Class 12 Notes - Application of Derivatives

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Application of Derivatives

The application of derivatives is a crucial topic in Class 12 mathematics. It involves using the concept of derivatives to solve real-world and mathematical problems such as finding the rate of change, maxima and minima, tangents and normals, and approximations.

1. Rate of Change

Derivatives are used to determine the rate at which one quantity changes with respect to another. For example, velocity is the rate of change of displacement with respect to time.

  • If \( y = f(x) \), then the rate of change of \( y \) with respect to \( x \) is \( \frac{dy}{dx} \).
  • Applications: Physics (velocity, acceleration), Economics (marginal cost, marginal revenue), Biology (population growth).

2. Increasing and Decreasing Functions

The derivative helps determine where a function is increasing or decreasing.

  • If \( f'(x) > 0 \) in an interval, \( f(x) \) is increasing there.
  • If \( f'(x) < 0 \) in an interval, \( f(x) \) is decreasing there.

3. Tangents and Normals

The derivative at a point gives the slope of the tangent to the curve at that point. The normal is perpendicular to the tangent.

  • Equation of tangent at \( x = a \): \( y - f(a) = f'(a)(x - a) \)
  • Equation of normal at \( x = a \): \( y - f(a) = -\frac{1}{f'(a)}(x - a) \) (if \( f'(a) \neq 0 \))

4. Approximations

Derivatives are used to approximate small changes in functions using differentials.

  • If \( y = f(x) \), then a small change \( \Delta y \approx f'(x) \Delta x \).

5. Maxima and Minima

Derivatives help find the local maximum and minimum values of a function.

  • Critical points occur where \( f'(x) = 0 \) or \( f'(x) \) does not exist.
  • Second derivative test: If \( f''(x) > 0 \), it is a local minimum; if \( f''(x) < 0 \), it is a local maximum.

6. Practical Problems (Word Problems)

  • Finding dimensions that maximize area or volume.
  • Minimizing cost, distance, or time in real-life scenarios.
  • Optimization problems in economics, engineering, and science.

Example Problems

  1. Find the equation of the tangent to the curve \( y = x^2 \) at \( x = 1 \).
    Solution: \( f(x) = x^2 \), \( f'(x) = 2x \), so at \( x = 1 \), slope = 2.
    Equation: \( y - 1 = 2(x - 1) \) or \( y = 2x - 1 \).
  2. Find the maximum value of \( f(x) = -x^2 + 4x + 5 \).
    Solution: \( f'(x) = -2x + 4 \). Set \( f'(x) = 0 \) ⇒ \( x = 2 \).
    \( f''(x) = -2 < 0 \), so maximum at \( x = 2 \).
    Maximum value = \( f(2) = -4 + 8 + 5 = 9 \).

Summary

  • Derivatives are used to find rates of change, tangents, normals, maxima, minima, and for approximations.
  • They have wide applications in science, engineering, and economics.
  • Understanding the application of derivatives is essential for solving real-world optimization problems.