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Home » Evaluate both functions at $ x = 2 $ and equate them: - Belip

Evaluate both functions at $ x = 2 $ and equate them: - Belip

📅 May 6, 2026 👤 scraface

May 06, 2026

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📌 \frac{J(310)}{J(300)} = 2 \implies \frac{a(310)^n}{a(300)^n} = 2 \implies \left(\frac{310}{300}\right)^n = 2

📌 بأخذ الطبيع والت LOG:

📌 n \ln\left(\frac{310}{300}\right) = \ln 2 \implies n = \frac{\ln 2}{\ln\left(\frac{31}{30}\right)} \approx \frac{0.6931}{0.0324} \approx 21.42

📌 إذن، القيمة التقريبية لـ \( n \) هي:

📌 \boxed{21.42}

📌 السؤال: يقوم مطور برمجيات في واشنطن العاصمة بتحليل وقت معالجة \( T \) (بالملي ثانية) لخوارزمية تُمَثّل بالدالة التربيعية \( T(n) = 2n^2 - 12n + 20 \)، حيث \( n \) هو حجم المدخلات. أوجد حجم المدخلات \( n \) التي يُقلّل وقت المعالجة إلى أدنى مستوى له.

📌 الحل: دالة الوقت \( T(n) = 2n^2 - 12n + 20 \) هي دالة تربيعية فتحتها لأعلى (بما أن معامل \( n^2 \) موجب). تحدث القيمة الدنيا عند رأس القطع المكافئ، المعطى بالعلاقة:

📌 n = -\frac{b}{2a} = -\frac{-12}{2 \times 2} = \frac{12}{4} = 3

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