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Home » Calculate total time per segment with animation: - DNSFLEX

Calculate total time per segment with animation: - DNSFLEX

📅 March 11, 2026 👤 scraface

Mar 11, 2026

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📌 Expand: \( x^2 - 3x - 4 = 0 \).

📌 Factor: \( (x - 4)(x + 1) = 0 \), so \( x = 4 \) (since \( x > 3 \) for log to be defined).

📌 Find the derivative of \( y = \ln(x^2 + 1) \) at \( x = 1 \).

📌 Use chain rule: \( rac{dy}{dx} = rac{1}{x^2 + 1} \cdot 2x = rac{2x}{x^2 + 1} \).

📌 Evaluate at \( x = 1 \): \( rac{2(1)}{1^2 + 1} = rac{2}{2} = 1 \).

📌 The sum of an infinite geometric series is 12, and the first term is 3. Find the common ratio.

📌 Sum formula: \( S = rac{a}{1 - r} \), so \( 12 = rac{3}{1 - r} \).

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Find the minimum value of this expression for $ x \in (0, \frac{\pi}{2...
We expand both terms:
\sin^2 x + \cos^2 x + \sec^2 x + \csc^2 x + 2(\tan x + \cot x)
Use identity $ \sin^2 x + \cos^2 x = 1 $, and:
+ (1 + \tan^2 x) + (1 + \cot^2 x) + 2(\tan x + \cot x) = 3 + \tan^2 x...
Let $ t = \tan x + \cot x $. Then:
\tan^2 x + \cot^2 x = t^2 - 2
+ (t^2 - 2) + 2t = t^2 + 2t + 1 = (t + 1)^2
Now, $ t = \tan x + \cot x \geq 2 $ by AM-GM, with equality when $ \ta...
So minimum occurs at $ t = 2 $, giving:

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