Check Full Chapter Explained - Continuity and Differentiability - Continuity and Differentiability Class 12

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  1. Chapter 5 Class 12 Continuity and Differentiability
  2. Serial order wise

Transcript

Example 36 Find 𝑑𝑦/𝑑π‘₯ , if π‘₯ = π‘Ž (πœƒ+sinβ‘πœƒ), 𝑦 = π‘Ž (1 – cosβ‘πœƒ) 𝑑𝑦/𝑑π‘₯ = 𝑑𝑦/𝑑π‘₯ Γ— π‘‘πœƒ/π‘‘πœƒ 𝑑𝑦/𝑑π‘₯ = 𝑑𝑦/π‘‘πœƒ Γ— π‘‘πœƒ/𝑑π‘₯ 𝑑𝑦/𝑑π‘₯ = (𝑑𝑦/π‘‘πœƒ)/(𝑑π‘₯/π‘‘πœƒ) Calculating π’…π’š/π’…πœ½ 𝑦 = π‘Ž (1 – cosβ‘πœƒ) 𝑑𝑦/π‘‘πœƒ = 𝑑(π‘Ž (1 – cosβ‘πœƒ))/π‘‘πœƒ 𝑑𝑦/π‘‘πœƒ = π‘Ž (0 βˆ’(βˆ’sinβ‘πœƒ )) 𝑑𝑦/π‘‘πœƒ = π‘Ž (sinβ‘πœƒ ) Calculating 𝒅𝒙/π’…πœ½ π‘₯ = π‘Ž (πœƒ+π‘ π‘–π‘›β‘πœƒ) 𝑑π‘₯/π‘‘πœƒ = 𝑑(π‘Ž(πœƒ + π‘ π‘–π‘›β‘πœƒ))/π‘‘πœƒ 𝑑π‘₯/π‘‘πœƒ = π‘Ž (π‘‘πœƒ/π‘‘πœƒ+𝑑(sinβ‘πœƒ )/(π‘‘πœƒ )) 𝑑π‘₯/π‘‘πœƒ = π‘Ž (1+cosβ‘πœƒ ) Therefore 𝑑𝑦/𝑑π‘₯ = (𝑑𝑦/π‘‘πœƒ)/(𝑑π‘₯/π‘‘πœƒ) 𝑑𝑦/𝑑π‘₯ = π‘Ž" " (sinβ‘πœƒ )/π‘Ž" " (1 +γ€– cosγ€—β‘πœƒ ) 𝑑𝑦/𝑑π‘₯ = sinβ‘πœƒ/(1 +γ€– cosγ€—β‘πœƒ ) 𝑑𝑦/𝑑π‘₯ = (γ€–πŸ π’”π’Šπ’γ€—β‘γ€– 𝜽/πŸγ€— .γ€– 𝒄𝒐𝒔〗⁑〖 𝜽/πŸγ€—)/(1 + γ€–πŸ 〖𝒄𝒐𝒔〗^𝟐 γ€—β‘γ€–πœ½/πŸγ€— βˆ’ 𝟏) 𝑑𝑦/𝑑π‘₯ = (γ€–2 sin〗⁑〖 πœƒ/2γ€— .γ€– cos〗⁑〖 πœƒ/2γ€—)/γ€–2 cos^2 γ€—β‘γ€–πœƒ/2γ€— 𝑑𝑦/𝑑π‘₯ = (sin⁑〖 πœƒ/2γ€— )/γ€–cos γ€—β‘γ€–πœƒ/2γ€— π’…π’š/𝒅𝒙 = π­πšπ§β‘γ€–πœ½/πŸγ€— We know that sin 2ΞΈ = 2 sin ΞΈ cos ΞΈ Replacing ΞΈ by πœƒ/2 sin ΞΈ = 2 π’”π’Šπ’β‘γ€–πœ½/πŸγ€— π’„π’π’”β‘γ€–πœ½/πŸγ€— and cos 2ΞΈ = 2cos2 ΞΈ – 1 Replacing ΞΈ by πœƒ/2 cos ΞΈ = 2cos2 𝜽/𝟐 – 1

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 9 years. He provides courses for Maths and Science at Teachoo.