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Номер 16, страница 83, часть 2 - гдз по алгебре 10 класс учебник Пак, Ардакулы
Авторы: Пак О. В., Ардакулы Д., Ескендирова Е. В.
Тип: Учебник
Издательство: Алматыкітап баспасы
Год издания: 2019 - 2026
Часть: 2
ISBN: 978-601-01-3958-9
Рекомендовано Министерством образования и науки Республики Казахстан
Популярные ГДЗ в 10 классе
Часть 2. Глава 6. Применение производной. Параграф 1. Признаки монотонности. Задачи - номер 16, страница 83.
App\Models\Task {#1030 // resources/views/models/task/default.blade.php #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120331 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "83" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-1zad-16" "field_display_title" => "16" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#1037 #items: array:1 [ 0 => App\Models\Term {#1036 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #original: array:6 [ "id" => 26 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "номер" "field_cases" => array:6 [ "field_accusative_case" => "номер" "field_creative_case" => "номером" "field_dative_case" => "номеру" "field_genitive_case" => "номера" "field_nominative_case" => "номер" "field_prepositional_case" => "номере" ] "field_short_name" => "№" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_match" => null "breadcrumbs" => [] "edition_groups" => Illuminate\Database\Eloquent\Collection {#1035 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#1072 #items: array:1 [ 0 => App\Models\Branch {#1034 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039 #items: array:1 [ 0 => App\Models\Term {#1038 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040 #items: array:1 [ 0 => App\Models\Book {#1041 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1042 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1044 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1046 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1048 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1052 …2} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1053 …2} "field_country" => Illuminate\Database\Eloquent\Collection {#1055 …2} "field_city" => Illuminate\Database\Eloquent\Collection {#1057 …2} "field_series" => Illuminate\Database\Eloquent\Collection {#1059 …2} "field_umk" => Illuminate\Database\Eloquent\Collection {#1060 …2} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1061 …2} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1062 …2} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1063 …2} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1064 …2} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ …1] "field_cover_alts" => array:1 [ …1] "field_covers" => array:1 [ …1] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1065 …2} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1066 …2} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1067 …2} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1068 …2} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ …3] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1039} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1040} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1069} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "parent_branches" => Illuminate\Database\Eloquent\Collection {#1106 #items: array:4 [ 0 => App\Models\Branch {#1116 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1120 #items: array:1 [ 0 => App\Models\Term {#1117 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #original: array:7 [ "id" => 31 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Часть" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => "1" "field_short_name" => "ч." ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149 #items: array:1 [ 0 => App\Models\Book {#1119 #connection: "mysql" #table: "books" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1121 #items: array:1 [ 0 => App\Models\Term {#1118 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #original: array:10 [ "id" => 6471 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алгебра" "field_abbreviated_name" => null "field_cases" => array:6 [ …6] "field_foreign_lang_name" => null "field_short_name" => null "field_subject_type" => "technical_subject" "field_translit" => "algebra" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_class" => Illuminate\Database\Eloquent\Collection {#1122 #items: array:1 [ 0 => App\Models\Term {#1123 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #original: array:6 [ "id" => 5459 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "10" "field_cases" => array:6 [ …6] "field_translit" => "desjatyj" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_publisher" => Illuminate\Database\Eloquent\Collection {#1124 #items: array:1 [ 0 => App\Models\Term {#1125 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #original: array:6 [ "id" => 7006 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматыкітап баспасы" "field_cases" => array:6 [ …6] "field_translit" => "almatykitap baspasy" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author" => Illuminate\Database\Eloquent\Collection {#1126 #items: array:3 [ 0 => App\Models\Term {#1127 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #original: array:12 [ "id" => 7007 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Пак" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Олег" "field_patronymic" => "Владимирович" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Term {#1128 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #original: array:12 [ "id" => 7008 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ардакулы" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Дархан" "field_patronymic" => null "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Term {#1129 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #original: array:12 [ "id" => 7009 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Ескендирова" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Елена" "field_patronymic" => "Викторовна" "field_surname_rp" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1130 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1131 #items: array:1 [ 0 => App\Models\Term {#1132 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #original: array:10 [ "id" => 6671 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Учебник" "field_book_type_foreign" => null "field_cases" => array:6 [ …6] "field_plural_form" => null "field_short_name" => null "field_short_name_foreign" => null "field_translit" => "uchebnik" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_country" => Illuminate\Database\Eloquent\Collection {#1133 #items: array:1 [ 0 => App\Models\Term {#1134 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #original: array:6 [ "id" => 6994 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Казахстан" "field_cases" => array:6 [ …6] "field_translit" => "kazakhstan" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_city" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Term {#1136 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #original: array:6 [ "id" => 6995 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Алматы" "field_cases" => array:6 [ …6] "field_translit" => "almaty" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_series" => Illuminate\Database\Eloquent\Collection {#1137 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1138 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1139 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1140 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1141 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142 #items: [] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146 #items: [] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #original: array:50 [ "id" => 4295 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_tree_default_status" => "tasks" "field_pages_status" => null "field_subject" => Illuminate\Database\Eloquent\Collection {#1121} "field_class" => Illuminate\Database\Eloquent\Collection {#1122} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1124} "field_author" => Illuminate\Database\Eloquent\Collection {#1126} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1130} "field_book_type" => Illuminate\Database\Eloquent\Collection {#1131} "field_country" => Illuminate\Database\Eloquent\Collection {#1133} "field_city" => Illuminate\Database\Eloquent\Collection {#1135} "field_series" => Illuminate\Database\Eloquent\Collection {#1137} "field_umk" => Illuminate\Database\Eloquent\Collection {#1138} "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1139} "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1140} "field_publication_number" => null "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1141} "field_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1142} "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2019" "field_publication_year_until" => null "field_part" => "1, 2" "field_part_writing" => "Часть" "field_second_foreign_language" => null "field_for_whom" => "Учебник для учащихся 10 класса общеобразовательной школы общественно-гуманитарного направления" "field_allowed" => "Рекомендовано Министерством образования и науки Республики Казахстан" "field_reserve_field" => null "field_link_to_source" => null "field_tasks_count" => "1063" "field_priority" => "6" "field_default_folder" => "/algebra_10/baspasy-u19/" "field_isbn" => "978-601-01-3958-9" "field_cover" => array:1 [ 0 => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/baspasy-u19/covers/cover1.webp?ts=1752152865" "alt" => "" "width" => "1940" "height" => "2895" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1143} "field_new_book" => Illuminate\Database\Eloquent\Collection {#1144} "field_old_book" => Illuminate\Database\Eloquent\Collection {#1145} "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1146} "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "breadcrumbs" => array:3 [ "class" => 381 "subject" => 542 "class_subject" => 387 ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147 #items: [] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119426 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => null "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1120} "field_page_start" => "5" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1149} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1147} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Branch {#1148 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153 #items: array:1 [ 0 => App\Models\Term {#1150 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #original: array:7 [ "id" => 29 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Глава" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1152 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Branch {#1034} ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1153} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1152} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1151} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 2 => App\Models\Branch {#1154 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119443 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки монотонности" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Term {#1155 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #original: array:7 [ "id" => 32 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Параграф" "field_cases" => array:6 [ …6] "field_page_check_not_needed" => null "field_short_name" => "§" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1157 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166 #items: array:1 [ 0 => App\Models\Branch {#1165 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1168 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 …2} ] #original: array:24 [ "id" => 1119442 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Применение производной" "field_branch_order" => "6" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1167 …2} "field_page_start" => "76" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1168 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1169 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119443 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки монотонности" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1158} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1157} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1166} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 3 => App\Models\Branch {#1159 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156 #items: [] #escapeWhenCastingToString: false } "field_page_start" => "81" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1162 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "branch_parent" => Illuminate\Database\Eloquent\Collection {#1175 #items: array:1 [ 0 => App\Models\Branch {#1174 #connection: "mysql" #table: "branches" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1119443 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки монотонности" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1178 …2} ] #original: array:24 [ "id" => 1119443 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Признаки монотонности" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1176 …2} "field_page_start" => "77" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1177 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1178 …2} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } ] #original: array:24 [ "id" => 1119445 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Задачи" "field_branch_order" => null "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1156} "field_page_start" => "81" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" "field_display_task_interval" => "0" "field_display_branch_page" => "1" "field_branch_title_in_content" => "0" "field_navigation_title" => null "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1162} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1175} ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1070 #items: array:2 [ 0 => App\Models\Element {#1097 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 1301016 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087 #items: array:1 [ 0 => App\Models\Edition {#1096 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #original: array:21 [ "id" => 4962 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1094 …2} "field_content_type" => "free" "field_content_mode" => "text, image" "field_page_content_mode" => "image" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Автор" "field_moderator" => "pasha" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1093 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1091 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1092 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120331" "type" => "task" ] "text" => "<p><strong>16. (2)</strong> Для следующих функций определите интервалы монотонности (используя производную):</p><p><strong>а)</strong> $f(x)=5\sin(-x)+5\operatorname{tg}5$;</p><p><strong>б)</strong> $g(x)=-2\cos3x-6x\cos\frac{3\pi}{4}$;</p><p><strong>в)</strong> $h(x)=\sin\left(4x-\frac{\pi}{3}\right)-2x.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "16-1.jpg" "alt" => null "width" => "1662" "height" => 445 "path" => "/media/algebra_10/baspasy-u19/0-6-1zad/16-1.webp?ts=1753534893" ] ] ] #original: array:7 [ "id" => 1301016 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1087} "task" => array:2 [ "refs" => "1120331" "type" => "task" ] "text" => "<p><strong>16. (2)</strong> Для следующих функций определите интервалы монотонности (используя производную):</p><p><strong>а)</strong> $f(x)=5\sin(-x)+5\operatorname{tg}5$;</p><p><strong>б)</strong> $g(x)=-2\cos3x-6x\cos\frac{3\pi}{4}$;</p><p><strong>в)</strong> $h(x)=\sin\left(4x-\frac{\pi}{3}\right)-2x.$</p>" "img" => array:1 [ 0 => array:5 [ "name" => "16-1.jpg" "alt" => null "width" => "1662" "height" => 445 "path" => "/media/algebra_10/baspasy-u19/0-6-1zad/16-1.webp?ts=1753534893" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 1 => App\Models\Element {#1089 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1552052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081 #items: array:1 [ 0 => App\Models\Edition {#1090 #connection: "mysql" #table: "editions" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5659 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2 (rus)" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1086 …2} "field_content_type" => "free" "field_content_mode" => "text" "field_page_content_mode" => "" "field_content_text_checked" => null "field_page_content_text_checked" => "0" "field_solution_author" => "Gemini 2.5 Pro" "field_moderator" => "tolik" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1082 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1085 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1084 …2} "field_content_source" => null ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "task" => array:2 [ "refs" => "1120331" "type" => "task" ] "text" => "<p><strong>a) $f(x) = 5\sin(-x) + 5\tan(5)$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Сначала упростим выражение для функции. Так как синус — нечетная функция, $\sin(-x) = -\sin(x)$. Слагаемое $5\tan(5)$ является константой, так как не содержит переменной $x$.</p><p>$f(x) = -5\sin(x) + 5\tan(5)$</p><p>Найдем производную функции $f(x)$ по переменной $x$:</p><p>$f'(x) = (-5\sin(x) + 5\tan(5))' = -5(\sin(x))' + (5\tan(5))' = -5\cos(x) + 0 = -5\cos(x)$</p><p>2. <i>Определение знака производной.</i></p><p>Интервалы монотонности определяются знаком производной. Функция возрастает, когда $f'(x) > 0$, и убывает, когда $f'(x) < 0$.</p><p>Найдем интервалы, на которых $f'(x) > 0$ (возрастание):</p><p>$-5\cos(x) > 0$</p><p>$\cos(x) < 0$</p><p>Это неравенство выполняется для $x \in (\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.</p><p>Найдем интервалы, на которых $f'(x) < 0$ (убывание):</p><p>$-5\cos(x) < 0$</p><p>$\cos(x) > 0$</p><p>Это неравенство выполняется для $x \in (-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.</p><p>В точках $x = \frac{\pi}{2} + \pi k$ производная равна нулю, это стационарные точки.</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k]$ и убывает на интервалах $[-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k]$, где $k \in \mathbb{Z}$.</p><p><strong>б) $g(x) = -2\cos(3x) - 6x\cos(\frac{3\pi}{4})$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Сначала упростим выражение для функции. Вычислим значение константы $\cos(\frac{3\pi}{4})$:</p><p>$\cos(\frac{3\pi}{4}) = \cos(\pi - \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$</p><p>Подставим это значение в функцию:</p><p>$g(x) = -2\cos(3x) - 6x(-\frac{\sqrt{2}}{2}) = -2\cos(3x) + 3\sqrt{2}x$</p><p>Найдем производную функции $g(x)$:</p><p>$g'(x) = (-2\cos(3x) + 3\sqrt{2}x)' = -2(-\sin(3x) \cdot 3) + 3\sqrt{2} = 6\sin(3x) + 3\sqrt{2}$</p><p>2. <i>Определение знака производной.</i></p><p>Найдем интервалы возрастания ($g'(x) > 0$):</p><p>$6\sin(3x) + 3\sqrt{2} > 0$</p><p>$6\sin(3x) > -3\sqrt{2}$</p><p>$\sin(3x) > -\frac{\sqrt{2}}{2}$</p><p>Решая это тригонометрическое неравенство, получаем:</p><p>$-\frac{\pi}{4} + 2\pi k < 3x < \frac{5\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Разделим все части на 3:</p><p>$-\frac{\pi}{12} + \frac{2\pi k}{3} < x < \frac{5\pi}{12} + \frac{2\pi k}{3}$</p><p>Найдем интервалы убывания ($g'(x) < 0$):</p><p>$6\sin(3x) + 3\sqrt{2} < 0$</p><p>$\sin(3x) < -\frac{\sqrt{2}}{2}$</p><p>Решая это неравенство, получаем:</p><p>$\frac{5\pi}{4} + 2\pi k < 3x < \frac{7\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Разделим все части на 3:</p><p>$\frac{5\pi}{12} + \frac{2\pi k}{3} < x < \frac{7\pi}{12} + \frac{2\pi k}{3}$</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[-\frac{\pi}{12} + \frac{2\pi k}{3}, \frac{5\pi}{12} + \frac{2\pi k}{3}]$ и убывает на интервалах $[\frac{5\pi}{12} + \frac{2\pi k}{3}, \frac{7\pi}{12} + \frac{2\pi k}{3}]$, где $k \in \mathbb{Z}$.</p><p><strong>в) $h(x) = \sin(4x - \frac{\pi}{3}) - 2x$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Найдем производную функции $h(x)$, используя правило дифференцирования сложной функции и правило дифференцирования разности:</p><p>$h'(x) = (\sin(4x - \frac{\pi}{3}) - 2x)' = (\sin(4x - \frac{\pi}{3}))' - (2x)'$</p><p>$h'(x) = \cos(4x - \frac{\pi}{3}) \cdot (4x - \frac{\pi}{3})' - 2 = 4\cos(4x - \frac{\pi}{3}) - 2$</p><p>2. <i>Определение знака производной.</i></p><p>Найдем интервалы возрастания ($h'(x) > 0$):</p><p>$4\cos(4x - \frac{\pi}{3}) - 2 > 0$</p><p>$4\cos(4x - \frac{\pi}{3}) > 2$</p><p>$\cos(4x - \frac{\pi}{3}) > \frac{1}{2}$</p><p>Решим это тригонометрическое неравенство. Обозначим $u = 4x - \frac{\pi}{3}$. Неравенство $\cos(u) > \frac{1}{2}$ выполняется при:</p><p>$-\frac{\pi}{3} + 2\pi k < u < \frac{\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Подставим обратно $u = 4x - \frac{\pi}{3}$:</p><p>$-\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{\pi}{3} + 2\pi k$</p><p>Прибавим $\frac{\pi}{3}$ ко всем частям:</p><p>$2\pi k < 4x < \frac{2\pi}{3} + 2\pi k$</p><p>Разделим на 4:</p><p>$\frac{\pi k}{2} < x < \frac{\pi}{6} + \frac{\pi k}{2}$</p><p>Найдем интервалы убывания ($h'(x) < 0$):</p><p>$\cos(4x - \frac{\pi}{3}) < \frac{1}{2}$</p><p>Неравенство $\cos(u) < \frac{1}{2}$ выполняется при:</p><p>$\frac{\pi}{3} + 2\pi k < u < \frac{5\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Подставим обратно $u = 4x - \frac{\pi}{3}$:</p><p>$\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{5\pi}{3} + 2\pi k$</p><p>Прибавим $\frac{\pi}{3}$ ко всем частям:</p><p>$\frac{2\pi}{3} + 2\pi k < 4x < 2\pi + 2\pi k$</p><p>Разделим на 4:</p><p>$\frac{\pi}{6} + \frac{\pi k}{2} < x < \frac{\pi}{2} + \frac{\pi k}{2}$</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[\frac{\pi k}{2}, \frac{\pi}{6} + \frac{\pi k}{2}]$ и убывает на интервалах $[\frac{\pi}{6} + \frac{\pi k}{2}, \frac{\pi}{2} + \frac{\pi k}{2}]$, где $k \in \mathbb{Z}$.</p>" ] #original: array:6 [ "id" => 1552052 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1081} "task" => array:2 [ "refs" => "1120331" "type" => "task" ] "text" => "<p><strong>a) $f(x) = 5\sin(-x) + 5\tan(5)$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Сначала упростим выражение для функции. Так как синус — нечетная функция, $\sin(-x) = -\sin(x)$. Слагаемое $5\tan(5)$ является константой, так как не содержит переменной $x$.</p><p>$f(x) = -5\sin(x) + 5\tan(5)$</p><p>Найдем производную функции $f(x)$ по переменной $x$:</p><p>$f'(x) = (-5\sin(x) + 5\tan(5))' = -5(\sin(x))' + (5\tan(5))' = -5\cos(x) + 0 = -5\cos(x)$</p><p>2. <i>Определение знака производной.</i></p><p>Интервалы монотонности определяются знаком производной. Функция возрастает, когда $f'(x) > 0$, и убывает, когда $f'(x) < 0$.</p><p>Найдем интервалы, на которых $f'(x) > 0$ (возрастание):</p><p>$-5\cos(x) > 0$</p><p>$\cos(x) < 0$</p><p>Это неравенство выполняется для $x \in (\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.</p><p>Найдем интервалы, на которых $f'(x) < 0$ (убывание):</p><p>$-5\cos(x) < 0$</p><p>$\cos(x) > 0$</p><p>Это неравенство выполняется для $x \in (-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.</p><p>В точках $x = \frac{\pi}{2} + \pi k$ производная равна нулю, это стационарные точки.</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k]$ и убывает на интервалах $[-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k]$, где $k \in \mathbb{Z}$.</p><p><strong>б) $g(x) = -2\cos(3x) - 6x\cos(\frac{3\pi}{4})$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Сначала упростим выражение для функции. Вычислим значение константы $\cos(\frac{3\pi}{4})$:</p><p>$\cos(\frac{3\pi}{4}) = \cos(\pi - \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$</p><p>Подставим это значение в функцию:</p><p>$g(x) = -2\cos(3x) - 6x(-\frac{\sqrt{2}}{2}) = -2\cos(3x) + 3\sqrt{2}x$</p><p>Найдем производную функции $g(x)$:</p><p>$g'(x) = (-2\cos(3x) + 3\sqrt{2}x)' = -2(-\sin(3x) \cdot 3) + 3\sqrt{2} = 6\sin(3x) + 3\sqrt{2}$</p><p>2. <i>Определение знака производной.</i></p><p>Найдем интервалы возрастания ($g'(x) > 0$):</p><p>$6\sin(3x) + 3\sqrt{2} > 0$</p><p>$6\sin(3x) > -3\sqrt{2}$</p><p>$\sin(3x) > -\frac{\sqrt{2}}{2}$</p><p>Решая это тригонометрическое неравенство, получаем:</p><p>$-\frac{\pi}{4} + 2\pi k < 3x < \frac{5\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Разделим все части на 3:</p><p>$-\frac{\pi}{12} + \frac{2\pi k}{3} < x < \frac{5\pi}{12} + \frac{2\pi k}{3}$</p><p>Найдем интервалы убывания ($g'(x) < 0$):</p><p>$6\sin(3x) + 3\sqrt{2} < 0$</p><p>$\sin(3x) < -\frac{\sqrt{2}}{2}$</p><p>Решая это неравенство, получаем:</p><p>$\frac{5\pi}{4} + 2\pi k < 3x < \frac{7\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Разделим все части на 3:</p><p>$\frac{5\pi}{12} + \frac{2\pi k}{3} < x < \frac{7\pi}{12} + \frac{2\pi k}{3}$</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[-\frac{\pi}{12} + \frac{2\pi k}{3}, \frac{5\pi}{12} + \frac{2\pi k}{3}]$ и убывает на интервалах $[\frac{5\pi}{12} + \frac{2\pi k}{3}, \frac{7\pi}{12} + \frac{2\pi k}{3}]$, где $k \in \mathbb{Z}$.</p><p><strong>в) $h(x) = \sin(4x - \frac{\pi}{3}) - 2x$</strong></p><p>1. <i>Нахождение производной.</i></p><p>Найдем производную функции $h(x)$, используя правило дифференцирования сложной функции и правило дифференцирования разности:</p><p>$h'(x) = (\sin(4x - \frac{\pi}{3}) - 2x)' = (\sin(4x - \frac{\pi}{3}))' - (2x)'$</p><p>$h'(x) = \cos(4x - \frac{\pi}{3}) \cdot (4x - \frac{\pi}{3})' - 2 = 4\cos(4x - \frac{\pi}{3}) - 2$</p><p>2. <i>Определение знака производной.</i></p><p>Найдем интервалы возрастания ($h'(x) > 0$):</p><p>$4\cos(4x - \frac{\pi}{3}) - 2 > 0$</p><p>$4\cos(4x - \frac{\pi}{3}) > 2$</p><p>$\cos(4x - \frac{\pi}{3}) > \frac{1}{2}$</p><p>Решим это тригонометрическое неравенство. Обозначим $u = 4x - \frac{\pi}{3}$. Неравенство $\cos(u) > \frac{1}{2}$ выполняется при:</p><p>$-\frac{\pi}{3} + 2\pi k < u < \frac{\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Подставим обратно $u = 4x - \frac{\pi}{3}$:</p><p>$-\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{\pi}{3} + 2\pi k$</p><p>Прибавим $\frac{\pi}{3}$ ко всем частям:</p><p>$2\pi k < 4x < \frac{2\pi}{3} + 2\pi k$</p><p>Разделим на 4:</p><p>$\frac{\pi k}{2} < x < \frac{\pi}{6} + \frac{\pi k}{2}$</p><p>Найдем интервалы убывания ($h'(x) < 0$):</p><p>$\cos(4x - \frac{\pi}{3}) < \frac{1}{2}$</p><p>Неравенство $\cos(u) < \frac{1}{2}$ выполняется при:</p><p>$\frac{\pi}{3} + 2\pi k < u < \frac{5\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.</p><p>Подставим обратно $u = 4x - \frac{\pi}{3}$:</p><p>$\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{5\pi}{3} + 2\pi k$</p><p>Прибавим $\frac{\pi}{3}$ ко всем частям:</p><p>$\frac{2\pi}{3} + 2\pi k < 4x < 2\pi + 2\pi k$</p><p>Разделим на 4:</p><p>$\frac{\pi}{6} + \frac{\pi k}{2} < x < \frac{\pi}{2} + \frac{\pi k}{2}$</p><p><strong>Ответ:</strong> функция возрастает на интервалах $[\frac{\pi k}{2}, \frac{\pi}{6} + \frac{\pi k}{2}]$ и убывает на интервалах $[\frac{\pi}{6} + \frac{\pi k}{2}, \frac{\pi}{2} + \frac{\pi k}{2}]$, где $k \in \mathbb{Z}$.</p>" ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1120332" "type" => "task" ] "previous" => array:2 [ "refs" => "1120330" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1100 #items: array:1 [ 0 => App\Models\Book {#1119} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1075 #items: array:1 [ 0 => App\Models\BookPage {#1101 #connection: "mysql" #table: "book_pages" #primaryKey: 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Illuminate\Database\Eloquent\Collection {#1078 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1113 #items: array:1 [ 0 => App\Models\Element {#1111 #connection: "mysql" #table: "elements" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:6 [ "id" => 1262661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #original: array:6 [ "id" => 1262661 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1109 …2} "book_page" => array:2 [ …2] "img" => array:1 [ …1] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "1099237" "type" => "book_page" ] "previous" => array:2 [ "refs" => "1099235" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#1277 #items: array:8 [ 0 => App\Models\Task {#1291 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ "id" => 1120326 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "83" "field_page_end" => null "field_url" => "/10-klass/algebra/baspasy-pak-uchebnik/6-1zad-11" "field_display_title" => "11" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection 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true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 5 => App\Models\Task {#1346 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } 6 => App\Models\Task {#1358 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:24 [ …24] #original: array:24 [ …24] 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№16 (с. 83)
Условие. №16 (с. 83)
скриншот условия
16. (2) Для следующих функций определите интервалы монотонности (используя производную):
а) $f(x)=5\sin(-x)+5\operatorname{tg}5$;
б) $g(x)=-2\cos3x-6x\cos\frac{3\pi}{4}$;
в) $h(x)=\sin\left(4x-\frac{\pi}{3}\right)-2x.$
Решение 2 (rus). №16 (с. 83)
a) $f(x) = 5\sin(-x) + 5\tan(5)$
1. Нахождение производной.
Сначала упростим выражение для функции. Так как синус — нечетная функция, $\sin(-x) = -\sin(x)$. Слагаемое $5\tan(5)$ является константой, так как не содержит переменной $x$.
$f(x) = -5\sin(x) + 5\tan(5)$
Найдем производную функции $f(x)$ по переменной $x$:
$f'(x) = (-5\sin(x) + 5\tan(5))' = -5(\sin(x))' + (5\tan(5))' = -5\cos(x) + 0 = -5\cos(x)$
2. Определение знака производной.
Интервалы монотонности определяются знаком производной. Функция возрастает, когда $f'(x) > 0$, и убывает, когда $f'(x) < 0$.
Найдем интервалы, на которых $f'(x) > 0$ (возрастание):
$-5\cos(x) > 0$
$\cos(x) < 0$
Это неравенство выполняется для $x \in (\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.
Найдем интервалы, на которых $f'(x) < 0$ (убывание):
$-5\cos(x) < 0$
$\cos(x) > 0$
Это неравенство выполняется для $x \in (-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k)$, где $k \in \mathbb{Z}$.
В точках $x = \frac{\pi}{2} + \pi k$ производная равна нулю, это стационарные точки.
Ответ: функция возрастает на интервалах $[\frac{\pi}{2} + 2\pi k, \frac{3\pi}{2} + 2\pi k]$ и убывает на интервалах $[-\frac{\pi}{2} + 2\pi k, \frac{\pi}{2} + 2\pi k]$, где $k \in \mathbb{Z}$.
б) $g(x) = -2\cos(3x) - 6x\cos(\frac{3\pi}{4})$
1. Нахождение производной.
Сначала упростим выражение для функции. Вычислим значение константы $\cos(\frac{3\pi}{4})$:
$\cos(\frac{3\pi}{4}) = \cos(\pi - \frac{\pi}{4}) = -\cos(\frac{\pi}{4}) = -\frac{\sqrt{2}}{2}$
Подставим это значение в функцию:
$g(x) = -2\cos(3x) - 6x(-\frac{\sqrt{2}}{2}) = -2\cos(3x) + 3\sqrt{2}x$
Найдем производную функции $g(x)$:
$g'(x) = (-2\cos(3x) + 3\sqrt{2}x)' = -2(-\sin(3x) \cdot 3) + 3\sqrt{2} = 6\sin(3x) + 3\sqrt{2}$
2. Определение знака производной.
Найдем интервалы возрастания ($g'(x) > 0$):
$6\sin(3x) + 3\sqrt{2} > 0$
$6\sin(3x) > -3\sqrt{2}$
$\sin(3x) > -\frac{\sqrt{2}}{2}$
Решая это тригонометрическое неравенство, получаем:
$-\frac{\pi}{4} + 2\pi k < 3x < \frac{5\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.
Разделим все части на 3:
$-\frac{\pi}{12} + \frac{2\pi k}{3} < x < \frac{5\pi}{12} + \frac{2\pi k}{3}$
Найдем интервалы убывания ($g'(x) < 0$):
$6\sin(3x) + 3\sqrt{2} < 0$
$\sin(3x) < -\frac{\sqrt{2}}{2}$
Решая это неравенство, получаем:
$\frac{5\pi}{4} + 2\pi k < 3x < \frac{7\pi}{4} + 2\pi k$, где $k \in \mathbb{Z}$.
Разделим все части на 3:
$\frac{5\pi}{12} + \frac{2\pi k}{3} < x < \frac{7\pi}{12} + \frac{2\pi k}{3}$
Ответ: функция возрастает на интервалах $[-\frac{\pi}{12} + \frac{2\pi k}{3}, \frac{5\pi}{12} + \frac{2\pi k}{3}]$ и убывает на интервалах $[\frac{5\pi}{12} + \frac{2\pi k}{3}, \frac{7\pi}{12} + \frac{2\pi k}{3}]$, где $k \in \mathbb{Z}$.
в) $h(x) = \sin(4x - \frac{\pi}{3}) - 2x$
1. Нахождение производной.
Найдем производную функции $h(x)$, используя правило дифференцирования сложной функции и правило дифференцирования разности:
$h'(x) = (\sin(4x - \frac{\pi}{3}) - 2x)' = (\sin(4x - \frac{\pi}{3}))' - (2x)'$
$h'(x) = \cos(4x - \frac{\pi}{3}) \cdot (4x - \frac{\pi}{3})' - 2 = 4\cos(4x - \frac{\pi}{3}) - 2$
2. Определение знака производной.
Найдем интервалы возрастания ($h'(x) > 0$):
$4\cos(4x - \frac{\pi}{3}) - 2 > 0$
$4\cos(4x - \frac{\pi}{3}) > 2$
$\cos(4x - \frac{\pi}{3}) > \frac{1}{2}$
Решим это тригонометрическое неравенство. Обозначим $u = 4x - \frac{\pi}{3}$. Неравенство $\cos(u) > \frac{1}{2}$ выполняется при:
$-\frac{\pi}{3} + 2\pi k < u < \frac{\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.
Подставим обратно $u = 4x - \frac{\pi}{3}$:
$-\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{\pi}{3} + 2\pi k$
Прибавим $\frac{\pi}{3}$ ко всем частям:
$2\pi k < 4x < \frac{2\pi}{3} + 2\pi k$
Разделим на 4:
$\frac{\pi k}{2} < x < \frac{\pi}{6} + \frac{\pi k}{2}$
Найдем интервалы убывания ($h'(x) < 0$):
$\cos(4x - \frac{\pi}{3}) < \frac{1}{2}$
Неравенство $\cos(u) < \frac{1}{2}$ выполняется при:
$\frac{\pi}{3} + 2\pi k < u < \frac{5\pi}{3} + 2\pi k$, где $k \in \mathbb{Z}$.
Подставим обратно $u = 4x - \frac{\pi}{3}$:
$\frac{\pi}{3} + 2\pi k < 4x - \frac{\pi}{3} < \frac{5\pi}{3} + 2\pi k$
Прибавим $\frac{\pi}{3}$ ко всем частям:
$\frac{2\pi}{3} + 2\pi k < 4x < 2\pi + 2\pi k$
Разделим на 4:
$\frac{\pi}{6} + \frac{\pi k}{2} < x < \frac{\pi}{2} + \frac{\pi k}{2}$
Ответ: функция возрастает на интервалах $[\frac{\pi k}{2}, \frac{\pi}{6} + \frac{\pi k}{2}]$ и убывает на интервалах $[\frac{\pi}{6} + \frac{\pi k}{2}, \frac{\pi}{2} + \frac{\pi k}{2}]$, где $k \in \mathbb{Z}$.
Другие задания:
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