Rates of Change and Monotonic Functions: Quick Revision

A brief note on Rates of Change and Monotonic Functions. Read it before a test. Try each question at the end before you open its answer.

Rates of change

• The derivative: $\dfrac{dy}{dx}$ is the rate of change of $y$ with respect to $x$. Its unit is the unit of $y$ per unit of $x$.
• Average and instant: the average rate is $\dfrac{\Delta y}{\Delta x}$. The derivative is its limit as $\Delta x \to 0$.
• Motion: velocity $v = \dfrac{ds}{dt}$ and acceleration $a = \dfrac{dv}{dt}$. The particle is at rest when $v = 0$.
• Marginal cost: $C'(x)$, the extra cost of one more item.

Related rates

• Method: draw and name the quantities. Write one equation true at every moment. Differentiate with respect to $t$. Only then put in the numbers.
• Chain rule: $\dfrac{d}{dt}f(r) = f'(r)\,\dfrac{dr}{dt}$.
• Common links: circle $A = \pi r^2$, sphere $V = \tfrac{4}{3}\pi r^3$ and $S = 4\pi r^2$, ladder $x^2 + y^2 = L^2$, cone with similar triangles.
• Sign: a quantity that shrinks has a negative rate.

Increasing and decreasing

• Test: $f' > 0$ on an interval means strictly increasing. $f' < 0$ means strictly decreasing.
• Method: solve $f'(x) = 0$, test the sign in each gap, write the intervals.
• Isolated zeros: $f' \ge 0$ with zeros at isolated points still gives a strictly increasing function, as with $x^3$.
• Roots: a strictly monotonic function takes each value at most once, so $f(x) = k$ has at most one root.

A worked example

A circular patch of oil on water has its radius growing at $2\ \text{cm/s}$. How fast does its area grow when $r = 9\ \text{cm}$?

1. The link. $A = \pi r^2$.
2. Differentiate. $\dfrac{dA}{dt} = 2\pi r\,\dfrac{dr}{dt}$.
3. Put in the numbers. $2\pi \times 9 \times 2 = 36\pi \approx 113.10\ \text{cm}^2/\text{s}$.

Practice with new numbers

A program to try

The program checks the main ideas on one function: the derivative at a point, whether the function always increases, and an average rate against the rate at an instant.

Mistakes to avoid

• Putting numbers in too early. Differentiate first. A fixed number has derivative $0$.
• Leaving out units or the sign of a rate.
• Testing the sign of $f$ instead of $f'$ to find where a function rises.
• Thinking $f'(c) = 0$ stops the rise. For $x^3$ it keeps rising through $x = 0$.

Check yourself

1. The side of a square grows at $3\ \text{cm/s}$. How fast does the area grow when the side is $8\ \text{cm}$?

2. A particle has $s = t^2 - 4t$ metres. When is it at rest?

3. On which interval is $f(x) = x^2 - 8x$ decreasing?

4. Is $f(x) = x^3 + x$ strictly increasing?

5. The cost of $x$ items is $C = 0.1x^2 + 4x$ dollars. Find the marginal cost at $x = 20$.

Sources

• NCERT, Mathematics Textbook for Class XII, Part I, Chapter 6 (Application of Derivatives).
• OpenStax, Calculus Volume 1, Section 3.4 (Derivatives as Rates of Change) and Section 4.1 (Related Rates).