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Maharashtra State Board · HSC Commerce · Class 12 (Std XII)

Class 12 Maths and Statistics Part 1 Chapter 4: Applications of Derivatives

Short notes, key concepts, quick revision and practice MCQs with answers

Derivatives describe slopes and rates. The chapter finds equations of tangents and normals, tests whether functions are increasing or decreasing, locates maxima and minima, and applies derivatives to economics: demand, supply, cost, revenue, profit, elasticity and the propensities to consume and save.

Key concepts: Applications of Derivatives

Tangent and normal
The slope of the tangent to y = f(x) at P(a, b) is f′(a). Tangent: y − b = f′(a)(x − a). Normal: y − b = (−1/f′(a))(x − a), f′(a) ≠ 0.
Increasing and decreasing functions
f is increasing at c if f′(c) > 0 and decreasing at c if f′(c) < 0.
Maxima and minima
If f′(c) = 0 and f″(c) < 0, f has a local maximum at c; if f′(c) = 0 and f″(c) > 0, a local minimum. Points where f′(x) = 0 are stationary points.
Cost and revenue
Total cost C = f(x); average cost = C/x; marginal cost = dC/dx. Revenue R = P × D; average revenue RA = R/D = P; marginal revenue Rm = dR/dD.
Profit
Profit π = R − C. It is maximum where dπ/dx = 0 and d²π/dx² < 0.
Elasticity of demand
η = −(P/D)(dD/dP). Demand is elastic if η > 1, unitary if η = 1 and inelastic if η < 1.
Revenue and elasticity
Rm = P(1 − 1/η) = RA(1 − 1/η). When η = 1, Rm = 0.
Consumption and saving
For income x = Ec + S: MPC = dEc/dx, MPS = dS/dx, APC = Ec/x, APS = S/x. MPC + MPS = 1 and APC + APS = 1.

Quick revision: important points

Practice MCQs with answers

  1. The slope of the tangent to y = x² + 4x + 1 at x = −1 is:

    1. 0
    2. 2
    3. 4
    4. −2
    Show answer

    Answer: B. 2
    dy/dx = 2x + 4 = 2 at x = −1.

  2. The equation of the tangent to y = f(x) at P(a, b) is:

    1. y − b = f(a)(x − a)
    2. y − b = f′(a)(x − a)
    3. y − a = f′(b)(x − b)
    4. x − a = f′(a)(y − b)
    Show answer

    Answer: B. y − b = f′(a)(x − a)
    Slope-point form with slope f′(a).

  3. The slope of the normal to y = f(x) at x = a is:

    1. f′(a)
    2. −f′(a)
    3. −1/f′(a)
    4. 1/f′(a)
    Show answer

    Answer: C. −1/f′(a)
    The normal is perpendicular to the tangent.

  4. A function f is increasing at x = c if:

    1. f′(c) < 0
    2. f′(c) > 0
    3. f′(c) = 0
    4. f(c) > 0
    Show answer

    Answer: B. f′(c) > 0
    A positive derivative means rising.

  5. f(x) = x³ + 3x is:

    1. Increasing for all real x
    2. Decreasing for all real x
    3. Increasing only for x > 0
    4. Constant
    Show answer

    Answer: A. Increasing for all real x
    f′(x) = 3x² + 3 > 0 always.

All Class 12 Maths and Statistics chapters

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