Problems on Radius of Curvature Problem No 10 - Application of Derivatives - Diploma Maths - II | Summary and Q&A

TL;DR
Find the radius of curvature of the curve y = log(sin x) at x = pi/2, which is equal to -1.
Key Insights
- 🎮 In this video, the problem of finding the radius of curvature is solved for a given curve.
- 📏 The chain rule is used to find the derivative of the function.
- ❓ The second derivative is calculated by differentiating the first derivative.
- ❓ The formula for radius of curvature involves the first and second derivatives of the function.
- 💠 The radius of curvature is found to be -1 unit, indicating a concave shape.
- ☺️ The value of the first derivative at x = pi/2 is 0, indicating a horizontal tangent.
- ☺️ The value of the second derivative at x = pi/2 is -1, showing concavity.
Transcript
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Questions & Answers
Q: What is the given curve and what is required to be found?
The given curve is y = log(sin x), and we need to find the radius of curvature at x = pi/2.
Q: How is the derivative of the given function calculated?
The derivative of y = log(sin x) is found using the chain rule, giving us dy/dx = cot x.
Q: What is the second derivative of the function?
The second derivative of y = log(sin x) is d^2y/dx^2 = -cosec^2 x.
Q: How is the radius of curvature calculated?
The formula for the radius of curvature is (1 + (dy/dx)^2)^(3/2) / |d^2y/dx^2|. By substituting the previously found values, the radius of curvature is determined to be -1 unit.
Summary & Key Takeaways
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The problem involves finding the radius of curvature of a curve represented by y = log(sin x) at x = pi/2.
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The derivative of the function is calculated using the chain rule, resulting in dy/dx = cot x.
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The second derivative of the function is found to be d^2y/dx^2 = -cosec^2 x.
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The formula for the radius of curvature is applied, resulting in a radius of curvature of -1 unit.
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