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üniversite matematiği derslerinden calculus-I dersine ait \"Parçalı Fonksiyon Grafiği Çizme Örnek Soru-1 \" videosudur. 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Eğitmen, parçalı fonksiyon grafiği çizerken dikkat edilmesi gereken noktaları açıklayarak başlar, ardından parçalanma noktasının belirlenmesi, grafiklerin çizilmesi ve parçalı fonksiyonun tam grafiğinin oluşturulması adımlarını gösterir. Video, parçalı fonksiyonların nasıl çizileceğini adım adım gösteren bir eğitim içeriğidir."]},"endTime":374,"title":"Parçalı Fonksiyon Grafiği Çizimi Örneği","beginTime":0}],"fullResult":[{"index":0,"title":"Parçalı Fonksiyon Grafiği Çizme Örneği","list":{"type":"unordered","items":["Video, parçalı fonksiyon grafiği çizme konusunda bir örnek soru inceleyecektir.","Soruda verilen fonksiyon f(x) = -2x (x ≤ 0) ve f(x) = √x + 1 (x > 0) şeklindedir.","Parçalı fonksiyon grafiğini çizmede ilk adım, parçalanma noktasını belirlemektir ve bu nokta koordinat düzlemi üzerinde kesikli çizgi ile gösterilmelidir."]},"beginTime":1,"endTime":70,"href":"/video/preview/2212934160768735757?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=1&ask_summarization=1"},{"index":1,"title":"Parçalanma Noktası ve İlk Fonksiyon","list":{"type":"unordered","items":["Bu örnekte parçalanma noktası x = 0'dır ve bu nokta y ekseninde yer alır.","İlk fonksiyon y = -2x şeklindedir ve x ≤ 0 koşulunda geçerlidir.","y = -2x grafiği, x = 0 noktasından (0,0) başlayarak sol tarafta çizilir ve bu nokta dolu bir nokta ile gösterilir."]},"beginTime":70,"endTime":237,"href":"/video/preview/2212934160768735757?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=70&ask_summarization=1"},{"index":2,"title":"İkinci Fonksiyon","list":{"type":"unordered","items":["İkinci fonksiyon y = √x + 1 şeklindedir ve x > 0 koşulunda geçerlidir.","y = √x grafiği orijinden başlar, y = √x + 1 ise y ekseninde 1 birim yukarı kayar.","Bu fonksiyonun grafiği, x = 0 noktasından (0,1) başlayarak sağ tarafta çizilir ve eğrisel bir şekilde devam 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üniversite matematiği derslerinden calculus-II dersine ait \"Vektör Alanlarının Çizgi İntegralini Hesaplama\" videosudur. Hazırlayan: Kemal Duran (Matematik Öğretmeni)...","preview":{"posterSrc":"//avatars.mds.yandex.net/get-vthumb/2835853/7f831b954899f8ad8fd3ac750e82e358/564x318_1","videoSrc":"https://video-preview.s3.yandex.net/H5naZwAAAAA.mp4","videoType":"video/mp4"},"target":"_self","position":"5","reqid":"1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL","summary":{"isFull":true,"fullTextUrl":"/video/result?ask_summarization=1&numdoc=1&noreask=1&nomisspell=1&parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=videoid:9867457546522917982","teaser":[{"list":{"type":"unordered","items":["Bu video, bir eğitmen tarafından sunulan matematik dersi formatındadır. Eğitmen, vektör alanlarının çizgi integralinin nasıl hesaplanacağını adım adım anlatmaktadır.","Videoda, vektör alanının tanımı hatırlatılarak başlanmakta ve vektör alanının çizgi integralinin formülleri (C∫f·dr) açıklanmaktadır. Eğitmen, parametrize etme işlemi yaparak vektör alanının çizgi integralini hesaplama sürecini iki farklı örnek üzerinden göstermektedir. Video, vektör alanının çizgi integralinin nasıl hesaplanacağını gösteren bir örnekle sona ererken, bir sonraki videoda vektör alanı konsorveti (korumacı vektör alanı) konusunun ele alınacağı belirtilmektedir.","Eğitmen, bu konuyla ilgili örneklerden oluşan beş serilik bir video serisi hazırlayacağını da eklemektedir."]},"endTime":1302,"title":"Vektör Alanlarının Çizgi İntegrali Hesaplama Dersi","beginTime":0}],"fullResult":[{"index":0,"title":"Vektör Alanlarının Çizgi İntegrali","list":{"type":"unordered","items":["Bu videoda vektör alanlarının çizgi integralinin nasıl hesaplanacağı anlatılacaktır.","Vektör alanı, koordinatları değişkenlerden oluşan vektörlere verilen genel isimdir ve büyük F harfi ile gösterilir.","Vektör alanı iki değişkenli (p(x,y), q(x,y)) veya üç değişkenli (p(x,y,z), q(x,y,z), r(x,y,z)) olabilir."]},"beginTime":0,"endTime":94,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=0&ask_summarization=1"},{"index":1,"title":"Vektör Alanlarının Çizgi İntegrali Formülü","list":{"type":"unordered","items":["Vektör alanının çizgi integralinin sorulduğunu, çizgi integralinin içinde büyük F harfi ve dot product (iç çarpım) işaretiyle çarpım gördüğümüzde anlarız.","Vektör alanının çizgi integrali formülü: ∫_c F·dr'dir.","Bu integral hesaplanırken, r parametresi t'ye göre parametrize edilir ve integral t'ye göre alınır."]},"beginTime":94,"endTime":388,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=94&ask_summarization=1"},{"index":2,"title":"Vektör Alanlarının Çizgi İntegrali Hesaplama Adımları","list":{"type":"unordered","items":["Vektör alanının çizgi integralini hesaplamak için önce f(x,y,z) içindeki x, y, z değerleri r parametresine göre değiştirilir.","İkinci adım olarak r'nin t'ye göre türevi alınır.","Üçüncü adım olarak f(r) ile r'nin türevinin iç çarpımı yapılır ve vektörsel durumdan kurtulunur.","Son adım olarak elde edilen ifade t'ye göre integral alınır ve sınırlar belirlenir."]},"beginTime":388,"endTime":439,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=388&ask_summarization=1"},{"index":3,"title":"Örnek Çözüm","list":{"type":"unordered","items":["Örnek olarak f(x,y,z) = 8x²yz + 5zj - 4xyk vektör alanı ve r(t) = t i + t²j + t³k eğrisi verilmiştir.","İlk adım olarak f(r) = 8t⁴i + 5t³j - 4t³k olarak hesaplanır.","İkinci adım olarak r'(t) = i + 2j + 3k olarak bulunur.","Üçüncü adım olarak f(r)·r'(t) = 8t⁷ + 10t⁴ - 12t⁵ olarak hesaplanır.","Dördüncü adım olarak integral ∫₀¹ (8t⁷ + 10t⁴ - 12t⁵) dt alınır ve sonuç 1 olarak bulunur."]},"beginTime":439,"endTime":825,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=439&ask_summarization=1"},{"index":4,"title":"Parametreze İşlemi Gerektiren Örnek","list":{"type":"unordered","items":["İkinci örnek olarak f(x,y,z) = x k vektör alanı ve doğru parçası verilmiştir.","Doğru parçasının parametrize edilmesi için x = -1 + 4t, y = 2 - 2t, z = t olarak bulunur.","r(t) = -1 + 4t i + 2 - 2t j + t k olarak hesaplanır.","Bu örnekte parametreze işlemi gereklidir çünkü r parametresi doğrudan verilmemiştir."]},"beginTime":825,"endTime":1025,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=825&ask_summarization=1"},{"index":5,"title":"Vektör Alanının Çizgi İntegrali Hesaplama","list":{"type":"unordered","items":["Vektör alanının çizgi integrali hesaplanırken, r'nin türevi t olarak belirleniyor.","f(r(t)) ifadesi hesaplanırken, x yerine -1+4t, y yerine 2-2t, z yerine t değerleri yerleştiriliyor.","r(t) ve r'(t) vektörleri iç çarpım yapılarak 18t²-6t sonucu elde ediliyor."]},"beginTime":1031,"endTime":1205,"href":"/video/preview/9867457546522917982?parent-reqid=1769025279041994-6530599311729037499-balancer-l7leveler-kubr-yp-klg-317-BAL&text=Calcworkshop.com+-+Calculus+Videos&t=1031&ask_summarization=1"},{"index":6,"title":"İntegral Hesaplama ve Sonuç","list":{"type":"unordered","items":["18t²-6t ifadesi 0'dan 1'e kadar t'ye göre integral 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