صفحة 185 - كتاب الرياضيات - الصف 12 - الفصل 2 - المملكة العربية السعودية

{ "language": "ar", "direction": "rtl", "page_context": { "page_title": "اختبار الفصل 8", "page_type": "exercises", "main_topics": [ "النهايات والاشتقاق", "التكامل والمساحة تحت المنحنى", "السرعة المتجهة اللحظية", "ميل المماس" ], "headers": [ "اختبار الفصل 8", "قدّر كل نهاية مما يأتي:", "احسب كل نهاية مما يأتي باستعمال التعويض المباشر إذا كان ممكناً، وإلا فاذكر السبب:", "احسب كل نهاية مما يأتي (إن وجدت):", "أوجد ميل مماس منحنى كل دالة مما يأتي عند النقاط المعطاة:", "أوجد السرعة المتجهة اللحظية v(t) لجسم يُعطى موقعه عند أي زمن بالدالة h(t) في كل مما يأتي:", "أوجد مشتقة كل دالة مما يأتي:", "استعمل النهايات؛ لتقريب مساحة المنطقة المحصورة بين منحنى الدالة والمحور x، والمعطاة بالتكامل المحدد في كل مما يأتي:", "أوجد جميع الدوال الأصلية لكل دالة مما يأتي:", "احسب كل تكامل مما يأتي:" ], "has_questions": true, "has_formulas": true, "has_examples": false, "has_visual_elements": true }, "sections": [ { "order": 1, "type": "header", "content": "اختبار الفصل 8", "content_classification": "EDUCATIONAL_CONTENT" }, { "order": 2, "type": "header", "content": "قدّر كل نهاية مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 3, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "1" }, "content": "1) lim_{x \to 0^+} \sqrt{x + 4} - 8", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 4, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "2" }, "content": "2) lim_{x \to 4} \frac{x^2 - 16}{x - 4}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 5, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "3" }, "content": "3) lim_{x \to 7} \frac{6}{x - 7}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 6, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "4" }, "content": "4) lim_{x \to \infty} x^3 + 5x^2 - 2x + 21", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 7, "type": "exercise", "title": "إلكترونيات", "question_indicators": { "has_numbering": true, "question_number": "5" }, "content": "5) إلكترونيات: يُعطى متوسط تكلفة إنتاج جهاز إلكتروني بالريال عند إنتاج x جهاز بالدالة C(x) = \frac{100x + 7105}{x}.", "content_classification": "QUESTION_HOMEWORK", "sub_questions": [ { "number": "a", "question": "احسب نهاية الدالة عندما تقترب x من المالانهاية." }, { "number": "b", "question": "فسّر الناتج في الفرع a." } ] }, { "order": 8, "type": "header", "content": "احسب كل نهاية مما يأتي باستعمال التعويض المباشر إذا كان ممكناً، وإلا فاذكر السبب:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 9, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "6" }, "content": "6) lim_{x \to 5} \frac{x^2}{\sqrt{x - 4} - 2}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 10, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "7" }, "content": "7) lim_{x \to 9} (2x^3 - 12x + 3)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 11, "type": "exercise", "title": "نادٍ رياضي", "question_indicators": { "has_numbering": true, "question_number": "8" }, "content": "8) نادٍ رياضي: تُمثّل الدالة S(t) = \frac{2000t^2 + 4}{1 + 10t^2} عدد المشتركين في نادٍ رياضي بعد t يوم من افتتاحه.", "content_classification": "QUESTION_HOMEWORK", "sub_questions": [ { "number": "a", "question": "ما عدد المشتركين في البداية؟" }, { "number": "b", "question": "ما أكبر عدد ممكن لمشتركي النادي؟" } ] }, { "order": 12, "type": "header", "content": "احسب كل نهاية مما يأتي (إن وجدت):", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 13, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "9" }, "content": "9) lim_{x \to \infty} (x^2 - 7x + 2)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 14, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "10" }, "content": "10) lim_{x \to \infty} (2x^3 - 8x^2 - 5)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 15, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "11" }, "content": "11) lim_{x \to \infty} \frac{2x^3 - x - 1}{-x^4 + 7x^3 + 4}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 16, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "12" }, "content": "12) lim_{x \to \infty} \frac{\sqrt{25 + x} - 4}{x}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 17, "type": "exercise", "title": "اختيار من متعدد", "question_indicators": { "has_numbering": true, "question_number": "13", "has_multiple_choice": true }, "content": "13) اختيار من متعدد: ما قيمة lim_{x \to 0} \frac{\frac{1}{x+3} - \frac{1}{3}}{x} ؟", "content_classification": "QUESTION_HOMEWORK", "format": "multiple_choice", "options": [ "A) -1/9", "B) 0", "C) 1/9", "D) غير موجودة" ] }, { "order": 18, "type": "header", "content": "أوجد ميل مماس منحنى كل دالة مما يأتي عند النقاط المعطاة:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 19, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "14" }, "content": "14) y = x^2 + 2x - 8, (-5, 7), (-2, -8)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 20, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "15" }, "content": "15) y = \frac{4}{x^3} + 2, (-1, -2), (2, \frac{5}{2})", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 21, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "16" }, "content": "16) y = (2x + 1)^2, (-3, 25), (0, 1)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 22, "type": "header", "content": "أوجد السرعة المتجهة اللحظية v(t) لجسم يُعطى موقعه عند أي زمن بالدالة h(t) في كل مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 23, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "17" }, "content": "17) h(t) = 9t + 3t^2", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 24, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "18" }, "content": "18) h(t) = 10t^2 - 7t^3", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 25, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "19" }, "content": "19) h(t) = 3t^3 - 2 + 4t", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 26, "type": "header", "content": "أوجد مشتقة كل دالة مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 27, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "20" }, "content": "20) f(x) = -3x - 7", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 28, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "21" }, "content": "21) b(c) = 4c^{\frac{1}{2}} - 8c^{\frac{2}{3}} + 5c^{\frac{4}{5}}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 29, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "22" }, "content": "22) w(y) = 3y^{\frac{4}{3}} + 6y^{\frac{1}{2}}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 30, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "23" }, "content": "23) g(x) = (x^2 - 4)(2x - 5)", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 31, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "24" }, "content": "24) h(t) = \frac{t^3 + 4t^2 + t}{t^2}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 32, "type": "exercise", "title": "صناعة", "question_indicators": { "has_numbering": true, "question_number": "25" }, "content": "25) صناعة: تُعطى التكلفة الحدية c بالريال لإنتاج x كرة قدم يومياً بالدالة c(x) = 15 - 0.005x.", "content_classification": "QUESTION_HOMEWORK", "sub_questions": [ { "number": "a", "question": "أوجد دالة تمثّل التكلفة الحقيقية." }, { "number": "b", "question": "أوجد تكلفة زيادة الإنتاج اليومي من 1500 كرة إلى 2000 كرة." } ] }, { "order": 33, "type": "header", "content": "استعمل النهايات؛ لتقريب مساحة المنطقة المحصورة بين منحنى الدالة والمحور x، والمعطاة بالتكامل المحدد في كل مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 34, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "26" }, "content": "26) \int_1^4 (x^2 - 3x + 4) dx", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 35, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "27" }, "content": "27) \int_3^8 10x^4 dx", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 36, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "28" }, "content": "28) \int_2^5 (7 - 2x + 4x^2) dx", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 37, "type": "header", "content": "أوجد جميع الدوال الأصلية لكل دالة مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 38, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "29" }, "content": "29) d(a) = 4a^3 + 9a^2 - 2a + 8", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 39, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "30" }, "content": "30) w(z) = \frac{3}{4}z^4 + \frac{1}{2}z^2 - \frac{2}{5}", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 40, "type": "header", "content": "احسب كل تكامل مما يأتي:", "content_classification": "EDUCATIONAL_CONTENT", "question_indicators": { "has_instruction_words": true } }, { "order": 41, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "31" }, "content": "31) \int (5x^3 - 6x^2 + 4x - 3) dx", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 42, "type": "exercise", "question_indicators": { "has_numbering": true, "question_number": "32" }, "content": "32) \int_1^4 (x^2 + 4x - 2) dx", "content_classification": "QUESTION_HOMEWORK", "format": "short_answer" }, { "order": 43, "type": "exercise", "title": "مساحات", "question_indicators": { "has_numbering": true, "question_number": "33", "has_multiple_choice": true }, "content": "33) مساحات: ما مساحة المنطقة المحصورة بين منحنيي f(x), g(x) في الفترة 2 ≤ x ≤ 4 في الشكل أدناه؟", "content_classification": "QUESTION_HOMEWORK", "format": "multiple_choice", "associated_visual_elements": [ 0 ], "options": [ "A) 17 5/12 وحدة مساحة", "B) 17 1/3 وحدة مساحة", "C) 15 1/3 وحدة مساحة", "D) 16 وحدة مساحة" ] } ], "visual_elements": [ { "index": 0, "label": "الشكل أدناه", "question_number": "33", "type": "graph", "location": "bottom left", "coordinate_system": "Standard Cartesian grid, major grid lines every 2 units on x-axis and 10 units on y-axis.", "shape": "Two intersecting power functions with a shaded region between them.", "title": "مساحة المنطقة المحصورة بين منحنيين", "function": "f(x) = x^2/4, g(x) = x^3/3", "description": "The graph shows two curves, f(x) and g(x), on a Cartesian plane. A region is shaded between the two curves from x = 2 to x = 4. The curve g(x) is above f(x) in this interval.", "axes_labels": { "x_axis": "x", "y_axis": "y" }, "axes_ranges": { "x_min": 0, "x_max": 6, "y_min": 0, "y_max": 30 }, "key_points": [ "(0, 0) - Origin", "(2, 1) - Point on f(x)", "(4, 4) - Point on f(x)", "(2, 2.67) - Point on g(x)", "(4, 21.33) - Point on g(x)" ], "educational_context": "Visual representation for calculating the area between two curves using definite integrals." } ] }