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Номер 68, страница 39 - гдз по алгебре 10-11 класс учебник Колмогоров, Абрамов
Авторы: Колмогоров А. Н., Абрамов А. М., Дудницын Ю. П.
Тип: Учебник
Издательство: Просвещение
Год издания: 2008 - 2026
Цвет обложки: зелёный, чёрный
ISBN: 978-5-09-019513-3
Рекомендовано Министерством образования и науки Российской Федерации
Алгебра и начала математического анализа
Популярные ГДЗ в 10 классе
Глава 1. Тригонометрические функции. Параграф 2. Основные свойства функций - номер 68, страница 39.
App\Models\Task {#1024 // 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" => 103858 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "39" "field_page_end" => null "field_url" => "/10-klass/algebra/kolmogorov/68" "field_display_title" => "68" "field_outside_task" => null "field_task_type" => Illuminate\Database\Eloquent\Collection {#331 #items: array:1 [ 0 => App\Models\Term {#1029 #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 {#1026 #items: [] #escapeWhenCastingToString: false } "top_parent_branch" => Illuminate\Database\Eloquent\Collection {#342 #items: array:1 [ 0 => App\Models\Branch {#1023 #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" => 103762 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Тригонометрические функции" "field_branch_order" => "1" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#333 #items: array:1 [ 0 => App\Models\Term {#1033 #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" => "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 {#1030 #items: array:1 [ 0 => App\Models\Book {#1032 #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" => 30 "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 {#1034 #items: array:1 [ 0 => App\Models\Term {#1035 #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 {#1036 #items: array:2 [ 0 => App\Models\Term {#1037 #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 => "*" ] } 1 => 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:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #original: array:6 [ "id" => 5460 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "11" "field_cases" => array:6 [ …6] "field_translit" => "odinadcatyj" ] #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 {#1039 #items: array:1 [ 0 => App\Models\Term {#1040 #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" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #original: array:6 [ "id" => 5153 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Просвещение" "field_cases" => null "field_translit" => "prosveschenie" ] #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 {#1041 #items: array:3 [ 0 => App\Models\Term {#1042 #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" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #original: array:12 [ "id" => 4632 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Колмогоров" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Андрей" "field_patronymic" => "Николаевич" "field_surname_rp" => "Колмогорова" ] #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 {#1043 #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" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "field_surname_rp" => null ] #original: array:12 [ "id" => 3650 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Абрамов" "field_bio" => null "field_degree_rp" => null "field_foreign_lang_name" => null "field_foreign_lang_patronymic" => null "field_foreign_lang_surname" => null "field_name" => "Александр" "field_patronymic" => "Михайлович" "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 {#1044 #connection: "mysql" #table: "terms" #primaryKey: "id" #keyType: "int" +incrementing: false #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 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#hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } ] #escapeWhenCastingToString: false } "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1045 #items: [] #escapeWhenCastingToString: false } "field_book_type" => Illuminate\Database\Eloquent\Collection {#1046 #items: array:1 [ 0 => App\Models\Term {#1047 #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 {#1048 #items: array:1 [ 0 => App\Models\Term {#1049 #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" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #original: array:6 [ "id" => 9 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Россия" "field_cases" => array:6 [ …6] "field_translit" => "rossiya" ] #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 {#1050 #items: array:1 [ 0 => App\Models\Term {#1051 #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" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #original: array:6 [ "id" => 15 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "Москва" "field_cases" => array:6 [ …6] "field_translit" => "moskva" ] #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 {#1052 #items: [] #escapeWhenCastingToString: false } "field_umk" => Illuminate\Database\Eloquent\Collection {#1053 #items: [] #escapeWhenCastingToString: false } "field_level_of_education" => Illuminate\Database\Eloquent\Collection {#1054 #items: [] #escapeWhenCastingToString: false } "field_standart_of_education" => Illuminate\Database\Eloquent\Collection {#1055 #items: [] #escapeWhenCastingToString: false } "field_publication_number" => "17" "field_publication_type" => Illuminate\Database\Eloquent\Collection {#1056 #items: array:1 [ 0 => App\Models\Term {#1057 #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" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #original: array:6 [ "id" => 33 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "стереотипное" "field_cases" => array:6 [ …6] "field_translit" => "stereotipnoe" ] #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_under_the_edition" => Illuminate\Database\Eloquent\Collection {#1058 #items: array:1 [ 0 => App\Models\Term {#1042} ] #escapeWhenCastingToString: false } "field_under_the_edition_degree" => null "field_cover_description" => null "field_publication_year" => "2008" "field_publication_year_until" => null "field_part" => "" "field_part_writing" => "" "field_second_foreign_language" => null "field_for_whom" => "Учебник для 10-11 классов общеобразовательных учреждений" "field_allowed" => "Рекомендовано Министерством образования и науки Российской Федерации" "field_reserve_field" => "Алгебра и начала математического анализа" "field_link_to_source" => null "field_tasks_count" => "1202" "field_priority" => "2" "field_default_folder" => "/algebra_10/kolmogorov/" "field_isbn" => "978-5-09-019513-3" "field_cover" => array:1 [ 0 => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" ] "field_cover_alts" => array:1 [ 0 => "" ] "field_covers" => array:1 [ 0 => array:4 [ "path" => "/media/algebra_10/kolmogorov/covers/cover1.webp?ts=1733838489" "alt" => "" "width" => "680" "height" => "915" ] ] "field_popular_book" => null "field_recommended_books" => Illuminate\Database\Eloquent\Collection {#1059 #items: [] #escapeWhenCastingToString: false } "field_new_book" => Illuminate\Database\Eloquent\Collection {#1060 #items: [] #escapeWhenCastingToString: false } "field_old_book" => Illuminate\Database\Eloquent\Collection {#1061 #items: [] #escapeWhenCastingToString: false } "field_url" => "/10-klass/algebra/kolmogorov" "field_cover_color" => Illuminate\Database\Eloquent\Collection {#1062 #items: array:2 [ 0 => App\Models\Term {#1063 #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" => 47 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"field_translit" => "chyornyy" ] #original: array:6 [ "id" => 56 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "name" => "чёрный" "field_cases" => array:6 [ …6] "field_translit" => "chyornyy" ] #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" => "<!--Текст в низу страницы--> <p> Выбирая в качестве онлайн-консультанта <strong>ГДЗ по алгебре 10-11 класс учебник Колмогоров, Абрамов, Дудницын (Просвещение)</strong>, каждый школьник может быть уверен, что справится со всеми учебными задачами текущего года. Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. Но его результат будет зависеть от того, насколько хорошо ребята освоят следующие разделы: </p> <ol> <li>Тригонометрические функции.</li> <li>Производная и её применения.</li> <li>Первообразная и интеграл.</li> <li>Показательная и логарифмическая функции.</li> <li>Действительные числа.</li> <li>Функции. </ol> <p> Материал очень насыщенный и достаточно сложный. При этом недостаточно будет просто вызубрить массу формул и правил. Простое заучивание - это лишь часть работы. Главное научиться применять полученные знания на практике. Именно с этим у большинства ребят возникают проблемы. Работа вместе с <strong>решебником по алгебре за 10-11 классы авторов Колмогорова, Абрамова, Дудницына</strong> позволит преодолеть все препятствия в учебе. Детально расписанная в пособии информация поможет прекрасно понять материал и научиться применять теорию на практике. </p> <h2> Что представляет собой решебник к учебнику Колмогорова (Просвещение) </h2> <p> Верные ответы — это далеко не все, что содержится во вспомогательном пособии. Сборник с готовыми домашними заданиями структурно и по содержанию полностью копирует учебник под редакцией Колмогорова и включает: </p> <ul> <li>тематические параграфы с соответствующими им упражнениями;</li> <li>номера к разделу «Задачи на повторение»;</li> <li>задачи повышенной трудности.</li> </ul> <p> Каждое решение расписано максимально подробно, с выкладками по теории, графиками и выводами. В особо сложных вопросах авторы дают дополнительные комментарии. Материал изложен простым языком, доступным для понимания абсолютно всем старшеклассникам. Электронный формат существенно облегчает работу с пособием, позволяет выбрать удобный ритм для учебы, экономит время и силы ребенка. </p> <h2> Зачем использовать ГДЗ по алгебре </h2> <p> Даже в выпускном классе нельзя опускать руки, если вдруг освоение алгебры не задалось. Еще есть время поправить свои оценки и знания. Главное правильно распределить свое время, найти верного помощника и вернуть интерес к предмету. Решебник позволит все эти элементы совместить воедино для достижения блестящего результата. Предложенное пособие станет тем консультантом, который даст возможность подростку быстро и качественно выполнить домашнее задание. Это в свою очередь позволит вернуть интерес к предмету и, что немало важно, выкроить ценные часы на отдых. Главное - не прибегать к списыванию готовых решений, ведь простое копирование поможет лишь на время, однако на первой же проверочной в классе обман вскроется. 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Помимо идеальной подготовки домашних заданий, подросток сможет самостоятельно осваивать новые темы, выполнять диагностику своих знаний и восполнять в них пробелы, повторять и закреплять изученный материал. При этом ребенку не понадобятся дополнительные консультации педагога. Со всеми вопросами и темами текущего учебного года он справится самостоятельно. </p> <h2> Что готовит школьникам алгебра в 10-11 классах </h2> <p> Алгебру можно назвать лидером среди всех учебных дисциплин по количеству изучаемых тем. В десятом и одиннадцатом классах все усложняется тем, что помимо новых разделов, ребятам предстоит огромная работа по повторению ранее пройденного материала, ведь в конце года всех их ждет сложнейшее испытание - итоговый экзамен по предмету. 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null "field_description_top" => null "field_description_bottom" => null "field_branch_cover" => [] "field_branch_covers" => [] "book" => Illuminate\Database\Eloquent\Collection {#1075 …2} "branch_parent" => Illuminate\Database\Eloquent\Collection {#1108 …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" => 103764 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_display_title" => "Основные свойства функций" "field_branch_order" => "2" "field_url" => null "field_branch_type" => Illuminate\Database\Eloquent\Collection {#1068} "field_page_start" => "21" "field_branch_display" => "0" "field_branch_expanded" => "0" "field_display_branch_in_title" => "1" 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#with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:7 [ "id" => 156015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1111 #items: array:1 [ 0 => App\Models\Edition {#1112 #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" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1117 …2} "field_content_source" => null ] #original: array:21 [ "id" => 644 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Условие" "field_order" => "1" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1113 …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" => "vadim" "field_edition_type" => "statement" "field_root_dir" => "0-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1115 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1116 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1117 …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" => "103858" "type" => "task" ] "text" => "<p><strong>68.</strong> Для функции $f$ ученик проверил справедливость двух равенств и сделал вывод, что $T$ является периодом $f$. Прав ли ученик, если:</p><p><strong>а)</strong> $f(x) = \sin x$, $\sin \frac{\pi}{6} = \frac{1}{2}$, $\sin \left(\frac{\pi}{6} + \frac{2\pi}{3}\right) = \frac{1}{2}$, $T = \frac{2\pi}{3}$;</p><p><strong>б)</strong> $f(x) = \cos x$, $\cos \frac{\pi}{2} = 0$, $\cos \left(\frac{\pi}{2} + \pi\right) = 0$, $T = \pi$;</p><p><strong>в)</strong> $f(x) = \begin{cases} x+1, \text{ если } x \leq 1, \\ 3-x, \text{ если } x > 1, \end{cases}$</p><p>$f\left(-\frac{1}{2}\right)=0,5$, $f\left(-\frac{1}{2}+3\right)=0,5$, $T=3$;</p><p><strong>г)</strong> $f(x) = x+|x|$, $f(-4)=0$, $f(-4+3)=0$, $T=3$? </p>" "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1366" "height" => 693 "path" => "/media/algebra_10/kolmogorov/0-00/68-1.webp?ts=1733864664" ] ] ] #original: array:7 [ "id" => 156015 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1111} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "text" => "<p><strong>68.</strong> Для функции $f$ ученик проверил справедливость двух равенств и сделал вывод, что $T$ является периодом $f$. Прав ли ученик, если:</p><p><strong>а)</strong> $f(x) = \sin x$, $\sin \frac{\pi}{6} = \frac{1}{2}$, $\sin \left(\frac{\pi}{6} + \frac{2\pi}{3}\right) = \frac{1}{2}$, $T = \frac{2\pi}{3}$;</p><p><strong>б)</strong> $f(x) = \cos x$, $\cos \frac{\pi}{2} = 0$, $\cos \left(\frac{\pi}{2} + \pi\right) = 0$, $T = \pi$;</p><p><strong>в)</strong> $f(x) = \begin{cases} x+1, \text{ если } x \leq 1, \\ 3-x, \text{ если } x > 1, \end{cases}$</p><p>$f\left(-\frac{1}{2}\right)=0,5$, $f\left(-\frac{1}{2}+3\right)=0,5$, $T=3$;</p><p><strong>г)</strong> $f(x) = x+|x|$, $f(-4)=0$, $f(-4+3)=0$, $T=3$? </p>" "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1366" "height" => 693 "path" => "/media/algebra_10/kolmogorov/0-00/68-1.webp?ts=1733864664" ] ] ] #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 {#1118 #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" => 156264 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1119 #items: array:1 [ 0 => App\Models\Edition {#1120 #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" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 …2} "field_content_source" => null ] #original: array:21 [ "id" => 645 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 1" "field_order" => "2" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1121 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "1-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1123 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1124 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1125 …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" => "103858" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "68-1.png" "alt" => null "width" => "767" "height" => 1072 "path" => "/media/algebra_10/kolmogorov/1-00/68-1.webp?ts=1733864682" ] 1 => array:5 [ "name" => "68-2.png" "alt" => null "width" => "766" "height" => 1026 "path" => "/media/algebra_10/kolmogorov/1-00/68-2.webp?ts=1733864682" ] ] ] #original: array:6 [ "id" => 156264 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1119} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "68-1.png" "alt" => null "width" => "767" "height" => 1072 "path" => "/media/algebra_10/kolmogorov/1-00/68-1.webp?ts=1733864682" ] 1 => array:5 [ "name" => "68-2.png" "alt" => null "width" => "766" "height" => 1026 "path" => "/media/algebra_10/kolmogorov/1-00/68-2.webp?ts=1733864682" ] ] ] #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\Element {#1126 #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" => 156834 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127 #items: array:1 [ 0 => App\Models\Edition {#1128 #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" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133 …2} "field_content_source" => null ] #original: array:21 [ "id" => 646 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 2" "field_order" => "3" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1129 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "2-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1131 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1132 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1133 …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" => "103858" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1125" "height" => 378 "path" => "/media/algebra_10/kolmogorov/2-00/68-1.webp?ts=1733995476" ] ] ] #original: array:6 [ "id" => 156834 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1127} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1125" "height" => 378 "path" => "/media/algebra_10/kolmogorov/2-00/68-1.webp?ts=1733995476" ] ] ] #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\Element {#1134 #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" => 168270 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135 #items: array:1 [ 0 => App\Models\Edition {#1136 #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" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …2} "field_content_source" => null ] #original: array:21 [ "id" => 647 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 3" "field_order" => "4" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1137 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "3-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1139 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1140 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1141 …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" => "103858" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1125" "height" => 378 "path" => "/media/algebra_10/kolmogorov/3-00/68-1.webp?ts=1733995499" ] 1 => array:5 [ "name" => "68-2.jpg" "alt" => null "width" => "1125" "height" => 611 "path" => "/media/algebra_10/kolmogorov/3-00/68-2.webp?ts=1733995499" ] ] ] #original: array:6 [ "id" => 168270 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1135} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "img" => array:2 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1125" "height" => 378 "path" => "/media/algebra_10/kolmogorov/3-00/68-1.webp?ts=1733995499" ] 1 => array:5 [ "name" => "68-2.jpg" "alt" => null "width" => "1125" "height" => 611 "path" => "/media/algebra_10/kolmogorov/3-00/68-2.webp?ts=1733995499" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 4 => App\Models\Element {#1142 #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" => 167789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1143 #items: array:1 [ 0 => App\Models\Edition {#1144 #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" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149 …2} "field_content_source" => null ] #original: array:21 [ "id" => 648 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 4" "field_order" => "5" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1145 …2} "field_content_type" => "free" "field_content_mode" => "image" "field_page_content_mode" => "" "field_content_text_checked" => "0" "field_page_content_text_checked" => "0" "field_solution_author" => "Неизвестный" "field_moderator" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "4-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1147 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1148 …2} "field_process_formula" => "" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1149 …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" => "103858" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1372" "height" => 236 "path" => "/media/algebra_10/kolmogorov/4-00/68-1.webp?ts=1733992357" ] ] ] #original: array:6 [ "id" => 167789 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1143} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "img" => array:1 [ 0 => array:5 [ "name" => "68-1.jpg" "alt" => null "width" => "1372" "height" => 236 "path" => "/media/algebra_10/kolmogorov/4-00/68-1.webp?ts=1733992357" ] ] ] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ 0 => "*" ] } 5 => App\Models\Element {#1150 #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" => 1340753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1151 #items: array:1 [ 0 => App\Models\Edition {#1152 #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" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 …2} "field_content_source" => null ] #original: array:21 [ "id" => 5114 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "title" => "Решение 5" "field_order" => "6" "field_publisher" => Illuminate\Database\Eloquent\Collection {#1153 …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" => "vadim" "field_edition_type" => "solution" "field_root_dir" => "5-" "field_responsible" => Illuminate\Database\Eloquent\Collection {#1155 …2} "field_comment" => null "field_similar_book" => Illuminate\Database\Eloquent\Collection {#1156 …2} "field_process_formula" => "katex" "field_edition_group" => Illuminate\Database\Eloquent\Collection {#1157 …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" => "103858" "type" => "task" ] "text" => "<p><strong>а)</strong> Утверждение ученика основано на проверке равенства $f(x_0) = f(x_0+T)$ для одного конкретного значения $x_0 = \frac{\pi}{6}$. Для того чтобы число $T$ являлось периодом функции, равенство $f(x+T) = f(x)$ должно выполняться для <strong>всех</strong> значений $x$ из области определения функции.<br>Проверка ученика: $f(\frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2}$. Значение функции в точке $x_0+T$: $f(\frac{\pi}{6} + \frac{2\pi}{3}) = \sin(\frac{\pi+4\pi}{6}) = \sin \frac{5\pi}{6} = \frac{1}{2}$. Для данной точки равенство выполняется.<br>Однако, это не доказывает, что $T=\frac{2\pi}{3}$ является периодом. Найдем контрпример. Возьмем $x=0$:<br>$f(0) = \sin 0 = 0$.<br>$f(0 + \frac{2\pi}{3}) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$.<br>Поскольку $f(0) \neq f(0+T)$, то есть $0 \neq \frac{\sqrt{3}}{2}$, число $T = \frac{2\pi}{3}$ не является периодом функции $f(x) = \sin x$. Таким образом, ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>б)</strong> Ученик проверил равенство для точки $x = \frac{\pi}{2}$ и $T=\pi$ для функции $f(x)=\cos x$.<br>$f(\frac{\pi}{2}) = \cos \frac{\pi}{2} = 0$.<br>$f(\frac{\pi}{2} + \pi) = \cos \frac{3\pi}{2} = 0$.<br>Для этой точки равенство $f(x)=f(x+T)$ выполняется. Однако, как и в предыдущем пункте, этого недостаточно.<br>Проверим для другой точки, например, $x=0$:<br>$f(0) = \cos 0 = 1$.<br>$f(0 + \pi) = \cos \pi = -1$.<br>Так как $1 \neq -1$, равенство не выполняется для всех $x$. На самом деле, верна формула $\cos(x+\pi) = -\cos x$, то есть равенство $\cos(x+\pi)=\cos x$ выполняется только для тех $x$, где $\cos x = 0$. Следовательно, $T = \pi$ не является периодом функции $f(x)=\cos x$, и ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>в)</strong> Дана кусочно-заданная функция $f(x) = \begin{cases} x+1, & \text{если } x \le 1 \\ 3-x, & \text{если } x > 1 \end{cases}$ и предполагаемый период $T=3$.<br>Ученик проверил равенство для $x = -\frac{1}{2}$:<br>Поскольку $-\frac{1}{2} \le 1$, то $f(-\frac{1}{2}) = -\frac{1}{2} + 1 = 0,5$.<br>Точка $x+T = -\frac{1}{2} + 3 = 2,5$. Поскольку $2,5 > 1$, то $f(2,5) = 3 - 2,5 = 0,5$.<br>Равенство для данной точки выполняется.<br>Проверим для другой точки, например, $x=0$:<br>Поскольку $0 \le 1$, то $f(0) = 0+1=1$.<br>Точка $x+T = 0+3=3$. Поскольку $3 > 1$, то $f(3) = 3-3=0$.<br>Так как $f(0) \neq f(0+3)$ ($1 \neq 0$), число $T=3$ не является периодом функции. Ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>г)</strong> Дана функция $f(x) = x+|x|$ и предполагаемый период $T=3$. Эту функцию можно записать в виде: $f(x) = \begin{cases} 0, & \text{если } x < 0 \\ 2x, & \text{если } x \ge 0 \end{cases}$.<br>Ученик проверил равенство для $x=-4$:<br>Поскольку $-4 < 0$, то $f(-4)=0$.<br>Точка $x+T = -4+3=-1$. Поскольку $-1 < 0$, то $f(-1)=0$.<br>Равенство для данной точки выполняется.<br>Проверим для другой точки, например, $x=1$:<br>Поскольку $1 \ge 0$, то $f(1) = 2 \cdot 1 = 2$.<br>Точка $x+T = 1+3=4$. Поскольку $4 \ge 0$, то $f(4) = 2 \cdot 4 = 8$.<br>Так как $f(1) \neq f(1+3)$ ($2 \neq 8$), число $T=3$ не является периодом функции. Ученик не прав.</p><p>Ответ: Ученик не прав.</p>" ] #original: array:6 [ "id" => 1340753 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "edition" => Illuminate\Database\Eloquent\Collection {#1151} "task" => array:2 [ "refs" => "103858" "type" => "task" ] "text" => "<p><strong>а)</strong> Утверждение ученика основано на проверке равенства $f(x_0) = f(x_0+T)$ для одного конкретного значения $x_0 = \frac{\pi}{6}$. Для того чтобы число $T$ являлось периодом функции, равенство $f(x+T) = f(x)$ должно выполняться для <strong>всех</strong> значений $x$ из области определения функции.<br>Проверка ученика: $f(\frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2}$. Значение функции в точке $x_0+T$: $f(\frac{\pi}{6} + \frac{2\pi}{3}) = \sin(\frac{\pi+4\pi}{6}) = \sin \frac{5\pi}{6} = \frac{1}{2}$. Для данной точки равенство выполняется.<br>Однако, это не доказывает, что $T=\frac{2\pi}{3}$ является периодом. Найдем контрпример. Возьмем $x=0$:<br>$f(0) = \sin 0 = 0$.<br>$f(0 + \frac{2\pi}{3}) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$.<br>Поскольку $f(0) \neq f(0+T)$, то есть $0 \neq \frac{\sqrt{3}}{2}$, число $T = \frac{2\pi}{3}$ не является периодом функции $f(x) = \sin x$. Таким образом, ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>б)</strong> Ученик проверил равенство для точки $x = \frac{\pi}{2}$ и $T=\pi$ для функции $f(x)=\cos x$.<br>$f(\frac{\pi}{2}) = \cos \frac{\pi}{2} = 0$.<br>$f(\frac{\pi}{2} + \pi) = \cos \frac{3\pi}{2} = 0$.<br>Для этой точки равенство $f(x)=f(x+T)$ выполняется. Однако, как и в предыдущем пункте, этого недостаточно.<br>Проверим для другой точки, например, $x=0$:<br>$f(0) = \cos 0 = 1$.<br>$f(0 + \pi) = \cos \pi = -1$.<br>Так как $1 \neq -1$, равенство не выполняется для всех $x$. На самом деле, верна формула $\cos(x+\pi) = -\cos x$, то есть равенство $\cos(x+\pi)=\cos x$ выполняется только для тех $x$, где $\cos x = 0$. Следовательно, $T = \pi$ не является периодом функции $f(x)=\cos x$, и ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>в)</strong> Дана кусочно-заданная функция $f(x) = \begin{cases} x+1, & \text{если } x \le 1 \\ 3-x, & \text{если } x > 1 \end{cases}$ и предполагаемый период $T=3$.<br>Ученик проверил равенство для $x = -\frac{1}{2}$:<br>Поскольку $-\frac{1}{2} \le 1$, то $f(-\frac{1}{2}) = -\frac{1}{2} + 1 = 0,5$.<br>Точка $x+T = -\frac{1}{2} + 3 = 2,5$. Поскольку $2,5 > 1$, то $f(2,5) = 3 - 2,5 = 0,5$.<br>Равенство для данной точки выполняется.<br>Проверим для другой точки, например, $x=0$:<br>Поскольку $0 \le 1$, то $f(0) = 0+1=1$.<br>Точка $x+T = 0+3=3$. Поскольку $3 > 1$, то $f(3) = 3-3=0$.<br>Так как $f(0) \neq f(0+3)$ ($1 \neq 0$), число $T=3$ не является периодом функции. Ученик не прав.</p><p>Ответ: Ученик не прав.</p><p><strong>г)</strong> Дана функция $f(x) = x+|x|$ и предполагаемый период $T=3$. Эту функцию можно записать в виде: $f(x) = \begin{cases} 0, & \text{если } x < 0 \\ 2x, & \text{если } x \ge 0 \end{cases}$.<br>Ученик проверил равенство для $x=-4$:<br>Поскольку $-4 < 0$, то $f(-4)=0$.<br>Точка $x+T = -4+3=-1$. Поскольку $-1 < 0$, то $f(-1)=0$.<br>Равенство для данной точки выполняется.<br>Проверим для другой точки, например, $x=1$:<br>Поскольку $1 \ge 0$, то $f(1) = 2 \cdot 1 = 2$.<br>Точка $x+T = 1+3=4$. Поскольку $4 \ge 0$, то $f(4) = 2 \cdot 4 = 8$.<br>Так как $f(1) \neq f(1+3)$ ($2 \neq 8$), число $T=3$ не является периодом функции. Ученик не прав.</p><p>Ответ: Ученик не прав.</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" => "103859" "type" => "task" ] "previous" => array:2 [ "refs" => "103857" "type" => "task" ] "book" => Illuminate\Database\Eloquent\Collection {#1158 #items: array:1 [ 0 => App\Models\Book {#1032} ] #escapeWhenCastingToString: false } "page" => Illuminate\Database\Eloquent\Collection {#1173 #items: array:1 [ 0 => App\Models\BookPage {#1172 #connection: "mysql" #table: "book_pages" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false #attributes: array:21 [ "id" => 863398 "created_at" => "2026-04-10 13:58:26" "updated_at" => null "field_page_start" => "39" "field_url" => "/10-klass/algebra/kolmogorov/page-39" "field_display_title" => "39" "field_folder" => "1" "field_image_name" => "39" "field_branch_parent" => Illuminate\Database\Eloquent\Collection {#1174 #items: [] #escapeWhenCastingToString: false } "field_weight" => "0" "field_book_parent" => Illuminate\Database\Eloquent\Collection {#1209 #items: array:1 [ 0 => App\Models\Book {#1175 #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" => 30 "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 {#1177 …2} "field_class" => Illuminate\Database\Eloquent\Collection {#1178 …2} "field_publisher" => Illuminate\Database\Eloquent\Collection {#1181 …2} "field_author" => Illuminate\Database\Eloquent\Collection {#1183 …2} "field_author_foreign" => Illuminate\Database\Eloquent\Collection {#1187 …2} …40 ] #original: array:50 [ …50] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "field_metatags_title" => null "field_metatags_description" => null "field_h1" => null "field_description_top" => null "field_description_bottom" => null "edition_groups" => Illuminate\Database\Eloquent\Collection {#1207 #items: [] #escapeWhenCastingToString: false } "content" => Illuminate\Database\Eloquent\Collection {#1218 #items: array:1 [ 0 => App\Models\Element {#1217 #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 [ …6] #original: array:6 [ …6] #changes: [] #casts: [] #classCastCache: [] #attributeCastCache: [] #dateFormat: null #appends: [] #dispatchesEvents: [] #observables: [] #relations: [] #touches: [] +timestamps: true +usesUniqueIds: false #hidden: [] #visible: [] #fillable: [] #guarded: array:1 [ …1] } ] #escapeWhenCastingToString: false } "next" => array:2 [ "refs" => "863399" "type" => "book_page" ] "previous" => array:2 [ "refs" => "863397" "type" => "book_page" ] "tasks" => Illuminate\Database\Eloquent\Collection {#2133 #items: array:7 [ 0 => App\Models\Task {#1210 #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] } 1 => 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] } 2 => App\Models\Task {#1473 #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] } 3 => App\Models\Task {#1608 #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] } 4 => App\Models\Task {#1743 #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] } 5 => App\Models\Task {#1878 #connection: "mysql" #table: "tasks" #primaryKey: "id" #keyType: "int" +incrementing: true #with: [] #withCount: [] +preventsLazyLoading: false #perPage: 15 +exists: true +wasRecentlyCreated: false #escapeWhenCastingToString: false 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№68 (с. 39)
Условие. №68 (с. 39)
скриншот условия
68. Для функции $f$ ученик проверил справедливость двух равенств и сделал вывод, что $T$ является периодом $f$. Прав ли ученик, если:
а) $f(x) = \sin x$, $\sin \frac{\pi}{6} = \frac{1}{2}$, $\sin \left(\frac{\pi}{6} + \frac{2\pi}{3}\right) = \frac{1}{2}$, $T = \frac{2\pi}{3}$;
б) $f(x) = \cos x$, $\cos \frac{\pi}{2} = 0$, $\cos \left(\frac{\pi}{2} + \pi\right) = 0$, $T = \pi$;
в) $f(x) = \begin{cases} x+1, \text{ если } x \leq 1, \\ 3-x, \text{ если } x > 1, \end{cases}$
$f\left(-\frac{1}{2}\right)=0,5$, $f\left(-\frac{1}{2}+3\right)=0,5$, $T=3$;
г) $f(x) = x+|x|$, $f(-4)=0$, $f(-4+3)=0$, $T=3$?
Решение 1. №68 (с. 39)
Решение 2. №68 (с. 39)
Решение 3. №68 (с. 39)
Решение 4. №68 (с. 39)
Решение 5. №68 (с. 39)
а) Утверждение ученика основано на проверке равенства $f(x_0) = f(x_0+T)$ для одного конкретного значения $x_0 = \frac{\pi}{6}$. Для того чтобы число $T$ являлось периодом функции, равенство $f(x+T) = f(x)$ должно выполняться для всех значений $x$ из области определения функции.
Проверка ученика: $f(\frac{\pi}{6}) = \sin \frac{\pi}{6} = \frac{1}{2}$. Значение функции в точке $x_0+T$: $f(\frac{\pi}{6} + \frac{2\pi}{3}) = \sin(\frac{\pi+4\pi}{6}) = \sin \frac{5\pi}{6} = \frac{1}{2}$. Для данной точки равенство выполняется.
Однако, это не доказывает, что $T=\frac{2\pi}{3}$ является периодом. Найдем контрпример. Возьмем $x=0$:
$f(0) = \sin 0 = 0$.
$f(0 + \frac{2\pi}{3}) = \sin \frac{2\pi}{3} = \frac{\sqrt{3}}{2}$.
Поскольку $f(0) \neq f(0+T)$, то есть $0 \neq \frac{\sqrt{3}}{2}$, число $T = \frac{2\pi}{3}$ не является периодом функции $f(x) = \sin x$. Таким образом, ученик не прав.
Ответ: Ученик не прав.
б) Ученик проверил равенство для точки $x = \frac{\pi}{2}$ и $T=\pi$ для функции $f(x)=\cos x$.
$f(\frac{\pi}{2}) = \cos \frac{\pi}{2} = 0$.
$f(\frac{\pi}{2} + \pi) = \cos \frac{3\pi}{2} = 0$.
Для этой точки равенство $f(x)=f(x+T)$ выполняется. Однако, как и в предыдущем пункте, этого недостаточно.
Проверим для другой точки, например, $x=0$:
$f(0) = \cos 0 = 1$.
$f(0 + \pi) = \cos \pi = -1$.
Так как $1 \neq -1$, равенство не выполняется для всех $x$. На самом деле, верна формула $\cos(x+\pi) = -\cos x$, то есть равенство $\cos(x+\pi)=\cos x$ выполняется только для тех $x$, где $\cos x = 0$. Следовательно, $T = \pi$ не является периодом функции $f(x)=\cos x$, и ученик не прав.
Ответ: Ученик не прав.
в) Дана кусочно-заданная функция $f(x) = \begin{cases} x+1, & \text{если } x \le 1 \\ 3-x, & \text{если } x > 1 \end{cases}$ и предполагаемый период $T=3$.
Ученик проверил равенство для $x = -\frac{1}{2}$:
Поскольку $-\frac{1}{2} \le 1$, то $f(-\frac{1}{2}) = -\frac{1}{2} + 1 = 0,5$.
Точка $x+T = -\frac{1}{2} + 3 = 2,5$. Поскольку $2,5 > 1$, то $f(2,5) = 3 - 2,5 = 0,5$.
Равенство для данной точки выполняется.
Проверим для другой точки, например, $x=0$:
Поскольку $0 \le 1$, то $f(0) = 0+1=1$.
Точка $x+T = 0+3=3$. Поскольку $3 > 1$, то $f(3) = 3-3=0$.
Так как $f(0) \neq f(0+3)$ ($1 \neq 0$), число $T=3$ не является периодом функции. Ученик не прав.
Ответ: Ученик не прав.
г) Дана функция $f(x) = x+|x|$ и предполагаемый период $T=3$. Эту функцию можно записать в виде: $f(x) = \begin{cases} 0, & \text{если } x < 0 \\ 2x, & \text{если } x \ge 0 \end{cases}$.
Ученик проверил равенство для $x=-4$:
Поскольку $-4 < 0$, то $f(-4)=0$.
Точка $x+T = -4+3=-1$. Поскольку $-1 < 0$, то $f(-1)=0$.
Равенство для данной точки выполняется.
Проверим для другой точки, например, $x=1$:
Поскольку $1 \ge 0$, то $f(1) = 2 \cdot 1 = 2$.
Точка $x+T = 1+3=4$. Поскольку $4 \ge 0$, то $f(4) = 2 \cdot 4 = 8$.
Так как $f(1) \neq f(1+3)$ ($2 \neq 8$), число $T=3$ не является периодом функции. Ученик не прав.
Ответ: Ученик не прав.
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