Application of Derivatives Notes: Class 12 Mathematics Chapter 6
Class 12 Mathematics Chapter 6 Application of Derivatives covers the rate of change of quantities, increasing and decreasing functions, tangents and normals to curves, and finding maxima and minima using derivatives.
Overview of Topics & Subtopics (Click to Jump)
1. Rate of Change and Monotonicity
Short Note:
• Derivatives measure the instantaneous rate of change of one quantity with respect to another, useful in problems involving a changing radius, volume, or other physical quantities over time.
• A function is increasing on an interval where f'(x) ≥ 0 throughout, and decreasing on an interval where f'(x) ≤ 0 throughout.
2. Tangents and Normals
Short Note:
• The slope of the tangent to a curve y = f(x) at a point (x1, y1) is given by f'(x1), and the tangent line's equation is y - y1 = f'(x1)(x - x1).
• The normal to the curve at the same point is perpendicular to the tangent, with slope -1/f'(x1) (when f'(x1) ≠ 0).
3. Maxima, Minima and Optimization
Short Note:
• Critical points, where f'(x) = 0 or f'(x) is undefined, are candidates for local maxima or minima and are analyzed using the first or second derivative test.
• Optimization problems (maximizing area, minimizing cost, etc.) are solved by expressing the quantity as a function of a single variable and applying derivative tests to find the optimal value.
NCERT Textbook Exercise Questions
How do you determine whether a function is increasing or decreasing on a given interval?
Answer: A function f(x) is said to be increasing on an interval if its derivative f'(x) is greater than or equal to zero throughout that interval, meaning the function's values do not decrease as x increases. Conversely, the function is decreasing on an interval if f'(x) is less than or equal to zero throughout, indicating the function's values do not increase as x increases.
Find the equation of the tangent to the curve y = x² at the point (2, 4).
Answer: First, differentiate y = x² to get dy/dx = 2x, so the slope of the tangent at x = 2 is 2(2) = 4. Using the point-slope form of a line, the equation of the tangent is y - 4 = 4(x - 2), which simplifies to y = 4x - 4.
Explain the second derivative test for finding local maxima and minima.
Answer: After finding the critical points where f'(x) = 0, the second derivative test evaluates f''(x) at each of these points. If f''(x) is positive, the critical point corresponds to a local minimum; if f''(x) is negative, it corresponds to a local maximum; and if f''(x) equals zero, the test is inconclusive and a different method must be used.
How are derivatives used to solve related rates problems?
Answer: In related rates problems, two or more quantities are connected by an equation, and both quantities are changing with respect to time. By differentiating the connecting equation implicitly with respect to time, a relationship between the rates of change (derivatives) of the quantities is obtained, which can then be solved for the unknown rate using the given values at a specific instant.
