Applications of differentiation

The equation of a curve is given by $4x^3+3x^2y=y^3-2$. Find $\dfrac{dy}{dx}$ in terms of $x$ and $y$, simplifying your answer.

The equation of a closed curve is given by $(nx+y)^2+4(nx-y)^2=20$, where $n$ is a constant. Find

MI Prelim 9758/2023/02/Q4. The curve $C$ has equation $2x^2+y^2=20$.

A curve $C$ is given by the equation $e^{x^2}y=\ln x^2$, where $x\ne0$.

A curve $C$ is defined by the parametric equations $x=\theta-\cos\theta$, $y=\theta+\cos\theta$, $0\le\theta\le2\pi$.

The diagram shows a sketch of the curve $y=xe^{-x}$.

A curve is defined parametrically by $x=\dfrac{at}{t+1}$, $y=\dfrac{at^2}{t+1}$, where $a$ is a non-zero constant and $t\in\mathbb R$, $t\ne-1$.

The curve $C$ has parametric equations $x=4\cos\theta$, $y=3\sin\theta$, $0<\theta\le\pi$.

A particle is moving along a curve, $C$, such that its position at time $t$ seconds after it is set into motion is given by the parametric equations $x=t+e^{-2t}$, $y=t-e^{-2t}$.

The diagram shows the trajectory of a cannonball fired off from an origin $O$ with an initial speed of $v\,\mathrm{m\,s^{-1}}$ and at an angle of $\theta^\circ$ above the ground. At time $t$ seconds, its position can be modelled by $x=(v\cos\theta)t$, $y=(v\sin\theta)t-5t^2$, where $x$ m is horizontal distance with respect to $O$ and $y$ m is vertical…

A manufacturer wants to design a lidless container with a square base and a capacity of $1000\,\mathrm{cm}^3$. Given that the uniform thickness of the container is 1 cm, using differentiation, find the internal length of the tank, $x$ cm, such that the volume of the material used is minimised.

[It is given that a right circular cone with base radius $r$ and height $h$ has volume $\dfrac13\pi r^2h$ and curved surface area $\pi r\ell$, where $\ell$ is the slanted height of the cone.]

A circular cylinder is inscribed in a right circular cone of base radius $2c$ and vertical height $c$. One of the circular ends of the cylinder lies on the base of the cone and the other end is in contact with the inner surface of the cone. If the cylinder has base radius $x$, find expressions for the volume $V$ and the total surface area $S$ of the cylinder…

A movie theatre screen which is 5 m high has its lower edge 1 m above an observer’s eye. The visual angle $\theta$ of the observer seated $x$ m away is as shown in the diagram below.

A curve $C$ has equation $x^3+y^3-xy=A$, where $A$ is a non-zero constant.

The diagram shows a vertical cross-section of a container in the form of an inverted cone of height 24 cm and base radius 9 cm. The container is initially empty. Water is poured into it at a constant rate of $30\,\mathrm{cm^3\,s^{-1}}$ and the height of water level in the container at time $t$ is $h$ cm.

The curve $C$ and the line $L$ have equations $y=x^2$ and $y=\dfrac12x-2$ respectively.

A capsule made of two similar hemispheres of radius $r$ cm, one at each end of a cylinder, is inscribed into a fixed cone of height 12 cm and base radius 9 cm. The two figures have a common axis of symmetry and the cross-section is shown below.