Example 33 - Chapter 5 Class 12 Continuity and Differentiability

Transcript

Example 33 Find 𝑑𝑦/𝑑𝑥 , if 𝑥 = 𝑎 (𝜃+sin⁡𝜃), 𝑦 = 𝑎 (1 – cos⁡𝜃)Here 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝜃)/(𝑑𝑥/𝑑𝜃) Calculating 𝒅𝒚/𝒅𝜽 𝑦 = 𝑎 (1 – cos⁡𝜃) 𝑑𝑦/𝑑𝜃 = 𝑑(𝑎 (1 – cos⁡𝜃))/𝑑𝜃 𝑑𝑦/𝑑𝜃 = 𝑎 (0 −(−sin⁡𝜃 )) 𝑑𝑦/𝑑𝜃 = 𝒂 (𝒔𝒊𝒏⁡𝜽 ) Calculating 𝒅𝒙/𝒅𝜽 𝑥 = 𝑎 (𝜃+𝑠𝑖𝑛⁡𝜃) 𝑑𝑥/𝑑𝜃 = 𝑑(𝑎(𝜃 + 𝑠𝑖𝑛⁡𝜃))/𝑑𝜃 𝑑𝑥/𝑑𝜃 = 𝑎 (𝑑𝜃/𝑑𝜃+𝑑(sin⁡𝜃 )/(𝑑𝜃 )) 𝑑𝑥/𝑑𝜃 = 𝒂 (𝟏+𝐜𝐨𝐬⁡𝜽 ) Therefore 𝑑𝑦/𝑑𝑥 = (𝑑𝑦/𝑑𝜃)/(𝑑𝑥/𝑑𝜃) 𝑑𝑦/𝑑𝑥 = 𝑎" " (sin⁡𝜃 )/𝑎" " (1 +〖 cos〗⁡𝜃 ) 𝑑𝑦/𝑑𝑥 = sin⁡𝜃/(1 +〖 cos〗⁡𝜃 ) 𝑑𝑦/𝑑𝑥 = (〖𝟐 𝒔𝒊𝒏〗⁡〖 𝜽/𝟐〗 .〖 𝒄𝒐𝒔〗⁡〖 𝜽/𝟐〗)/(1 + 〖𝟐 〖𝒄𝒐𝒔〗^𝟐 〗⁡〖𝜽/𝟐〗 − 𝟏) 𝑑𝑦/𝑑𝑥 = (sin⁡〖 𝜃/2〗 )/〖cos 〗⁡〖𝜃/2〗 𝒅𝒚/𝒅𝒙 = 𝐭𝐚𝐧⁡〖𝜽/𝟐〗 Rough We know that sin 2θ = 2 sin θ cos θ Replacing θ by 𝜃/2 sin θ = 2 𝒔𝒊𝒏⁡〖𝜽/𝟐〗 𝒄𝒐𝒔⁡〖𝜽/𝟐〗 and cos 2θ = 2cos2 θ – 1 Replacing θ by 𝜃/2 cos θ = 2cos2 𝜽/𝟐 – 1