Unit 4: Contextual Applications of Differentiation

This unit translates derivative ideas into real-world language. The math is usually not harder than earlier differentiation, but the interpretation must be precise.

Rates of change in context

If \(Q(t)\) is a quantity depending on time, then:

  • \(Q'(t)\) is the instantaneous rate of change of \(Q\),
  • units of \(Q'(t)\) are units of \(Q\) per unit of \(t\).

Always interpret both sign and units.

Average vs instantaneous rate

Average rate on \([a,b]\):

\[\frac{Q(b)-Q(a)}{b-a}\]

Instantaneous rate at \(t=a\):

\[Q'(a).\]

Position, velocity, acceleration

If position is \(s(t)\), then

\[v(t) = s'(t), \qquad a(t) = v'(t) = s''(t).\]

Interpret carefully:

  • \(v(t) > 0\) means motion in the positive direction,
  • \(v(t) < 0\) means motion in the negative direction,
  • speed is \(\lvert v(t) \rvert\).

Speed increasing:

  • \(v(t) > 0\) and \(a(t) > 0\), or
  • \(v(t) < 0\) and \(a(t) < 0\).

Rate in, rate out, and accumulation

If a quantity changes because something enters and leaves, then:

\[\text{net change rate} = \text{rate in} - \text{rate out}.\]

If \(R(t)\) is the rate entering a tank and \(L(t)\) is the rate leaving, then:

\[V'(t) = R(t) - L(t).\]

Interpreting graphs in context

Given a graph of a function:

  • slope tells rate of change,
  • steep positive slope means rapid increase,
  • slope near zero means little short-term change,
  • concavity tells whether the rate itself is increasing or decreasing.

Given a graph of a derivative:

  • positive derivative means original function is increasing,
  • negative derivative means decreasing,
  • derivative crossing zero may indicate an extremum in the original function.

[Image Placeholder: contextual graph with slope interpretation at several labeled points]

Related rates problems are mostly about translation. The key source equations usually come from:

  • Pythagorean theorem,
  • volume formulas,
  • area formulas,
  • similar triangles.

If the problem asks how fast a quantity is changing, the final answer should usually be a value of a derivative with units.

Linearization and differentials

Near \(x=a\),

\[L(x) = f(a) + f'(a)(x-a)\]

approximates \(f(x)\).

Differentials use the same idea:

\[dy = f'(x)\,dx.\]

If a measured input has small error \(dx\), then the output error is approximately \(dy\).

Marginal analysis

In economics-flavored problems:

  • cost function \(C(x)\),
  • revenue function \(R(x)\),
  • profit \(P(x) = R(x)-C(x)\).

Then:

  • marginal cost is \(C'(x)\),
  • marginal revenue is \(R'(x)\),
  • marginal profit is \(P'(x)\).

At large production levels, these are interpreted as approximate change from one additional unit.

Common contextual verbs

  • increasing means derivative positive,
  • decreasing means derivative negative,
  • at what rate means derivative value,
  • how fast often means magnitude, but read carefully,
  • changing more rapidly compares derivative magnitudes or second derivatives depending on context.

Common mistakes

  • Reporting velocity when the question asks for speed.
  • Giving a derivative without units.
  • Using the wrong variable as the independent variable.
  • Forgetting to evaluate at the specified time or input.
Made by Joshua Bie