1. Sketch the graph of the cubic function f(x) = 6+12x + 3x² − 2x³...

Question
1. Sketch the graph of the cubic function f(x) = 6+12x + 3x² − 2x³ for −2 ≤ x ≤3.
2. Sketch the graph of the cubic function f(x) = 8x³ +30x² +13x−15 for −4 ≤ x ≤ 2.
1. The fixed costs of producing a good are 12 and the variable costs are 7 per unit. Find expressions for TC and AC. Evaluate TC and AC when Q = 4 and Q = 12. Sketch the graph of AC as a function of Q.
2. The fixed costs of producing a good are 9 and the variable costs are 4+3Q per unit. Find expressions for TC and AC. Evaluate TC and AC when Q = 5 and Q = 10. Sketch the graph of AC as a function of Q.
3. Suppose that the total cost to a furniture company of producing Q desks is TC = 50Q + 40 000. Obtain an expression for the average cost function, AC. What value does AC approach when Q is very large?

1. Find the inverses of the following functions:
a) f(x) = −3x + 2,
b) f(x) = 5x + 3,
c) f(x) = (x − 3)² + 2, 3 ≤ x.
2. Sketch the functions y = 2x and y = 3x on the same graph for −2 ≤ x ≤ 2.
3. Evaluate (a) log3 9, (b) log4 2, (c) log7(1/7).
4. An economy is forecast to grow continuously at an annual rate of 3% so that the gross national product (GNP), measured in billions of euros, after t years is given by: GNP = 60e⁰·⁰³ᵗ
a) Calculate the current value of GNP and its future value in four years’time.
b) After how many years is GNP forecast to be 90 billion euros?
5. The percentage, y, of households possessing dishwashers t years after they have been introduced in a country is modelled by y = 30 − 25ᵉˉ⁰·²ᵗ

a) Find the percentage of households that have dishwashers
i. at their launch;
ii. after 1 year;
iii. after 10 years;
iv. after 20 years.
b) After how many years will the percentage of households possessing dishwashers first reach 15%?

6. A firm’s turnover, y, measured in millions of pounds, after t years is given by: y = 8e⁰·⁰⁹ᵗ
a) What is its turnover in its initial year of trading and after two years oftrading?
b) After how many years will its turnover have doubled since it started trading?
Extra exercises
Solve the equation ln(𝑥 + 4)² = 3 𝑓 or 𝑥
Express log𝑎 3 + log𝑎 4 − log𝑎6 as a single logarithm
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      • Illuminate\Cache\Events\CacheHit (130μs)
      • Illuminate\Cache\Events\CacheHit (118μs)
      • Illuminate\Cache\Events\CacheHit (252μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (198μs)
      • Illuminate\Cache\Events\CacheHit (586μs)
      • Illuminate\Cache\Events\CacheHit (248μs)
      • Illuminate\Cache\Events\CacheHit (299μs)
      • Illuminate\Cache\Events\CacheHit (218μs)
      • Illuminate\Cache\Events\CacheHit (113μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (95μs)
      • Illuminate\Cache\Events\CacheHit (107μs)
      • Illuminate\Cache\Events\CacheHit (182μs)
      • Illuminate\Cache\Events\CacheHit (259μs)
      • Illuminate\Cache\Events\CacheHit (244μs)
      • Illuminate\Cache\Events\CacheHit (136μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (178μs)
      • Illuminate\Cache\Events\CacheHit (105μs)
      • Illuminate\Cache\Events\CacheMissed (185μs)
      • Illuminate\Database\Events\StatementPrepared (1.63ms)
      • Illuminate\Database\Events\QueryExecuted (1.23ms)
      • eloquent.retrieved: App\Models\SubjectCat (66μs)
      • Illuminate\Cache\Events\KeyWritten (270μs)
      • Illuminate\Cache\Events\CacheHit (163μs)
      • Illuminate\Cache\Events\CacheHit (286μs)
      • Illuminate\Cache\Events\CacheHit (131μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (262μs)
      • Illuminate\Cache\Events\CacheHit (307μs)
      • Illuminate\Cache\Events\CacheHit (245μs)
      • Illuminate\Cache\Events\CacheHit (152μs)
      • Illuminate\Cache\Events\CacheHit (100μs)
      • Illuminate\Cache\Events\CacheHit (116μs)
      • Illuminate\Cache\Events\CacheHit (98μs)
      • Illuminate\Cache\Events\CacheHit (111μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • Illuminate\Cache\Events\CacheHit (110μs)
      • Illuminate\Cache\Events\CacheHit (96μs)
      • Illuminate\Cache\Events\CacheHit (109μs)
      • Illuminate\Cache\Events\CacheHit (94μs)
      • creating: site.layouts.app (323μs)
      • composing: site.layouts.app (16μs)
      • creating: components.canonical (366μs)
      • composing: components.canonical (826μs)
      • creating: components.open-graph (212μs)
      • composing: components.open-graph (678μs)
      • creating: site.headers.header (307μs)
      • composing: site.headers.header (57μs)
      • Illuminate\Cache\Events\CacheHit (1.59ms)
      • creating: components.footer (61μs)
      • composing: components.footer (150μs)
      • Illuminate\Cache\Events\CacheHit (699μs)
      • Illuminate\Cache\Events\CacheMissed (1.72ms)
      • Illuminate\Database\Events\StatementPrepared (848μs)
      • Illuminate\Database\Events\QueryExecuted (956μs)
      • eloquent.retrieved: App\Models\SubjectCat (64μs)
      • Illuminate\Cache\Events\KeyWritten (322μs)
      • Illuminate\Cache\Events\CacheHit (235μs)
      • Illuminate\Cache\Events\CacheHit (134μs)
      • Illuminate\Cache\Events\CacheHit (164μs)
      • Illuminate\Cache\Events\CacheHit (167μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (188μs)
      • Illuminate\Cache\Events\CacheHit (268μs)
      • Illuminate\Cache\Events\CacheHit (126μs)
      • Illuminate\Cache\Events\CacheHit (122μs)
      • Illuminate\Cache\Events\CacheHit (115μs)
      • Illuminate\Cache\Events\CacheHit (129μs)
      • Illuminate\Cache\Events\CacheHit (108μs)
      • creating: components.forms.contact-us (165μs)
      • composing: components.forms.contact-us (86μs)
      • creating: components.forms.get-started (107μs)
      • composing: components.forms.get-started (43μs)
      • creating: components.forms.free-tool-download (80μs)
      • composing: components.forms.free-tool-download (39μs)
      • creating: components.forms.claim-free-worksheet (76μs)
      • composing: components.forms.claim-free-worksheet (38μs)
      • creating: components.forms.tutor-subscription-waitlist (75μs)
      • composing: components.forms.tutor-subscription-waitlist (36μs)
      • creating: components.forms.tutor-subscription-join (162μs)
      • composing: components.forms.tutor-subscription-join (48μs)
      • creating: components.forms.tutor-support (86μs)
      • composing: components.forms.tutor-support (38μs)
      • 311 x Illuminate\Cache\Events\CacheHit (17.01%)
        61.80ms
        20 x Illuminate\Database\Events\StatementPrepared (9.54%)
        34.69ms
        20 x Illuminate\Database\Events\QueryExecuted (6.9%)
        25.08ms
        11 x Illuminate\Cache\Events\CacheMissed (2.3%)
        8.36ms
        11 x Illuminate\Cache\Events\KeyWritten (1.42%)
        5.14ms
        1 x Illuminate\Foundation\Events\LocaleUpdated (0.99%)
        3.60ms
        1 x Illuminate\Routing\Events\Routing (0.31%)
        1.12ms
        1 x Illuminate\Database\Events\ConnectionEstablished (0.23%)
        854μs
        1 x composing: components.canonical (0.23%)
        826μs
        12 x eloquent.retrieved: App\Models\SubjectCat (0.21%)
        756μs
        1 x composing: components.open-graph (0.19%)
        678μs
        1 x Illuminate\Routing\Events\RouteMatched (0.1%)
        372μs
        1 x creating: components.canonical (0.1%)
        366μs
        1 x eloquent.booting: App\Models\SubjectCat (0.09%)
        340μs
        1 x creating: site.layouts.app (0.09%)
        323μs
        1 x creating: site.headers.header (0.08%)
        307μs
        1 x creating: homework.show (0.07%)
        258μs
        1 x creating: components.breadcrumbs (0.07%)
        243μs
        7 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        232μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibrary (0.06%)
        216μs
        1 x creating: components.open-graph (0.06%)
        212μs
        8 x eloquent.retrieved: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.05%)
        187μs
        2 x eloquent.retrieved: App\Models\Subject (0.05%)
        181μs
        1 x creating: components.forms.contact-us (0.05%)
        165μs
        1 x creating: components.forms.tutor-subscription-join (0.04%)
        162μs
        1 x composing: components.footer (0.04%)
        150μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibrary (0.04%)
        139μs
        1 x composing: homework.show (0.03%)
        114μs
        1 x composing: components.breadcrumbs (0.03%)
        113μs
        1 x creating: components.forms.get-started (0.03%)
        107μs
        1 x eloquent.booting: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.03%)
        98μs
        1 x eloquent.booting: App\Models\Subject (0.03%)
        96μs
        1 x composing: components.forms.contact-us (0.02%)
        86μs
        1 x creating: components.forms.tutor-support (0.02%)
        86μs
        1 x creating: components.forms.free-tool-download (0.02%)
        80μs
        1 x creating: components.forms.claim-free-worksheet (0.02%)
        76μs
        1 x creating: components.forms.tutor-subscription-waitlist (0.02%)
        75μs
        1 x creating: components.footer (0.02%)
        61μs
        1 x composing: site.headers.header (0.02%)
        57μs
        1 x composing: components.forms.tutor-subscription-join (0.01%)
        48μs
        1 x eloquent.booted: App\Models\Subject (0.01%)
        44μs
        1 x eloquent.booted: App\Models\HomeworkLibrary\HomeworkLibraryFile (0.01%)
        43μs
        1 x composing: components.forms.get-started (0.01%)
        43μs
        1 x eloquent.booted: App\Models\SubjectCat (0.01%)
        39μs
        1 x composing: components.forms.free-tool-download (0.01%)
        39μs
        1 x composing: components.forms.claim-free-worksheet (0.01%)
        38μs
        1 x composing: components.forms.tutor-support (0.01%)
        38μs
        1 x composing: components.forms.tutor-subscription-waitlist (0.01%)
        36μs
        1 x composing: site.layouts.app (0%)
        16μs
      14 templates were rendered
      • 1x homework.showshow.blade.phpblade
      • 1x components.breadcrumbsbreadcrumbs.blade.phpblade
      • 1x site.layouts.appapp.blade.phpblade
      • 1x components.canonicalcanonical.blade.phpblade
      • 1x components.open-graphopen-graph.blade.phpblade
      • 1x site.headers.headerheader.blade.phpblade
      • 1x components.footerfooter.blade.phpblade
      • 1x components.forms.contact-uscontact-us.blade.phpblade
      • 1x components.forms.get-startedget-started.blade.phpblade
      • 1x components.forms.free-tool-downloadfree-tool-download.blade.phpblade
      • 1x components.forms.claim-free-worksheetclaim-free-worksheet.blade.phpblade
      • 1x components.forms.tutor-subscription-waitlisttutor-subscription-waitlist.blade.phpblade
      • 1x components.forms.tutor-subscription-jointutor-subscription-join.blade.phpblade
      • 1x components.forms.tutor-supporttutor-support.blade.phpblade
      uri
      GET college-homework-library/{category}/{subject}/{id}
      middleware
      web, utm.parameters
      controller
      App\Http\Controllers\HomeworkLibraryController@show
      namespace
      where
      as
      homework.show
      file
      app/Http/Controllers/HomeworkLibraryController.php:79-176
      20 statements were executed, 5 of which were duplicates, 15 unique. Show only duplicated45.93ms
      • Connection Establishedtwenty4_siteHomeworkLibraryController.php#91
        Backtrace
        • 13. app/Http/Controllers/HomeworkLibraryController.php:91
        • 14. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 15. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 16. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary` where `status` = 'published' and `price` > 0 and `solutionslibrary`.`id` = '26546' limit 1
        18.33mstwenty4_siteHomeworkLibraryController.php#97
        Bindings
        • 0: published
        • 1: 0
        • 2: 26546
        Backtrace
        • 16. app/Http/Controllers/HomeworkLibraryController.php:97
        • 17. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 18. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 19. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 20. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subjects` where `subjects`.`id` in (9)
        960μstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` in (26546)
        1.6mstwenty4_siteHomeworkLibraryController.php#97
        Backtrace
        • 21. app/Http/Controllers/HomeworkLibraryController.php:97
        • 22. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 23. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 24. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
        • 25. vendor/laravel/framework/src/Illuminate/Routing/Route.php:205
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        890μstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 26. app/Http/Controllers/HomeworkLibraryController.php:105
        • 27. vendor/laravel/framework/src/Illuminate/Routing/Controller.php:54
        • 28. vendor/laravel/framework/src/Illuminate/Routing/ControllerDispatcher.php:43
        • 29. vendor/laravel/framework/src/Illuminate/Routing/Route.php:260
      • select * from `solutionslibrary` where `id` <> 26546 and `subject` = 9 and `status` = 'published' and `price` > 0 order by RAND() limit 6
        8.79mstwenty4_siteHomeworkLibraryRepository.php#30
        Bindings
        • 0: 26546
        • 1: 9
        • 2: published
        • 3: 0
        Backtrace
        • 14. app/Repositories/HomeworkLibraryRepository.php:30
        • 15. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 16. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 18. app/Repositories/HomeworkLibraryRepository.php:39
        • 19. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `subjects` where `subjects`.`id` in (9)
        880μstwenty4_siteHomeworkLibraryRepository.php#30
        Backtrace
        • 19. app/Repositories/HomeworkLibraryRepository.php:30
        • 20. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 21. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 23. app/Repositories/HomeworkLibraryRepository.php:39
        • 24. app/Http/Controllers/HomeworkLibraryController.php:139
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 26546 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'question' order by `order` asc, `id` asc
        870μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26546
        • 1: question
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:58
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 26546 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'teaser' order by `order` asc, `id` asc
        770μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26546
        • 1: teaser
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:69
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `solutionslibrary_files` where `solutionslibrary_files`.`solutionlib_id` = 26546 and `solutionslibrary_files`.`solutionlib_id` is not null and `publish` = 'solution' order by `order` asc, `id` asc
        870μstwenty4_siteHomeworkLibrary.php#260
        Bindings
        • 0: 26546
        • 1: solution
        Backtrace
        • 15. app/Models/HomeworkLibrary/HomeworkLibrary.php:260
        • 16. app/Transformers/HomeworkLibrary/HomeworkLibraryTransformer.php:80
        • 19. vendor/league/fractal/src/TransformerAbstract.php:128
        • 20. vendor/league/fractal/src/TransformerAbstract.php:107
        • 21. vendor/league/fractal/src/Scope.php:383
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        1mstwenty4_siteHomeworkLibrary.php#201
        Bindings
        • 0: 1
        Backtrace
        • 20. app/Models/HomeworkLibrary/HomeworkLibrary.php:201
        • 32. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
        • 33. vendor/laravel/framework/src/Illuminate/View/Engines/PhpEngine.php:58
        • 34. vendor/livewire/livewire/src/ComponentConcerns/RendersLivewireComponents.php:69
        • 35. vendor/laravel/framework/src/Illuminate/View/Engines/CompilerEngine.php:70
      • select * from `subject_cats` where `subject_cats`.`id` = 1 limit 1
        950μstwenty4_siteSubject.php#100
        Bindings
        • 0: 1
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 3 limit 1
        980μstwenty4_siteSubject.php#100
        Bindings
        • 0: 3
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 4 limit 1
        900μstwenty4_siteSubject.php#100
        Bindings
        • 0: 4
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 10 limit 1
        1mstwenty4_siteSubject.php#100
        Bindings
        • 0: 10
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 33 limit 1
        980μstwenty4_siteSubject.php#100
        Bindings
        • 0: 33
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 11 limit 1
        910μstwenty4_siteSubject.php#100
        Bindings
        • 0: 11
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 5 limit 1
        950μstwenty4_siteSubject.php#100
        Bindings
        • 0: 5
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 34 limit 1
        1.01mstwenty4_siteSubject.php#100
        Bindings
        • 0: 34
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 31 limit 1
        2.04mstwenty4_siteSubject.php#100
        Bindings
        • 0: 31
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 34. vendor/laravel/framework/src/Illuminate/Filesystem/Filesystem.php:110
      • select * from `subject_cats` where `subject_cats`.`id` = 36 limit 1
        1.25mstwenty4_siteSubject.php#100
        Bindings
        • 0: 36
        Backtrace
        • 18. app/Models/Subject.php:100
        • 19. vendor/laravel/framework/src/Illuminate/Cache/Repository.php:397
        • 20. vendor/laravel/framework/src/Illuminate/Cache/CacheManager.php:419
        • 22. app/Models/Subject.php:101
        • 28. view::components.footer:170
      App\Models\SubjectCat
      12SubjectCat.php
      App\Models\HomeworkLibrary\HomeworkLibraryFile
      8HomeworkLibraryFile.php
      App\Models\HomeworkLibrary\HomeworkLibrary
      7HomeworkLibrary.php
      App\Models\Subject
      2Subject.php
          _token
          W22q9B75CxH1DJytEheKptP5xv4n9M0naOU0InEB
          utm_source
          direct
          redirectUrl
          /college-homework-library/Mathematics/Linear-Algebra/26546
          _previous
          array:1 [ "url" => "https://staging.dev.24houranswers.com/college-homework-library/Mathematics/Lin...
          _flash
          array:2 [ "old" => [] "new" => [] ]
          PHPDEBUGBAR_STACK_DATA
          []
          path_info
          /college-homework-library/Mathematics/Linear-Algebra/26546
          status_code
          200
          
          status_text
          OK
          format
          html
          content_type
          text/html; charset=UTF-8
          request_query
          []
          
          request_request
          []
          
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