlık analizi, maksimum-minimum problemlerini çözmek için de kullanılır."]},"beginTime":1671,"endTime":1777,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1671&ask_summarization=1"},{"index":11,"title":"İkinci Türev İncelemesi","list":{"type":"unordered","items":["f(x) fonksiyonunun artan veya azalan olduğu sorulduğunda, f'(x) türevinin işaret incelemesi yapılır.","f''(x) ikinci türevin işaret incelemesi, f(x) fonksiyonunun artan veya azalan olduğu aralıkları belirler.","İkinci dereceden eşitsizliklerde olduğu gibi, ikinci türevin kökleri bulunup işaret tablosu oluşturulur."]},"beginTime":1777,"endTime":1970,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1777&ask_summarization=1"},{"index":12,"title":"Örnek Problemler","list":{"type":"unordered","items":["f(x) = x³ fonksiyonunun grafiği daima artan bir parabol olup, türevi her zaman pozitiftir.","f(x) = x² fonksiyonunun grafiği ise tepe noktası olan bir parabol olup, türevi sıfır olan noktalarda işaret değiştirir.","Türev analizi, fonksiyonların ekstremum noktalarını ve artan-azalan aralıklarını belirlemek için kullanılır."]},"beginTime":1970,"endTime":2152,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1970&ask_summarization=1"},{"index":13,"title":"Fonksiyonun Artan Azalanlığı","list":{"type":"unordered","items":["Fonksiyonun artan azalandığını belirlemek için türevini alırız.","Türevin kökü sıfır çift katlı olduğunda, artanlık azalanlık değişmez.","Birinci türevin çift katlı köklerinde artanlık azalanlık bozulmaz."]},"beginTime":2154,"endTime":2256,"href":"/video/preview/9951096995897381327?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2154&ask_summarization=1"},{"index":14,"title":"Türev Verilmiş Fonksiyonun 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Öğretmen, öğrencilerine hitap ederek konuyu adım adım açıklamaktadır.","Video, türevin geometrik yorumundan başlayarak fonksiyonların artan ve azalan olduğu aralıkların nasıl belirleneceğini anlatmaktadır. Öğretmen önce temel kavramları açıklayıp, ardından doğrusal ve polinom fonksiyonları üzerinden örnekler vermektedir. Ayrıca, fonksiyonların grafiklerinden türev bilgilerini çıkarma ve f'(x) grafiğinden f(x) fonksiyonu hakkında bilgi çıkarma yöntemleri detaylı şekilde gösterilmektedir.","Videoda 191-202 numaralı sorular çözülmekte ve çeşitli fonksiyon tipleri (doğrusal, polinom, kesirli vb.) üzerinden örnekler verilmektedir. Öğretmen, fonksiyonların türevlerinin işaret tablosu oluşturma, ekstrem noktaların belirlenmesi ve grafiklerin yorumlanması gibi konuları pekiştirmek için pratik yöntemler sunmaktadır. Video, ekstrem noktalar konusuna geçiş yapılacağı bilgisiyle sonlanmaktadır."]},"endTime":3789,"title":"Türev Kavramı ve Fonksiyonların Artan-Azalan Olduğu Aralıklar Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Türev Konusuna Giriş","list":{"type":"unordered","items":["Öğrencilere türevin alımı ve geometrik yorumları konusunda ilerledikleri belirtiliyor.","Soru çözerek zorlukları aşabilecekleri ve yapamadıkları sorular için video çözüm uygulamasını kullanabilecekleri vurgulanıyor.","Türev konusunda video sayısını artırdığı ve videoları kısa tutmak için düzenlediği belirtiliyor."]},"beginTime":0,"endTime":50,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=0&ask_summarization=1"},{"index":1,"title":"Ödev ve Video Düzenlemesi","list":{"type":"unordered","items":["Artanlık-azalantı konusunda 18-19. videoların konumunu değiştirdiği ve ödevleri güncellediği açıklanıyor.","Artanlık-azalantı konusundan sonra Test 12'nin yapılacağı belirtiliyor.","Türev konusunda 18-19. video olduğunu ve 65. gün olduklarını söylüyor."]},"beginTime":50,"endTime":143,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=50&ask_summarization=1"},{"index":2,"title":"Artanlık-Azalantı Konusunun Özeti","list":{"type":"unordered","items":["Artanlık-azalantı sorularında önce fonksiyonun türevini alıp sıfıra eşitlemek gerekiyor.","Fonksiyonun türevinin sıfır olduğu yerler eğimin olduğu yerlerdir.","Türev işaret tablosu yapıldıktan sonra f(x) ile ilgili artan-azalan oklar konularak soru çözülür."]},"beginTime":143,"endTime":182,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=143&ask_summarization=1"},{"index":3,"title":"Doğrusal Fonksiyonların Artan ve Azalan Olduğu","list":{"type":"unordered","items":["Doğrusal fonksiyonların eğimi pozitifse artan, negatifse azalan olduğu belirtiliyor.","f(x) fonksiyonunun eğimi 1 olduğu için artan, g(x) fonksiyonunun eğimi -2 olduğu için azalan olduğu gösteriliyor.","Türev alarak da fonksiyonların artan veya azalan olup olmadığına karar verilebilir."]},"beginTime":185,"endTime":270,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=185&ask_summarization=1"},{"index":4,"title":"Türev Kullanarak Artan ve Azalan Fonksiyonlar","list":{"type":"unordered","items":["h(x) fonksiyonunun türevi alınıp, çift katlı kökü -2 olduğu bulunuyor ve h(x) fonksiyonunun hep azalan olduğu belirtiliyor.","m(x) fonksiyonunun türevi alınıp, çift katlı kökü 2 olduğu bulunuyor ve m(x) fonksiyonunun hep artan olduğu gösteriliyor.","Sorunun doğru cevabı A şıkkı olarak belirtiliyor."]},"beginTime":270,"endTime":422,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=270&ask_summarization=1"},{"index":5,"title":"Fonksiyonun Tanım Kümesinde Artan Olması","list":{"type":"unordered","items":["Bir fonksiyonun bir aralıkta artan olması için o aralıkta birinci türevin sıfırdan büyük olması gerekir.","f(x) fonksiyonunun türevi alınarak eşitsizlik oluşturuluyor ve m'nin alabileceği en büyük tam sayı değeri 6'dan küçük olduğu bulunuyor.","Sorunun doğru cevabı tekrar A şıkkı olarak belirtiliyor."]},"beginTime":422,"endTime":579,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=422&ask_summarization=1"},{"index":6,"title":"Fonksiyonun Özellikleri ve Doğruluk Kontrolü","list":{"type":"unordered","items":["f(x) fonksiyonunun türevi alınarak ekstremum noktaları bulunuyor ve fonksiyonun grafiği inceleniyor.","f(5) > f(4), (-1,3) aralığında azalan, f(-2) > f(-5), (3,∞) aralığında artan ifadelerin doğruluğu kontrol ediliyor.","f'(0) > f'(-1) ifadesinin yanlış olduğu gösteriliyor ve sorunun doğru cevabı E şıkkı olarak belirtiliyor."]},"beginTime":579,"endTime":813,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=579&ask_summarization=1"},{"index":7,"title":"Matematik Sorusu Çözümü","list":{"type":"unordered","items":["194. soru, soru bankalarında sıkça karşılaşılan ve öğrencinin yorum yapması gereken bir matematik sorusudur.","Soruda f(x) fonksiyonunun [a,b] aralığında pozitif değerli ve azalan olduğu belirtilmiştir.","Pozitif değerli fonksiyon, koordinat sisteminde sonuçlarının hep pozitif olduğu, yani x ekseni altında inmediği anlamına gelir."]},"beginTime":815,"endTime":889,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=815&ask_summarization=1"},{"index":8,"title":"Fonksiyonun Özellikleri","list":{"type":"unordered","items":["Azalan fonksiyonun türevinin [a,b] aralığında sıfırdan küçük olduğu bilgisi elde edilmiştir.","Soruda x değerleri hakkında belirli bir bilgi verilmemiş, x'ler negatif veya pozitif olabilir.","Fonksiyonun pozitif değerli ve azalan olduğu, birinci türevinin sıfırdan küçük olduğu bilgileri soruyu çözmek için kullanılacaktır."]},"beginTime":889,"endTime":970,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=889&ask_summarization=1"},{"index":9,"title":"Şıkların İncelenmesi","list":{"type":"unordered","items":["Soruda aynı aralıkta kesinlikle azalan fonksiyon sorulmaktadır ve şıkların artanlık-azalanlık durumunu anlamak için türevleri alınacaktır.","A ve B şıkları için, fonksiyonun içine farklı değerler girildiği için (x/2 ve f(x)), aynı aralıkta nasıl davrandıkları bilinmediği için yorum yapılamaz.","C şıkkı için türevi -f'(x) olup, f'(x) negatif olduğundan -f'(x) pozitif olur ve fonksiyon artandır."]},"beginTime":970,"endTime":1286,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=970&ask_summarization=1"},{"index":10,"title":"Doğru Cevabın Bulunması","list":{"type":"unordered","items":["D şıkkı için türevi 2f(x)f'(x) olup, f(x) pozitif ve f'(x) negatif olduğundan çarpım negatif olur ve fonksiyon azalandır.","E şıkkı için türevi -3(f(x))^2f'(x) olup, f(x) pozitif ve f'(x) negatif olduğundan çarpım pozitif olur ve fonksiyon artandır.","Sorunun doğru cevabı D şıkkıdır, çünkü soruda daima azalan fonksiyon sorulmuştur."]},"beginTime":1286,"endTime":1366,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1286&ask_summarization=1"},{"index":11,"title":"Fonksiyonların Artan ve Azalan Olduğu Soruları","list":{"type":"unordered","items":["Pozitif değerli ve azalan bir fonksiyon için f(x) > 0 ve f'(x) \u003c 0 olmalıdır.","f(x) hakkında yorum yapılabileceği halde, f(x²) hakkında yorum yapılamaz çünkü aralık değişir.","e^f(x) hakkında yorum yapılabileceği halde, e^f(e^x) hakkında yorum yapılamaz çünkü sonuçlar bilinmemektedir."]},"beginTime":1370,"endTime":1413,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1370&ask_summarization=1"},{"index":12,"title":"Artan ve Azalan Fonksiyonların Türevi","list":{"type":"unordered","items":["Artan veya azalan fonksiyonlar için şıkların türevleri alınır; türevi negatifse azalan, pozitifse artandır.","D şıkkında türevi 2e^f(x)e'(x) olup, e^f(x) pozitif ve e'(x) negatif olduğundan fonksiyon azalandır.","Fonksiyonların artan veya azalan olup olmadığı sorulduğunda, şıkların türevleri alınarak karar verilir."]},"beginTime":1413,"endTime":1469,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1413&ask_summarization=1"},{"index":13,"title":"Fonksiyonların Özelliklerinin Analizi","list":{"type":"unordered","items":["x değerleri pozitif olduğunda, f(x) değerleri de pozitif olur çünkü x ekseninin üstünde yer alır.","Fonksiyonun artan olduğu aralıkta, türevi pozitiftir.","f(x) fonksiyonu AB aralığında artandır, ancak e^f(x) hakkında yorum yapılamaz."]},"beginTime":1469,"endTime":1523,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1469&ask_summarization=1"},{"index":14,"title":"Türev Alma ve Fonksiyonların Analizi","list":{"type":"unordered","items":["B şıkkının türevi alınarak pozitif olduğu görülür ve bu fonksiyon artandır.","C şıkkının türevi e^f(x) + 1 olup, e^f(x) pozitif olduğundan fonksiyon artandır.","D şıkkında çarpımın türevi alınarak pozitif olduğu görülür ve bu fonksiyon da artandır."]},"beginTime":1523,"endTime":1602,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1523&ask_summarization=1"},{"index":15,"title":"Azalan Fonksiyonların Analizi","list":{"type":"unordered","items":["E şıkkında türevi -3e^f(x)²e'(x) olup, e^f(x)² pozitif ve e'(x) negatif olduğundan fonksiyon azalandır.","AB aralığında daima artan fonksiyonlar için, x değerleri sol tarafta olduğunda negatiftir.","f(x) değerleri x ekseninin altında olduğundan sıfırdan küçüktür ve fonksiyon azalmıştır."]},"beginTime":1602,"endTime":1694,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1602&ask_summarization=1"},{"index":16,"title":"Fonksiyonların Türevlerinin Analizi","list":{"type":"unordered","items":["A şıkkında çarpımın türevi alınarak, sonuç pozitif veya negatif olabileceği için yorum yapılamaz.","B şıkkında x'in türevi 1 ve f'(x) negatif olduğundan, toplamın pozitif veya negatif olabileceği için yorum yapılamaz.","C şıkkında bölümün türevi alınarak negatif olduğu görülür ve bu fonksiyon azalandır."]},"beginTime":1694,"endTime":1844,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1694&ask_summarization=1"},{"index":17,"title":"Fonksiyonların Türevlerinin Son Analizi","list":{"type":"unordered","items":["D şıkkında bölümün türevi alınarak negatif olduğu görülür ve bu fonksiyon azalandır.","E şıkkında türevi 3x² - e'(x) olup, x² pozitif ve e'(x) negatif olduğundan fonksiyon artandır.","Artan-azalanlık sorularında türev alınarak karar verilir."]},"beginTime":1844,"endTime":1928,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1844&ask_summarization=1"},{"index":18,"title":"Grafik Sorularında Mantık","list":{"type":"unordered","items":["Grafik sorularında dikkatlice inceleme yapılması gerekir.","f'(x) türevinin mantıkları daha sonra ayrıntılı şekilde ele alınacaktır.","f(x) grafiği verildiğinde tablo yerine direkt şekle bakarak karar verilir."]},"beginTime":1928,"endTime":1973,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1928&ask_summarization=1"},{"index":19,"title":"Fonksiyonun Grafiğinden Türevin Yorumlanması","list":{"type":"unordered","items":["Bir fonksiyonun grafiğinde birinci türevi sıfır olduğu yerler, eğimin sıfır olduğu ve teğetin x eksenine paralel olduğu yerlerdir.","Fonksiyonun grafiğinde tepeler ve çukurlar (parabolün tepe noktası gibi) yerlerde birinci türev sıfırdır.","Fonksiyon artan olduğu aralıklarda birinci türev sıfırdan büyük, azalan olduğu aralıklarda birinci türev sıfırdan küçüktür."]},"beginTime":1980,"endTime":2128,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1980&ask_summarization=1"},{"index":20,"title":"Grafiğin Artan ve Azalan Alanlarının Belirlenmesi","list":{"type":"unordered","items":["Grafiğin artan kısımları (yokuş yukarı) yerlerde birinci türev sıfırdan büyüktür.","Grafiğin azalan kısımları (yokuş aşağı) yerlerde birinci türev sıfırdan küçüktür.","Grafiğin sabit gittiği yerlerde (eğim sıfır olan yerler) birinci türev sıfırdır."]},"beginTime":2128,"endTime":2192,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2128&ask_summarization=1"},{"index":21,"title":"Türevin Belirli Noktalardaki Değerleri","list":{"type":"unordered","items":["Belirli bir noktada fonksiyonun türevinin değeri, o noktada grafiğin artan veya azalan olmasına bağlı olarak sıfırdan büyük veya küçüktür.","Türevin sıfır olduğu noktalar, grafiğin tepeleri, çukurları ve sabit gittiği yerlerdir.","Türevin olmadığı noktalar, grafiğin kırılma yaptığı yerlerdir."]},"beginTime":2192,"endTime":2520,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2192&ask_summarization=1"},{"index":22,"title":"Fonksiyonun Grafiğinden Türevin İşaretlerinin Belirlenmesi","list":{"type":"unordered","items":["Fonksiyonun grafiğinde artan kısımlarda birinci türevin işareti artı (+) olur.","Fonksiyonun grafiğinde sabit gittiği yerlerde birinci türev sıfırdır.","Fonksiyonun grafiğinde azalan kısımlarda birinci türevin işareti eksi (-) olur."]},"beginTime":2520,"endTime":2596,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2520&ask_summarization=1"},{"index":23,"title":"Türevin Grafiğinden Fonksiyonun Analizi","list":{"type":"unordered","items":["Bir fonksiyonun artanlık ve azalanlık analizinde, fonksiyonun türevi alınır ve sıfıra eşitlenerek kökler bulunur.","Türevin işareti pozitif olduğunda fonksiyon artan, negatif olduğunda fonksiyon azalandır.","En büyük dereceli terimin işareti, fonksiyonun genel davranışını belirler."]},"beginTime":2596,"endTime":2674,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2596&ask_summarization=1"},{"index":24,"title":"Fonksiyon Türevi Problemleri","list":{"type":"unordered","items":["f'(x) fonksiyonunun grafiği verildiğinde, f(x) fonksiyonunun artan ve azalanlık bilgileri için f'(x)=0 olan değerler bulunmalıdır.","f'(x) grafiğinde x eksenine değen noktalar f'(x)=0 olan değerleri gösterir, bu örnekte -2 ve 4 değerleri bulunmuştur.","f'(x) grafiğinin işaretleri (pozitif/negatif) belirlenerek, f(x) fonksiyonunun artan ve azalan aralıkları bulunabilir."]},"beginTime":2677,"endTime":2839,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2677&ask_summarization=1"},{"index":25,"title":"Problemin Çözümü","list":{"type":"unordered","items":["f'(x) grafiğinin işaretleri belirlendikten sonra, f(x) fonksiyonunun artan ve azalan aralıkları oklarla gösterilir.","f(x) fonksiyonu (-∞,-2) ve (4,∞) aralıklarında artarken, (-2,4) aralığında azalmaktadır.","Verilen şıkların doğruluğu, f(x) fonksiyonunun grafiğine göre kontrol edilir ve yanlış olan şık bulunur."]},"beginTime":2839,"endTime":3097,"href":"/video/preview/13327479729205251147?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=2839&ask_summarization=1"},{"index":26,"title":"İkinci Problemin Çözümü","list":{"type":"unordered","items":["İkinci probleminde f'(x) fonksiyonunun grafiği verilmiş ve x eksenine değen noktalar (-5, -1, 3) 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KONU PDF https://drive.google.com/file/d/1GIXc... Platin Kitap İnternet Satiş LİNK: Miray Yayınları...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/2423607/bfe8837a3c39bf4dc20e730ca3e2119b/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/WrEidAEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"16","reqid":"1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=videoid:1250277190955854137","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik öğretmeninin öğrencilere yönelik hazırladığı eğitim içeriğidir. Öğretmen, üniversite mezunu olduğunu belirterek konuyu anlatmakta ve \"hocam\" olarak hitap edilmektedir.","Videoda fonksiyonların artan ve azalanlık özellikleri ile türevleri detaylı şekilde ele alınmaktadır. Öğretmen, fonksiyonların artan ve azalan olduğu aralıkları bulma yöntemlerini, ikinci türevin kullanımını, işaret tablosu oluşturma tekniklerini ve grafiksel temsilleri adım adım göstermektedir. Video boyunca çeşitli örnek sorular çözülerek konu pekiştirilmektedir.","Video, sınavlarda çıkabilecek soru kalıplarını içermekte ve öğrencilere pratik çözümler sunmaktadır. 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inceleniyor.","Fonksiyonun türevi alınarak işaret tablosu oluşturuluyor ve kökler -1 ve 3 bulunuyor.","İşaret tablosu kullanılarak fonksiyonun belirli aralıklardaki davranışları (artan/azalan) inceleniyor."]},"beginTime":818,"endTime":1035,"href":"/video/preview/1250277190955854137?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=818&ask_summarization=1"},{"index":6,"title":"Fonksiyonun Özelliklerini Kullanma","list":{"type":"unordered","items":["194. soruda, y=f(x) fonksiyonunun ab aralığında pozitif değerli ve azalan olduğu bilgisi veriliyor.","Verilen şıkların türevleri alınarak artan/azalanlık özellikleri inceleniyor.","Fonksiyonun ab aralığında tanımlandığı için, f(2x) veya f(b) gibi içi değişen fonksiyonlar hakkında yorum yapılamıyor."]},"beginTime":1035,"endTime":1411,"href":"/video/preview/1250277190955854137?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1035&ask_summarization=1"},{"index":7,"title":"Fonksiyonların Artan ve Azalan Olduğu Aralıklar","list":{"type":"unordered","items":["Soruda AB aralığında f(x) fonksiyonunun grafiği verilmiş ve azalan fonksiyonu soruluyor.","Şıklarda f(x) ile ilgili bilgi verilmiş, f(2x), f(x) bileşke f(x) gibi ifadelerle ilgili yorum yapılamaz.","Artan-azalanlık sorulduğunda fonksiyonların türevleri alınarak incelenir."]},"beginTime":1419,"endTime":1485,"href":"/video/preview/1250277190955854137?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1419&ask_summarization=1"},{"index":8,"title":"Fonksiyonların Türevleri ve İşaretleri","list":{"type":"unordered","items":["x ekseninin üstünde olduğu için f(x) değerleri sıfırdan büyük, x değerleri de sıfırdan büyük 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azalandır."]},"beginTime":1523,"endTime":1645,"href":"/video/preview/1250277190955854137?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1523&ask_summarization=1"},{"index":10,"title":"İkinci Örnek Soru","list":{"type":"unordered","items":["İkinci soruda da AB aralığında artan-azalanlık sorulmuş ve f(x) değerleri sıfırdan küçük, f'(x) de sıfırdan küçük olarak belirlenmiştir.","x·f(x) fonksiyonunun türevi f(x) + xf'(x) olup, işaretleri belirsiz olduğu için artan-azalan yorumu yapılamaz.","f(x)/x fonksiyonunun türevi (f'(x)·x - f(x))/x² olup, her iki terim de negatif olduğu için fonksiyon azalandır.","3x² - f(x) fonksiyonunun türevi 6x - f'(x) olup, her iki terim de pozitif olduğu için fonksiyon artandır."]},"beginTime":1645,"endTime":1881,"href":"/video/preview/1250277190955854137?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1645&ask_summarization=1"},{"index":11,"title":"Sonuç ve 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serisini takip edenler türev konusundan bir fasikülden fazlasını çözmüş olacaklardır. KONU PDF https://drive.google.com/file/d/1GIXc... Platin Kitap İnternet Satiş LİNK: Miray Yayınları...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/3503216/4dd77a58862d1165b6a7bf00e913669f/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/VfZKcgEAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"17","reqid":"1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=videoid:9090738249664957487","teaser":[{"list":{"type":"unordered","items":["Bu video, bir matematik öğretmeninin öğrencilere yönelik hazırladığı eğitim içeriğidir. Öğretmen, artanlık-azalanlık konusunu iki videoluk bir seri kapsamında anlatmaktadır.","Video, fonksiyon grafikleri ve türev grafikleri arasındaki ilişkiyi detaylı şekilde ele almaktadır. İlk olarak fonksiyon grafiğinden türevin nasıl yorumlanacağı, tepe ve çukur noktalarında türevin sıfır olduğu, artan ve azalan bölgelerde türevin işaretinin nasıl belirleneceği anlatılmaktadır. Ardından türev grafiğinden fonksiyonun artan-azalan durumunun nasıl çıkarılacağı, ekstremum noktaları, yerel maksimum ve minimum noktaları gibi konular örneklerle açıklanmaktadır.","Video, bileşke fonksiyonların türevlerini hesaplama yöntemlerini, polinom fonksiyonların grafiklerini ve kökler toplamı-kökler çarpımı formüllerini kullanarak soruları çözmektedir. Ayrıca, ekstremum noktaları konusuna daha ayrıntılı bir şekilde döneceği belirtilmektedir."]},"endTime":1960,"title":"Fonksiyon ve Türev Grafikleri Eğitim Videosu","beginTime":0}],"fullResult":[{"index":0,"title":"Fonksiyon Grafiği ve Türev İlişkisi","list":{"type":"unordered","items":["Artanlık azalanlık konusunda iki videoluk bir seri yapılmıştır.","Fonksiyon grafikleri ve türev grafikleri karşılaştığında temel mantıkların anlatılacağı belirtilmiştir.","Ekstremumlara geldiğimizde türev grafiği daha ayrıntılı incelenecektir."]},"beginTime":6,"endTime":53,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=6&ask_summarization=1"},{"index":1,"title":"Fonksiyon Grafiğinden Türev Yorumlama","list":{"type":"unordered","items":["Fonksiyon grafiğinden türevi yorumlamak için önce grafiğin doğrusal veya morusal olup olmadığına bakılır.","Fonksiyonun tepeleri ve çukurlarında çizilen teğetlerin eğimleri sıfırdır, bu nedenle f'(x) = 0'dır.","Fonksiyonun artan olduğu yerlerde f'(x) > 0, azalan olduğu yerlerde f'(x) \u003c 0'dır."]},"beginTime":53,"endTime":230,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=53&ask_summarization=1"},{"index":2,"title":"Örnek Soru Çözümü","list":{"type":"unordered","items":["Fonksiyonun grafiğinde azalan olduğu yerlere f'(x) = - yazılır, artan olduğu yerlere f'(x) = + yazılır.","Tepe ve çukur noktalarında f'(x) = 0'dır, bu noktalar dahil edilmez.","Fonksiyonun grafiğinde sabit olduğu yerlerde de f'(x) = 0'dır."]},"beginTime":230,"endTime":569,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=230&ask_summarization=1"},{"index":3,"title":"Türev Grafiği Yorumlama","list":{"type":"unordered","items":["Türev grafiği çıktığında önce fonksiyonun artan, azalan veya sabit olduğu yerlere bakılır.","Fonksiyonun artanlık azalanlığını bulmak için türevini alıp sıfıra eşitlemek gerekir.","Fonksiyonun türevini sıfır yapan yerler artanlık azalanlık için önemlidir."]},"beginTime":569,"endTime":640,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=569&ask_summarization=1"},{"index":4,"title":"Türev Grafiği Yorumlama","list":{"type":"unordered","items":["Türev grafiğinin sıfır yapan noktaları, fonksiyonun tepe ve çukur noktalarında bulunur.","Türev grafiğinin eksenleri kestiği noktalar, fonksiyonun artan ve azalan aralıklarını belirlemek için kullanılır.","Türev grafiğinin işaretleri, fonksiyonun artan ve azalan olduğu bölgeleri belirler."]},"beginTime":642,"endTime":774,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=642&ask_summarization=1"},{"index":5,"title":"Türev Grafiği Yorumlama Yöntemi","list":{"type":"unordered","items":["Türev grafiğinin eksenleri kestiği noktalar sırayla yazılır ve çat çizgisi çekilir.","Türev grafiğinin işaretleri, x ekseninin üstünde olduğu yerlere artı, altında olduğu yerlere eksi olarak yerleştirilir.","Artan ve azalan oklar, fonksiyonun artan ve azalan olduğu bölgeleri gösterir."]},"beginTime":774,"endTime":811,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=774&ask_summarization=1"},{"index":6,"title":"Örnek Soru Çözümü","list":{"type":"unordered","items":["Fonksiyonun artan ve azalan aralıkları, türev grafiğinin işaretlerine göre belirlenir.","Fonksiyonun belirli noktalardaki değerleri, grafiğin o noktalardaki konumuna göre karşılaştırılır.","Türevin kendisi sorulduğunda, türev grafiğinden doğrudan değer bulunabilir."]},"beginTime":811,"endTime":1283,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=811&ask_summarization=1"},{"index":7,"title":"Türev Grafiği Yorumlama Tekniği","list":{"type":"unordered","items":["Türev grafiği yorumlamak için x eksenini kestiği noktalar sıralanır ve tablo yapılır.","Tabloda işaretler, türev grafiğinin x ekseninin üstünde ve altında olduğu bölgelere göre belirlenir.","Ekstremum noktaları konusunda daha ayrıntılı grafik bilgisi gerekecektir."]},"beginTime":1283,"endTime":1332,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1283&ask_summarization=1"},{"index":8,"title":"Türev ve Fonksiyonların Artan-Azalanlık Özellikleri","list":{"type":"unordered","items":["Türevin sonlarına doğru ekstremum noktalarına (yerel max ve yerel min) geçilecek.","Gerçel sayılarda tanımlı f azalan, g artan fonksiyonlar için f'(x) \u003c 0 ve g'(x) > 0 şeklinde ifade edilir.","Fonksiyonların artan-azalanlık özelliklerini belirlemek için türev alınır; türev sıfırdan küçükse fonksiyon azalan, sıfırdan büyükse artandır."]},"beginTime":1339,"endTime":1434,"href":"/video/preview/9090738249664957487?parent-reqid=1769399349924490-57362572697506375-balancer-l7leveler-kubr-yp-vla-246-BAL&text=ARTAN&t=1339&ask_summarization=1"},{"index":9,"title":"Fonksiyonların Türevleri ve Özellikleri","list":{"type":"unordered","items":["Bileşke fonksiyonların türevleri kabuk yöntemiyle alınabilir.","f(g(x)) fonksiyonunun türevi f'(g(x)) × g'(x) şeklinde hesaplanır.","f(x) + g(x) fonksiyonunun türevi f'(x) + g'(x) olup, işaretini belirlemek için f'(x) ve g'(x) işaretlerinin karşılaştırılması 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