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Applications Of Derivatives – Tangents and Normal — Syllabus

Telangana Open School (TOSS) · Class 12 · Mathematics

What Applications Of Derivatives – Tangents and Normal covers in Telangana Open School (TOSS) Class 12 Mathematics: 4 topics, for the 2026-27 session.

42 questions23 flashcards16 formulas & key relations4 concepts

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4 Topics · Telangana Open School (TOSS) Class 12 Mathematics · 2026-27

Topics in Applications Of Derivatives – Tangents and Normal

1

Slope of Tangent and Normal

  • The derivative \( \frac{dy}{dx} \) at a point \( (x_1, y_1) \) gives the slope of the tangent to the curve \( y = f(x) \) at that point.
  • The slope of the normal is the negative reciprocal of the tangent's slope: \( -\frac{1}{\frac{dy}{dx}} = -\frac{dx}{dy} \).
  • If the tangent is parallel to the x-axis, then \( \frac{dy}{dx} = 0 \).
2

Equations of Tangent and Normal

  • The equation of a line with slope \( m \) passing through \( (x_1, y_1) \) is \( y - y_1 = m(x - x_1) \).
  • For tangent: \( m = \frac{dy}{dx} \) at \( (x_1, y_1) \).
  • For normal: \( m = -\frac{1}{\frac{dy}{dx}} \) at \( (x_1, y_1) \).
3

Lengths of Tangent, Normal, Subtangent, and Subnormal

  • These lengths describe geometric properties of the curve at a point.
  • Subtangent and subnormal lie along the x-axis.
  • All formulas depend on \( y_1 \) and \( \frac{dy}{dx} \) at \( (x_1, y_1) \).
4

Rolle's and Lagrange's Mean Value Theorems

  • Rolle’s Theorem applies when \( f(a) = f(b) \) and guarantees a point where derivative is zero.
  • Lagrange’s Mean Value Theorem generalizes Rolle’s Theorem for any continuous and differentiable function.
  • Both require continuity on \( [a,b] \) and differentiability on \( (a,b) \).

Key Concepts

Central concept: Tangents and Normals to Curves

Slope of TangentSlope of NormalEquations of Tangent and NormalLengths: Tangent, Normal, Subtangent, SubnormalRolle's TheoremLagrange's Mean Value Theorem

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Frequently Asked Questions

What are the important topics in Applications Of Derivatives – Tangents and Normal for Telangana Open School (TOSS) Class 12 Mathematics?
Key topics in Applications Of Derivatives – Tangents and Normal include Slope of Tangent and Normal, Equations of Tangent and Normal, Lengths of Tangent, Normal, Subtangent, and Subnormal, Rolle's and Lagrange's Mean Value Theorems. Study these first, then practise questions on each for the Telangana Open School (TOSS) Class 12 board exam.
How should I revise Applications Of Derivatives – Tangents and Normal for the Telangana Open School (TOSS) Class 12 board exam?
Learn the core ideas first, then work through the 42 practice questions on Applications Of Derivatives – Tangents and Normal. Revise definitions regularly and use flashcards for quick recall before the exam.

Sources & Official References

Content is aligned to the official syllabus. Refer to the board website for the latest curriculum.

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All Telangana Open School (TOSS) Class 12 Mathematics Syllabus

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