Applications: related rates and optimization

Derivatives earn their place in engineering by answering two kinds of question. The first is "how fast?": if one quantity is changing at a known rate, how fast is a connected quantity changing? The second is "what is best?": which dimensions use the least material, which load gives the most power, which operating point costs the least?

This lesson covers both, plus three tools that build on them: linear approximation for estimating errors, Newton's method for solving equations numerically, and L'Hôpital's rule for limits that resist algebra. It is a long lesson with many worked examples, because these methods are learned by doing.

Related rates

In a related-rates problem, several quantities change with time and are linked by an equation. You know the rate of some of them and need the rate of another. Because they are linked, their rates are linked too, and the chain rule connects them.

A reliable method works for almost every problem:

• Draw a diagram and label every quantity that changes with a variable, not a number.
• Write down the rate you know and the rate you want, as derivatives with respect to time.
• Find an equation linking the variables (geometry, a physical law, or a model).
• Differentiate both sides of that equation with respect to time, using the chain rule.
• Only now substitute the numbers for the particular instant, and solve for the unknown rate.

The order of the last two steps matters. If you substitute numbers before differentiating, the quantities that are changing become constants, and their derivatives vanish incorrectly.

Notice that \(dy/dt = -(x/y)(dx/dt)\) grows as \(y\) shrinks. Near the bottom of its fall the top of the ladder moves very fast, which is one reason ladders that start to slip are so dangerous.

Maximum and minimum values

Optimization means finding the largest or smallest value of a function. Two kinds of extreme value matter:

• An absolute (global) maximum or minimum is the largest or smallest value over the whole interval being considered.
• A local maximum or minimum is the largest or smallest value compared with nearby points only: the top of a hill or the bottom of a valley.

The Extreme Value Theorem guarantees that a continuous function on a closed interval \([a, b]\) has both an absolute maximum and an absolute minimum. They can only occur at two kinds of place:

• Critical points: interior points where \(f'(x) = 0\) or \(f'(x)\) does not exist.
• Endpoints of the interval.

The closed-interval method

To find the absolute extremes of a continuous function on \([a, b]\): find the critical points inside the interval, evaluate \(f\) at each critical point and at both endpoints, and pick the largest and smallest values. No derivative tests are needed, just a comparison.

Classifying critical points

On an open interval, or when you need to know what kind of point you have found, use a derivative test.

• First derivative test: if \(f'\) changes from positive to negative at a critical point, it is a local maximum. If it changes from negative to positive, it is a local minimum. If it does not change sign, it is neither.
• Second derivative test: at a critical point where \(f'(c) = 0\), if \(f''(c) < 0\) it is a local maximum (concave down, like a cap); if \(f''(c) > 0\) it is a local minimum (concave up, like a cup). If \(f''(c) = 0\), the test tells you nothing; use the first derivative test.

For the function above, \(f''(x) = 6x - 6\). At \(x = 0\), \(f''(0) = -6 < 0\): a local maximum. At \(x = 2\), \(f''(2) = 6 > 0\): a local minimum.

A point where the concavity changes is an inflection point. Here \(f''\) changes sign at \(x = 1\), so (1, −1) is an inflection point. In beam design, inflection points mark where the bending moment changes sign, which matters for where reinforcement is placed in concrete beams.

Optimization problems

Applied optimization problems are word problems. The mathematics is usually straightforward; the skill is in setting the problem up. Use these steps:

• Read the problem and identify the quantity to be maximized or minimized (the objective).
• Draw a diagram and name the variables.
• Write the objective as a formula. If it has more than one variable, use the constraint (a fixed perimeter, volume or budget) to eliminate all but one.
• State the domain: the range of values that make physical sense.
• Find the critical points, then compare them with the endpoints (or use a derivative test) to confirm which is the optimum.
• Answer the question asked, with units, and check that the answer makes sense.

Real cans are usually taller than this. The ideal shape ignores the cost of the seams and the extra-thick ends, and designers trade a little material for a can that fits the hand and the shelf. Optimization gives a starting point; engineering judgement finishes the design.

Linear approximation and error propagation

In Lesson 2 you met the linear approximation \(f(x) \approx f(a) + f'(a)(x - a)\). Writing the small change in input as \(dx\) and the resulting change in output as \(dy\) gives the differential form:

This is how engineers estimate how measurement errors carry through a calculation. If you measure \(x\) with an uncertainty \(dx\), the calculated value \(y = f(x)\) has an uncertainty of about \(|f'(x)|\,dx\).

The pattern in that example is general: for \(y = kx^n\), the relative error in \(y\) is about \(n\) times the relative error in \(x\). Cubing a measurement triples its percentage error; taking a square root halves it. This tells you which measurements in an experiment need the most care.

Newton's method

Many engineering equations cannot be solved with algebra: the friction factor in pipe flow, the depth of flow in a channel, the angle of a mechanism. Newton's method finds a numerical solution of \(f(x) = 0\) by following tangent lines.

Start with a guess \(x_0\). The tangent line at \(x_0\) crosses the x-axis at a point that is usually much closer to the root. Setting the tangent line \(y = f(x_0) + f'(x_0)(x - x_0)\) to zero gives the next estimate, and the process repeats:

Newton's method is fast when it works, but it can fail. If \(f'(x_n)\) is zero or very small, the tangent is nearly flat and the next estimate shoots far away. A poor starting guess can lead to a different root, or none. In practice, engineers sketch the function first, start close to the root, and fall back on bisection if the estimates misbehave.

L'Hôpital's rule

Derivatives also give a powerful tool for indeterminate limits. If \(\lim_{x \to a} f(x)/g(x)\) gives \(0/0\) or \(\infty/\infty\), then

provided the limit on the right exists. Note that you differentiate the top and bottom separately. This is not the quotient rule.

The third result says that for small angles, \(\cos x \approx 1 - \tfrac12 x^2\). That is the second-order small-angle approximation, used when the first-order version \(\cos x \approx 1\) is too crude.

Always check that the form really is \(0/0\) or \(\infty/\infty\) before applying the rule. Using it on a limit like \(\lim_{x \to 0} \dfrac{x + 1}{x + 2}\), which is simply 1/2 by substitution, gives a wrong answer.