Ex 6.1,13 - Chapter 6 Class 12 Application of Derivatives
Last updated at Dec. 16, 2024 by Teachoo
Finding rate of change
Example 1
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Chapter 6 Class 12 Application of Derivatives
Concept wise
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Finding rate of change
- To show increasing/decreasing in whole domain
- To show increasing/decreasing in intervals
- Find intervals of increasing/decreasing
- Finding slope of tangent/normal
- Finding point when tangent is parallel/ perpendicular
- Finding equation of tangent/normal when point and curve is given
- Finding equation of tangent/normal when slope and curve are given
- Finding approximate value of numbers
- Finding approximate value of function
- Finding approximate value- Statement questions
- Finding minimum and maximum values from graph
- Local maxima and minima
- Minima/ maxima (statement questions) - Number questions
- Minima/ maxima (statement questions) - Geometry questions
- Absolute minima/maxima
Transcript
Ex 6.1, 13 A balloon, which always remains spherical, has a variable diameter 3/2 (2𝑥 +1). Find the rate of change of its volume with respect to 𝑥.Let d be the diameter of the balloon Given that Diameter = d = 3/2 (2x + 1) Let r be the radius of the balloon r = 𝑑/2 = 𝟑/𝟒 (2x + 1) The balloon is a spherical Volume of the balloon = 4/3 𝜋𝑟^3 We need to find rate of change of volume with respect to x i.e. 𝑑𝑉/𝑑𝑥 Now, 𝑑𝑉/𝑑𝑥 = 𝑑/𝑑𝑥 (4/3 𝜋𝑟^3 ) = 4𝜋/3 × (𝑑𝑟^3)/𝑑𝑥 = 4𝜋/3 × 𝑑/𝑑𝑥 (27/64 (2𝑥+1)^3 ) = 9𝜋/16 × (𝑑(2𝑥 + 1)^3)/𝑑𝑥 = 9𝜋/16 × 3(2x + 1)2 × 2 = 𝟐𝟕𝝅/𝟖 (𝟐𝒙+𝟏)^𝟐
