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Applications Of Derivatives – Tangents and Normal

Telangana Open School (TOSS) · Class 12 · Mathematics

Complete topic list for Applications Of Derivatives – Tangents and Normal in Telangana Open School (TOSS) Class 12 Mathematics. Key concepts, sub-topics, and what to focus on for board exams.

42 questions23 flashcards4 concepts

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

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

Frequently Asked Questions

What are the important topics in Applications Of Derivatives – Tangents and Normal for Telangana Open School (TOSS) Class 12 Mathematics?
Applications Of Derivatives – Tangents and Normal covers several key topics that are frequently asked in Telangana Open School (TOSS) Class 12 board exams. Focus on the core concepts listed on this page and practise related questions to build confidence.
How to score full marks in Applications Of Derivatives – Tangents and Normal — Telangana Open School (TOSS) Class 12 Mathematics?
Understand the core concepts first, then work through the 42 practice questions available for this chapter. Revise formulas and 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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