Differentiation

A vacuum flask, used to keep drinks hot, is represented by a closed cylinder whose internal radius is $r\,\text{cm}$ and internal height is $h\,\text{cm}$. The flask has volume $1000\,\text{cm}^3$. It is most efficient when the total internal surface area, $A\,\text{cm}^2$, is as small as possible.

Feb/March 2016

The diagram depicts a water container shaped like an inverted pyramid, arranged so that when the water depth is $h$ cm, the water surface is a square with side $\frac{1}{2}h$ cm.

Feb/March 2017

For $x > 0$, the function $f$ is given by $f(x) = (4x + 1)^{\tfrac{3}{2}}$.

Feb/March 2017

Point $A\,(2, 2)$ is located on the curve $y = x^2 - 2x + 2$.

Feb/March 2017

The diagram depicts the curve whose equation is $y = 4x^{\frac{1}{2}}$.

Feb/March 2019

The graph of $y = f(x)$ includes the points $(0, 2)$ and $(3, -1)$.

Feb/March 2019

The curve is defined by $y = (2x - 1)^{-1} + 2x$.

Feb/March 2019

The function $f$ is specified by $f(x) = \frac{1}{3x + 2} + x^2$ for $x < -1$.

Feb/March 2020

For the curve, the gradient at $(x, y)$ is $\frac{dy}{dx} = 2\sqrt{x + 3} - x$. It has a stationary point at $(a, 14)$, with $a$ a positive constant.

Feb/March 2020

The diagram displays the curve whose equation is $y = 9\left(x^{\frac{1}{2}} - 4x^{-\frac{3}{2}}\right)$. This curve meets the $x$-axis at $A$.

Feb/March 2021

The curve satisfies $\frac{dy}{dx} = \frac{6}{(3x - 2)^3}$, and $A\,(1,-3)$ is on it. A point moves along the curve, and at $A$ the $y$-coordinate of the point is rising at $3$ units per second.

Feb/March 2021

A curve is defined by the equation $y = k(3x - k)^{-1} + 3x$, with $k$ taken as a constant.

Feb/March 2022

At the point $(4, -1)$ on the curve, the gradient is $-\frac{3}{2}$. It is also given that $\frac{dy}{dx} = x^{-\frac{1}{2}} + k$, where $k$ is constant.

Feb/March 2023

A curve is defined by $y = \frac{1}{60}(3x + 1)^2$, and a point moves along this curve.

Feb/March 2023

The equation of a curve is $y = \frac{3}{2x^2 - 5}$.

Feb/March 2024

The diagram shows the curve whose equation is $y = 2x^2 - \frac{5}{x} + 3$. The curve intersects the $x$-axis at $P(1, 0)$, and $M$ is a minimum point.

Feb/March 2025

A curve is defined by $\frac{d^2 y}{dx^2} = \frac{6}{x^4} - \frac{5}{x^3}$. It is stated that this curve has a stationary point at $\left(\frac{1}{2}, 9\right)$.

Feb/March 2025

A curve is defined by $\frac{dy}{dx} = 3x^2 - 6$, and the point $(9, 2)$ lies on it.

May/June 2010

The curve is defined by $y = \tfrac{1}{6}(2x-3)^3 - 4x$.

May/June 2010

A solid rectangular block is built with a square base of side $x$ cm. Its height is $h$ cm, and the block’s total surface area is $96\text{ cm}^2$.

May/June 2010

The domain of the function $f : x \mapsto 2x^2 - 8x + 14$ is $x \in \mathbb{R}$.

May/June 2010

The curve is described by the differential equation $\frac{dy}{dx} = \frac{6}{\sqrt{3x - 2}}$, and it goes through the point $P(2, 11)$.

May/June 2010

A spherical balloon's volume is rising at a steady rate of $50\text{ cm}^3$ per second.

May/June 2011

The variables $x$, $y$ and $z$ are restricted to positive values, and they satisfy $z = 3x + 2y$ together with $xy = 600$.

May/June 2011

The curve satisfies $\frac{dy}{dx} = \frac{3}{(1 + 2x)^2}$, and the point $(1, \frac{1}{2})$ lies on it.

May/June 2011

The curve is described by the equation $y = \frac{4}{3x - 4}$, and the point $P\,(2,2)$ lies on it.

May/June 2011

A curve satisfies $\frac{dy}{dx} = \frac{2}{\sqrt{x}} - 1$ and $P(9, 5)$ lies on the curve.

May/June 2011

It is stated that the curve is described by $y = f(x)$, where $f(x) = x^3 - 2x^2 + x$.

May/June 2012

A watermelon is taken to have a spherical shape while it grows. Its mass, $M$ kg, and radius, $r$ cm, are linked by $M = kr^3$, where $k$ is constant. The radius is also assumed to increase at a steady rate of $0.1$ centimetres per day. On one particular day, the radius is $10$ cm and the mass is $3.2$ kg.

May/June 2012

The diagram illustrates the curve $y = 7\sqrt{x}$ together with the line $y = 6x + k$, where $k$ is a constant. The curve and the line cross at the points $A$ and $B$.

May/June 2012

A curve has equation $y = 4\sqrt{x} + \frac{2}{\sqrt{x}}$.

May/June 2012

The function $f$ is given by $f(x) = 8 - (x - 2)^2$, for $x \in \mathbb{R}$. The function $g$ is given by $g(x) = 8 - (x - 2)^2$, for $k \leq x \leq 4$, where $k$ is a constant.

May/June 2012

The curve is defined by $\frac{d^2y}{dx^2} = -4x$. It reaches a maximum at $(2,12)$.

May/June 2012

It is stated that $f(x) = (2x - 5)^3 + x$, for $x \in \mathbb{R}$.

May/June 2013

The curve is given by $y = x^2 - 4x + 4$ and the line is given by $y = mx$, with $m$ as a constant.

May/June 2013

The curve is described by the equation $y = f(x)$, where $f'(x) = 3x^{\frac{1}{2}} + 3x^{-\frac{1}{2}} - 10$.

May/June 2013

A solid circular cylinder with radius $r$ cm has volume $250\pi\text{ cm}^3$.

May/June 2013

The function $f$ is given by $f(x) = \frac{5}{1 - 3x}$, where $x \geq 1$.

May/June 2013

The non-zero variables $x$, $y$, and $u$ satisfy $u = x^2y$. It is given that $y + 3x = 9$.

May/June 2013

A curve is defined by $\frac{dy}{dx} = x^2 - x^{\frac{1}{2}}$. The curve goes through the point $(4, \frac{2}{3})$. Find the equation of the curve.

May/June 2014

A curve is defined by $y = \frac{4}{(3x + 1)^2}$. Find the equation of the tangent to the curve at the point where the line $x = -1$ meets it.

May/June 2014

A curve is defined by $\frac{d^2y}{dx^2} = 2x - 1$. It has a minimum point at $(3, -10)$. Find the coordinates of the maximum point.

May/June 2014

The base of a cuboid has side lengths $x\text{ cm}$ and $3x\text{ cm}$. Its volume is $288\text{ cm}^3$.

May/June 2014

The curve $y = 2x^2$ is drawn with the points $X(-2,0)$ and $P(p,0)$ marked on it. Point $Q$ is on the curve, and $PQ$ is parallel to the $y$-axis.

May/June 2015

The curve is given by $y = x^3 + px^2$, where $p$ is a positive constant. A second curve is given by $y = x^3 + px^2 + px$.

May/June 2015

The variables $u$, $x$ and $y$ satisfy $u = 2x(y - x)$ together with $x + 3y = 12$.

May/June 2015

The points $A(2, 9)$ and $B(3, 0)$ are on the curve $y = 9 + 6x - 3x^2$, as the diagram shows. The tangent at $A$ meets the $x$-axis at $C$.

May/June 2015

The function $f$ takes the form $f(x) = \frac{1}{x+1} + \frac{1}{(x+1)^2}$ for $x > -1$.

May/June 2015

For the curve, $\frac{dy}{dx} = 2 - 8(3x + 4)^{-\frac{1}{2}}$.

May/June 2016

A farmer splits a rectangular plot of land into $8$ equal rectangular sheep pens, as the diagram shows. Each pen has dimensions $x$ m by $y$ m and is completely surrounded by metal fencing. Altogether, the farmer uses $480$ m of fencing.

May/June 2016

The curve is given by $y = 3x - \frac{4}{x}$ and it passes through $A(1, -1)$ and $B(4, 11)$. At the points $C$ and $D$ on the curve, the tangent is parallel to $AB$. Find the equation of the perpendicular bisector of $CD$.

May/June 2016

The curve is given by $y = 8x + (2x - 1)^{-1}$. Find the $x$-values where the curve has stationary points and determine the type of each stationary point, giving reasons for your answers.

May/June 2016

Point $P(x, y)$ moves along the curve $y = x^2 - \frac{10}{3}x^3 + 5x$ so that the rate at which $y$ changes stays constant.

May/June 2016

A solid prism has an equilateral triangle as its horizontal base, with side $x$ cm. The prism’s side faces are vertical. Its height is $h$ cm and its volume is $2000\text{ cm}^3$.

May/June 2017

The curve is given by $y = 3 + \frac{12}{2 - x}$.

May/June 2017

The curve is defined by $y = \frac{8}{\sqrt{x}} - 2x$.

May/June 2017

The function $f$ is defined for $x \geq 0$. It is given that $f$ reaches its minimum when $x = 2$, and that $f''(x) = (4x + 1)^{-\frac{1}{2}}$.

May/June 2017

The line $3y + x = 25$ is a normal to the curve $y = x^2 - 5x + k$. Determine the value of the constant $k$.

May/June 2017

The curve given by $y = x^3 - 2x^2 + 5x$ goes through the origin.

May/June 2018

A point travels along the curve $y = 2x + \frac{5}{x}$ so that the $x$-coordinate rises at a steady rate of $0.02$ units per second.

May/June 2018

The curve satisfies $\frac{dy}{dx} = \sqrt{4x + 1}$, and the point $(2, 5)$ lies on it.

May/June 2018

At the point $A$, the tangent to the curve $y = x^3 - 9x^2 + 24x - 12$ is parallel to the line $y = 2 - 3x$. Find the equation of the tangent at $A$.

May/June 2018

The curve satisfying $\frac{d^2 y}{dx^2} = 2x - 5$ has a stationary point at $(3, 6)$.

May/June 2019

The curve is defined by $\frac{dy}{dx} = x^3 - \frac{4}{x^2}$. The point $P(2, 9)$ is on the curve.

May/June 2019

The curve $C_1$ is given by the equation $y = x^2 - 4x + 7$. The curve $C_2$ is given by the equation $y^2 = 4x + k$, with $k$ constant. The tangent to $C_1$ at the point where $x = 3$ is also tangent to $C_2$ at the point $P$.

May/June 2019

The diagram presents a segment of the curve with equation $y = (3x + 4)^{\tfrac{1}{2}}$ together with the tangent to the curve at point $A$. The $x$-coordinate of $A$ is $4$.

May/June 2019

A curve is defined by $\frac{dy}{dx} = 3x^2 + ax + b$. Its stationary points are located at $(-1, 2)$ and $(3, k)$.

May/June 2019

The line has equation $y = mx + c$, with $m$ and $c$ as constants, and the curve is defined by $xy = 16$.

May/June 2020

The curve is described by $y = (3 - 2x)^3 + 24x$.

May/June 2020

The curve is given by the equation $y = 54x - (2x - 7)^3$.

May/June 2020

A spherical weather balloon is being inflated by a pump. Its volume is rising at a constant rate of $600\,\text{cm}^3$ per second. At the start of pumping, the balloon was empty.

May/June 2020

The curve is described by $y = 2x^2 + kx + k - 1$, with $k$ as a constant.

May/June 2020

A point $P$ is travelling on a curve so that the $x$-coordinate of $P$ is rising at a steady rate of $2$ units per minute. The curve is given by $y = \sqrt{5x - 1}$.

May/June 2020

The curve is described by $y = 2\sqrt{3x + 4} - x$.

May/June 2021

The curve has equation $y = (2k - 3)x^2 - kx - (k - 2)$, with $k$ as a constant. The straight line $y = 3x - 4$ touches the curve.

May/June 2021

The gradient of a curve is given by $\frac{dy}{dx} = 6(3x - 5)^3 - kx^2$, where $k$ is a constant. The curve passes through a stationary point at $(2, -3.5)$.

May/June 2021

A curve has equation $y = (x - 3)\sqrt{x + 1} + 3$. These points lie on the curve. Any non-exact values are rounded to 4 decimal places. $A\,(2, k)$, $B\,(2.9, 2.8025)$, $C\,(2.99, 2.9800)$, $D\,(2.999, 2.9980)$, $E\,(3, 3)$.

May/June 2021

The function $f$ is given by $f(x) = \frac{1}{3}(2x - 1)^{\frac{3}{2}} - 2x$ for $\frac{1}{2} < x < a$. It is stated that $f$ decreases as $x$ increases.

May/June 2021

The curve $y = x^2 - 4x + 3$ is touched by the line $y = mx - 6$ at a tangent.

May/June 2021

The curve is given by $\frac{d^2 y}{dx^2} = 6x^2 - \frac{4}{x^3}$. Its stationary point is at $(-1, \frac{9}{2})$.

May/June 2022

The curve is given by $y = 4x^2 - kx + \frac{1}{2}k^2$ and the line is given by $y = x - a$, where $k$ and $a$ are constants.

May/June 2022

The curve is given by $y = 3x + 1 - 4\sqrt{3x + 1}$ for $x > -\frac{1}{3}$.

May/June 2022

The point $P$ is on the line whose equation is $y = mx + c$, where $m$ and $c$ are positive constants. A curve is given by $y = -\tfrac{m}{x}$. There is one point $P$ on the curve for which the straight line is tangent to the curve at $P$.

May/June 2022

The curve is defined by $\frac{dy}{dx} = 6x^2 - 30x + 6a$, with $a$ being a positive constant. There is a stationary point on the curve at $(a, -15)$.

May/June 2023

For positive constant $k$, the line $y = kx - k$ is just tangent to the curve $y = -\frac{1}{2x}$.

May/June 2023

Water enters a tank at a steady rate of $500\,\text{cm}^3$ per second. The water depth in the tank is $h$ cm, measured $t$ seconds after filling begins. When the water depth is $h$ cm, the volume, $V\,\text{cm}^3$, of water in the tank is given by the formula $V = \dfrac{4}{3}(25 + h)^3 - \dfrac{62500}{3}$.

May/June 2023

The curve is given by $y=k\sqrt{4x+1}-x+5$, where $k$ is a positive constant.

May/June 2023

The function has rule $f(x) = \frac{4}{x^3} - \frac{3}{x} + 2$ for $x \neq 0$. Its graph, $y = f(x)$, is displayed in the diagram.

May/June 2024

The curve is defined by $y=f(x)$, where $f(x)=(2x-1)\sqrt{3x-2}-2$. The points below are on the curve. Values that are not exact have been rounded to $5$ decimal places. $A(2,4)$, $B(2.0001,k)$, $C(2.001,4.00625)$, $D(2.01,4.06261)$, $E(2.1,4.63566)$, $F(3,11.22876)$. The table gives the gradients of chords $AB$, $AC$, $AD$ and $AF$.

May/June 2024

A curve is given by $y = (5 - 2x)^{\tfrac{3}{2}} + 5$ for $x < \tfrac{5}{2}$.

May/June 2024

The function $f$ has derivative $f'(x) = 6(2x - 3)^2 - 6x$ for $x \in \mathbb{R}$.

May/June 2024

The curve is defined by $y = 2x^2 - \frac{1}{2x} + 3$.

May/June 2024

The curve is defined by the equation $\frac{dy}{dx} = 4(2x - 5)^3 - 9x^{\frac{1}{2}}$, and it passes through the point $A\left(4, -\frac{11}{2}\right)$.

May/June 2025

A curve is given by $y = 4x^2 + \tfrac{9}{x^2} - 8$.

May/June 2025

A point $P$ travels along the curve with equation $y = ax^{\frac{3}{2}} - 12x$ so that the $x$-coordinate of $P$ is increasing at a steady rate of 5 units per second.

May/June 2025

For the curve, $\frac{d^2 y}{dx^2} = -\frac{24}{x^3}$. It is also stated that the curve has a stationary point at $(-2, 19)$.

May/June 2025

The curve is given by $y = 2x + \frac{12}{x^2}$.

May/June 2025

A curve $C$ is given by the equation $y = \frac{9}{2x - 5} + 2x - 5$.

May/June 2025

The curve is defined by $\frac{dy}{dx} = 3x^2 + 10x - 8$.

May/June 2025

A curve is given by $y = x^3 + ax^2 + bx + 5$. It has a stationary point at $(1, 9)$.

May/June 2025