Applications of derivatives

Calculus I. Also known as: math 151, math 2413, math 1550, math 1910, math 140, calc 1, calc i, calculus 1, calculus i, differential calculus, calculus for science and engineering, math 121

Use derivatives for related rates, optimisation, curve sketching, linear approximation and the mean value theorem.

Related rates

When two quantities are linked by an equation and both change with time, differentiate the equation with respect to t, using the chain rule, and solve for the rate you want. The routine: draw a picture, name the variables, write the equation that links them, differentiate, then substitute the numbers last. Substituting early freezes a quantity that is really changing.

Example: a spherical balloon has volume V = (4/3)pir^3. If the radius is 6 cm and grows at 0.5 cm/s, then dV/dt = 4pir^2 * dr/dt = 4pi360.5 = 72pi, about 226 cm^3/s.

Optimisation

To find a largest or smallest value:

  1. Write the quantity to optimise as a function of one variable, using the constraint to eliminate the other variables.
  2. State the allowed interval.
  3. Find critical points, where f'(x) = 0 or f'(x) does not exist.
  4. Compare f at the critical points and at the endpoints. The biggest is the absolute maximum, and the smallest is the absolute minimum.

Example: cut a square of side x from each corner of a 30 cm square sheet and fold up the sides to make an open box. The volume is V(x) = x(30 - 2x)^2 for 0 < x < 15. Then V'(x) = (30 - 2x)(30 - 6x), which is zero at x = 5 (x = 15 gives no box). Then V(5) = 5 * 20^2 = 2000 cm^3, which is the largest volume.

Curve sketching

The sign of f' shows where f rises or falls. A critical point is a local maximum if f' changes from positive to negative, and a local minimum if it changes from negative to positive. The second derivative f'' gives concavity: positive means concave up (a cup) and negative means concave down (a cap). An inflection point is where concavity changes.

For f(x) = x^3 - 3x: f'(x) = 3x^2 - 3 = 0 at x = -1 and x = 1; f'' = 6x is negative at x = -1 (local maximum) and positive at x = 1 (local minimum), and the inflection is at x = 0.

Linear approximation and the mean value theorem

Near a, f(x) is approximately f(a) + f'(a)(x - a). The mean value theorem says a differentiable function on [a, b] has a point c where f'(c) equals the average slope (f(b) - f(a))/(b - a).

Common mistakes

  • Plugging numbers in before differentiating in related rates.
  • Forgetting to check endpoints.
  • Mixing up the sign of f'' with the direction of the function.

Practice questions

  1. A circular oil slick has radius 10 cm, growing at 2 cm/s. How fast is its area growing at that moment?

  2. A 13 m ladder leans against a vertical wall. Its base is 5 m from the wall and slides away at 0.6 m/s. How is the top moving at that moment?

  3. A farmer has 120 m of fence for three sides of a rectangular pen; a river forms the fourth side. What is the largest possible area?

  4. What are the local extrema of f(x) = x^3 - 12x + 5?

  5. What is the absolute maximum of f(x) = x^3 - 3x^2 on the interval [0, 4]?

  6. For which x is the graph of f(x) = x^3 - 6x^2 + 1 concave up?