Chapters

  • Chapter 6
  • Calculus
  • 23 questions

Application of Derivatives

Rate of change, increasing and decreasing functions, and maxima and minima.

A derivative measures how one quantity responds to a change in another. Once that is understood, three questions follow naturally: how fast is something changing, is it rising or falling, and where does it reach its largest or smallest value?

The optimisation problems at the end of this chapter are where the mathematics meets a real situation — the largest volume for a given sheet of metal, the least cost for a required capacity — and they are among the most reliably examined applications in the course.

What you should be able to do

  • Use derivatives to find the rate of change of one quantity with respect to another.
  • Determine the intervals on which a function is strictly increasing or strictly decreasing.
  • Find local maxima and minima using the first and second derivative tests.
  • Find the absolute maximum and minimum of a function on a closed interval.
  • Formulate and solve applied optimisation problems.

Topics

3 topics

  1. Rate of Change of Quantities

    Derivatives as rates, including related rates in geometric and physical contexts.

  2. Increasing and Decreasing Functions

    Using the sign of the first derivative to find intervals of monotonicity.

  3. Maxima and Minima

    First and second derivative tests, absolute extrema on a closed interval, and applied optimisation.

    • First derivative test
    • Second derivative test
    • Absolute maximum and minimum
    • Applied optimisation problems

Formulas

9 items · All formulas

  • Rate of change

    \dfrac{dy}{dx} \text{ is the rate of change of } y \text{ with respect to } x

    For related rates, use the chain rule: \dfrac{dA}{dt} = \dfrac{dA}{dr}\cdot\dfrac{dr}{dt}.

  • Strictly increasing function

    f'(x) > 0 \quad \text{for all } x \text{ in the interval}

  • Strictly decreasing function

    f'(x) < 0 \quad \text{for all } x \text{ in the interval}

  • Critical points

    f'(c) = 0 \quad \text{or} \quad f'(c) \text{ does not exist}

    The only candidates for a local extremum in the interior.

  • Second derivative test

    f'(c) = 0 \text{ and } \begin{cases} f''(c) < 0 & \Rightarrow \text{local maximum} \\ f''(c) > 0 & \Rightarrow \text{local minimum} \end{cases}
    Valid when
    Inconclusive when f''(c) = 0 — use the first derivative test instead.

  • Absolute extrema on a closed interval

    \max/\min \text{ of } \{\, f(a),\; f(b),\; f(c_i) \,\}

    Compare the values at the endpoints and at every critical point c_i in [a,b].

  • Volume and surface area of a sphere

    V = \dfrac{4}{3}\pi r^{3}, \qquad S = 4\pi r^{2}

    The most frequently used solid in rate-of-change questions.

  • Volume and curved surface area of a cylinder

    V = \pi r^{2} h, \qquad \text{CSA} = 2\pi r h, \qquad \text{TSA} = 2\pi r(r + h)

  • Volume and curved surface area of a cone

    V = \dfrac{1}{3}\pi r^{2} h, \qquad \text{CSA} = \pi r l, \qquad l = \sqrt{r^{2} + h^{2}}

Key points

7 items · All key points

  • f'(c) = 0 makes c a candidate, not automatically an extremum — it may be a point of inflection.

    f(x) = x^{3} has f'(0) = 0, yet x = 0 is neither a maximum nor a minimum.

  • For absolute extrema on a closed interval, always evaluate at the endpoints as well.

  • If f''(c) = 0, the second derivative test says nothing — switch to the first derivative test.

  • Reduce an optimisation problem to a single variable using the constraint before differentiating.

  • State the domain of the variable in an applied problem — lengths and radii are positive.

  • In related-rates questions, differentiate with respect to time and use the chain rule throughout.

  • 'Strictly increasing' needs f'(x) > 0; f'(x) \geq 0 only gives 'increasing' (non-decreasing).

Common mistakes

5 items · All common mistakes

  • Mistake

    Reporting a critical point as the answer when the question asked for the maximum or minimum value.

    Instead

    Substitute the critical point back into the original function to obtain the value.

    Why

    The hard work ends at solving f'(x)=0, and it feels like the answer.

  • Mistake

    Omitting endpoint values when finding absolute extrema on a closed interval.

    Instead

    Compare f at both endpoints and at every critical point inside the interval.

    Why

    The interior critical points are what the calculus produces, so the endpoints are forgotten.

  • Mistake

    Concluding that f'(c) = 0 guarantees a maximum or minimum at c.

    Instead

    Classify it. f(x) = x^3 at x = 0 has zero derivative but no extremum.

  • Mistake

    Differentiating an optimisation formula that still contains two variables.

    Instead

    Use the constraint to eliminate one variable first, then differentiate.

    Why

    The formula for the quantity is written down first, and the constraint is treated as a separate afterthought.

  • Mistake

    In related rates, substituting numerical values before differentiating.

    Instead

    Differentiate the general relation with respect to t first, then substitute the instantaneous values.

    Why

    Substituting early turns a variable into a constant, and its derivative wrongly becomes zero.

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