الاتصال والنهايات - كتاب الرياضيات - الصف 12 - الفصل 1 - المملكة العربية السعودية

--- SECTION: إرشادات للدراسة --- في المثال 6، أوجدت قيم تقريبية لـ (f(x لأن ما يهمنا هو استقصاء نهاية الدالة (f(x عندما تزداد |x| بلا حدود، وليس حساب القيم الدقيقة لـ (f(x. وكذلك في المثال 7. --- SECTION: المنحنيات التي تقترب من ما لانهاية --- المنحنيات التي تقترب من ما لانهاية --- SECTION: مثال 6 --- مثال 6 استعمل التمثيل البياني للدالة f(x) = -x⁴ + 8x³ + 3x² + 6x – 80 لوصف سلوك طرفي التمثيل البياني، ثم عزز إجابتك عددياً. --- SECTION: التحليل بيانياً: --- التحليل بيانياً: يتضح من التمثيل البياني أن lim f(x) = -∞ عندما x → -∞ ، وأن lim f(x) = -∞ عندما x → +∞ . --- SECTION: التعزيز عددياً: --- التعزيز عددياً: كون جدولاً لاستقصاء قيم (f(x عندما تزداد |x| بلا حدود أو تتناقص بلا حدود. --- SECTION: جدول قيم (f(x --- Table showing values of x and corresponding f(x) for the function f(x) = -x⁴ + 8x³ + 3x² + 6x - 80, demonstrating the function's behavior as x approaches positive and negative infinity. لاحظ أنه عندما x → -∞ ، فإن f(x) → -∞. وبالتمثيل البياني عندما x → +∞ ، فإن f(x) → -∞. وهذا يعزز ما توصلنا إليه من التمثيل البياني. --- SECTION: تحقق من فهمك --- تحقق من فهمك لاحظ أن بعض الدوال تقترب قيمها من ∞ أو ∞- عندما تزداد |x| بلا حدود، في حين تقترب قيم بعض الدوال من أعداد حقيقية دون أن تصل إليها بالضرورة. --- SECTION: منحنيات دوال تقترب من قيمة محددة --- منحنيات دوال تقترب من قيمة محددة --- SECTION: مثال 7 --- مثال 7 استعمل التمثيل البياني للدالة f(x) = x / (x² - 2x + 8) لوصف سلوك طرفي تمثيلها البياني. ثم عزز إجابتك عددياً. --- SECTION: التحليل بيانياً: --- التحليل بيانياً: يتضح من التمثيل البياني أن lim f(x) = 0 عندما x → -∞ ، وأن lim f(x) = 0 عندما x → +∞ . --- SECTION: التعزيز عددياً: --- التعزيز عددياً: --- SECTION: جدول قيم (f(x --- Table showing values of x and corresponding f(x) for the function f(x) = x / (x² - 2x + 8), demonstrating the function's behavior as x approaches positive and negative infinity. لاحظ أنه عندما x → -∞ ، فإن f(x) → 0 و عندما x → +∞ ، فإن f(x) → 0. وهذا يعزز ما توصلنا إليه من التمثيل البياني. وزارة التعليم الدرس 3-1 الاتصال والنهايات 33 2025 - 1447 --- VISUAL CONTEXT --- **GRAPH**: f(x) = -x⁴ + 8x³ + 3x² + 6x - 80 Description: A graph of the polynomial function f(x) = -x⁴ + 8x³ + 3x² + 6x - 80. The curve rises and then falls, with both ends approaching negative infinity. The x-axis ranges from approximately -8 to 8, and the y-axis ranges from approximately -200 to 400. X-axis: x Y-axis: y Data: The function starts from negative infinity, increases to a local maximum, then decreases, passing through the x-axis multiple times, and finally approaches negative infinity as x goes to positive infinity. The graph shows local maxima and minima within the visible range. Key Values: x-intercepts around -1, 0, 1, 7, y-intercept at -80, local maximum around x=6, y=400, local minimum around x=-0.5, y=-80 Context: Visually represents the end behavior of a polynomial function with an even degree and negative leading coefficient. (Note: Some details are estimated) **GRAPH**: g(x) = x³ - 9x + 2 Description: A graph of the cubic polynomial function g(x) = x³ - 9x + 2. The curve starts from negative infinity, rises to a local maximum, falls to a local minimum, and then rises to positive infinity. X-axis: x Y-axis: y Data: The function approaches negative infinity as x approaches negative infinity, and approaches positive infinity as x approaches positive infinity. It has a local maximum around x=-2, y=12 and a local minimum around x=2, y=-12. It crosses the x-axis multiple times. Key Values: y-intercept at 2, local maximum around (-2, 12), local minimum around (2, -12) Context: Practice problem for identifying the end behavior of a cubic polynomial function with a positive leading coefficient. (Note: Some details are estimated) **GRAPH**: f(x) = -x³/4 + 3x²/4 - x/2 Description: A graph of the cubic polynomial function f(x) = -x³/4 + 3x²/4 - x/2. The curve starts from positive infinity, falls to a local minimum, rises to a local maximum, and then falls to negative infinity. X-axis: x Y-axis: y Data: The function approaches positive infinity as x approaches negative infinity, and approaches negative infinity as x approaches positive infinity. It has a local maximum around x=2, y=1 and a local minimum around x=0, y=0. It crosses the x-axis at the origin. Key Values: y-intercept at 0, local maximum around (2, 1), local minimum around (0, 0) Context: Practice problem for identifying the end behavior of a cubic polynomial function with a negative leading coefficient. (Note: Some details are estimated) **GRAPH**: f(x) = x / (x² - 2x + 8) Description: A graph of the rational function f(x) = x / (x² - 2x + 8). The curve approaches the x-axis (y=0) as x approaches both positive and negative infinity. It has a local maximum and minimum. X-axis: x Y-axis: y Data: The function approaches 0 as x approaches negative infinity, passes through the origin, rises to a local maximum, then falls through a local minimum, and approaches 0 as x approaches positive infinity. The x-axis acts as a horizontal asymptote. Key Values: x-intercept at 0, y-intercept at 0, local maximum around x=4, y=0.4, local minimum around x=-2, y=-0.2 Context: Visually represents the end behavior of a rational function where the degree of the denominator is greater than the degree of the numerator, leading to a horizontal asymptote at y=0. (Note: Some details are estimated)