Applications of differentiation

The first derivative gives slope and the second describes how that slope changes. Stationary points satisfy f′(x)=0, but their nature needs a sign change or another test. In optimisation, reduce the quantity to one variable, preserve the physical domain, and compare candidates with boundaries. Connected rates differentiate a relation with respect to time.

Increasing versus strictly increasing

On an interval I, for every $a<b$, increasing means $f(a)\le f(b)$ and strictly increasing means $f(a)<f(b)$. Decreasing and strictly decreasing reverse those inequalities to $\ge$ and $>$.

If f is differentiable and $f^{\prime}(x)>0$ throughout I, f is strictly increasing there; a negative derivative throughout gives strict decrease. The source curve increases on $(a,b)$ and $(c,d)$ and decreases on $(b,c)$. These are sufficient tests: strict increase can still allow an isolated zero derivative, as with $x^3$ at zero.

Why applications need a model

The source introduces uses in physics (motion, electricity, heat, light, harmonics, acoustics, astronomy and dynamics), chemistry (reaction rates and radioactive decay), biology (birth/death rates), economics (marginal cost/revenue and profit), engineering, statistics and medicine.

Its supermarket example asks how to choose the height-to-diameter ratio of a cylindrical can to minimise manufacturing surface area at fixed volume.

The original can photograph is retained in the source archive, with its printed DataGenetics attribution; the linked external page has not been reviewed.

The full chapter outline also includes interpreting first/second derivative signs, relating a function’s graph to its derivative’s graph, tangents/normals including implicit curves, calculator extrema, analytical classification, optimisation and connected rates. The complete lecture now connects these ideas through sixteen examples, three self-practices, source diagrams and the linked Discussion/Review questions.