# Applications of Differentiation

Differentiation in calculus is a key concept used to determine instantaneous rates of change. Instantaneous rate of change is the measure of how quickly the new value is affecting the old value. Differentiation gives us the ability to isolate and measure values between two variables instead of just their sum or average.

##### 2020 NYJC CT2 P2 Q3

The diagram below shows a city map of two towns, $A$ and $B$ separated by a river. A bridge is to be built between the two towns, which are on opposite sides of a straight river of uniform width $r$km, and the two towns are $p$ km apart measured along the riverbank. Town $A$ is $1$km from the riverbank, and Town $B$ is $b$km away from riverbank.

A bridge is to be built perpendicular to the riverbank at a distance of $x$km from Town $B$, measured along the riverbank, allowing traffic to flow between the two towns.

Find the distance $x$, in terms of $b$ and $p$, such that the distance of travel between Town $A$ and Town $B$ can be minimised if $b>1$. (It is not necessary to verify that the distance is minimum.)

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##### 2019 DHS Promo Q12

[A circular cone with base radius $r$, vertical height $h$ and slant height $l$, has curved surfaced area $\pi rl$ and volume $\frac{1}{3}\pi r{{h}^{2}}$.]

A capsule made of metal sheet of fixed volume $p$cm$^{3}$ is made up of three parts.

• The top is modelled by the curved surface of a circular cone of radius $r$cm. The ratio of its height to its base radius is $4:3$.
• The body is modelled by the curved surface of a cylinder of radius $r$cm and height $H$cm.
• The base is modelled by a circular disc of radius $r$cm.

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