Applications of Differentiation (Cambridge (CIE) A Level Maths: Pure 1): Exam Questions

The function $\text{f}$ is defined by $\text{f}\left(x\right)=x^3-3x^2+6x-7$ for $x\in \mathrm{ℝ}$.

Show that $\text{f}$ is an increasing function.

The curve $C$ has equation $y=2x^3-3x^2+4x-3$.

Show that the point $P (2,9)$ lies on $C$.

Show that the value of $\frac{\text{d}y}{\text{d}x}$ at $P$ is 16.

Find the equation of the tangent to $C$ at $P$.

The curve $C$ has equation $y=3x^2-6+\frac{4}{x}$. The point $P (1,1)$ lies on $C$.

Find an expression for $\frac{\text{d}y}{\text{d}x}$.

Show that the equation of the normal to $C$ at $P$ is $x+2y=3$.

This normal cuts the $x$-axis at the point $Q$.

Find the length of $PQ$, giving your answer as an exact value.

Given that $y=2x^3-8\sqrt{x}$, find

$\frac{\text{d}y}{\text{d}x}$

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$

A curve has equation $y=x^3-12x+7$.

Find expressions for $\frac{\text{d}y}{\text{d}x}$ and $\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$.

Determine the coordinates of the minimum point of the curve.

The diagram below shows part of the curve with equation $y=x^3+11x^2+35x+25$.

The curve touches the $x$-axis at $A$ and cuts the $x$-axis at $C$.

The points $A$ and $B$ are stationary points on the curve.

Using calculus, and showing all your working, find the coordinates of $A$ and $B$.

Show that $(-1,0)$ is a point on the curve and explain why those must be the coordinates of point $C$.

A company manufactures food tins in the shape of cylinders which must have a constant volume of $150\pi {\text{ cm}}^3$. To lessen material costs the company would like to minimise the surface area of the tins.

By first expressing the height $h$ of the tin in terms of its radius $r$, show that the surface area of the cylinder is given by $S=2\pi r^2+\frac{300\pi }{r}$.

Use calculus to find the minimum value for the surface area of the tins. Give your answer correct to 2 decimal places.

A curve has equation $y=x^3-6x^2+9x-1$.

Find the $x$-coordinates of the stationary points of the curve.

Determine the nature of the stationary points found in part (a).

In a computer animation, the radius of a circle increases at a constant rate of $1{\text{ mm s}}^{-1}$.

Find the rate, per second, at which the area of the circle is increasing at the time when the radius is 8 millimetres. Give your answer as a multiple of $\pi$.

The side length of a cube increases at a rate of $0.1{\text{ cm s}}^{-1}$.

Find the rate of change of the volume of the cube at the instant the side length is 5 cm.

You may assume that the cube remains cubical at all times.

In the production process of a glass sphere, hot glass is blown such that the radius, $r$ cm, increases over time, $t$ seconds, in direct proportion to the temperature $T$ °C of the glass.

Find an expression, in terms of $r$ and $T$, for the rate of change of the volume, $V{\text{ cm}}^3$, of a glass sphere.

When the temperature of the glass is 1200 °C, a glass sphere has a radius of 2 cm and its volume is increasing at a rate of $5{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of increase of the radius at this time.

An ice cube, of side length $x$ cm, is melting at a constant rate of $0.8{\text{ cm}}^3{\text{ s}}^{-1}$.

Assuming that the ice cube remains in the shape of a cube whilst it melts, find the rate at which its surface area is decreasing at the point when its side length is 2 cm.

A bowl is in the shape of a hemisphere of radius 8 cm.

The volume of liquid in the bowl is given by the formula

$V=8\pi h^2-\frac{1}{3}\pi h^3$

where $h$ cm is the depth of the liquid (that is, the height between the bottom of the bowl and the level of the liquid).

Liquid is leaking through a small hole in the bottom of the bowl at a constant rate of $5{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of change of the depth of liquid in the bowl at the instant the depth of liquid is 3 cm.

The function $\text{f}$ is defined by $\text{f}\left(x\right)=x^3-5x^2+3x-2$ for $x\in \mathrm{ℝ}$.

Find the set of values of $x$ for which $\text{f}$ is a decreasing function.

The function $\text{f}$ is defined by $\text{f}\left(x\right)=7x^2-2x\left(x^2+5\right)$ for $x\in \mathrm{ℝ}$.

Show that $\text{f}$ is a decreasing function.

Given that $y=\frac{4}{x}-\sqrt[3]{}$, find

$\frac{\text{d}y}{\text{d}x}$

$\frac{{\text{d}}^{2}y}{\text{d}{x}^2}$

The curve $C$ has equation $y=3x^2-6x+\sqrt{2x}$. The point $P (2,2)$ lies on $C$.

Find the equation of the tangent to the curve at $P$.

The curve $C$ has equation $y=\frac{9}{\sqrt{3x}}-\frac{3}{x}$. The point $P (3,2)$ lies on $C$.

The normal to $C$ at $P$ intersects the $x$-axis at the point $Q$.

Find the coordinates of $Q$.

The diagram below shows a part of the curve with equation $y=\text{f}\left(x\right)$, where

$\text{f}\left(x\right)=460-\frac{x^3}{300}-\frac{8100}{x},x>0$

Point $A$ is the maximum point of the curve.

Find $\text{f}\text{'}\left(x\right)$.

Use your answer to part (a) to find the coordinates of point $A$.

A garden bed is to be divided by fencing into four identical isosceles triangles, arranged as shown in the diagram below:

The base of each triangle is $2x$ metres, and the equal sides are each $y$ metres in length.

Although $x$ and $y$ can vary, the total amount of fencing to be used is fixed at $P$ metres.

Explain why $0<x<\frac{P}{6}$.

Show that

$A^2=\frac{4}{9}{P}^2{x}^2-\frac{16}{3}P{x}^3$

where $A$ is the total area of the garden bed.

Using your answer to (b) find, in terms of $P$, the maximum possible area of the garden bed.

Describe the shape of the bed when the area has its maximum value.

A curve has equation $y=4x-x^2-2x^3$.

Find the coordinates of the stationary points of the curve, and determine their nature.

A spherical air bubble's surface area is increasing at a constant rate of $4\pi {\text{ cm}}^2{\text{ s}}^{-1}$.

Find an expression for the rate at which the radius is increasing per second.

(The surface area of a sphere is $4\pi r^2$.)

A cone, stood on its vertex, of radius 3 cm and height 9 cm, is being filled with sand at a constant rate of $0.2{\text{ cm}}^3{\text{ s}}^{-1}$.

Find the rate of change of the depth of sand in the cone at the instant the radius of the sand is 1.2 cm.

The material used to make a spherical balloon is designed so that it can be inflated at a maximum rate of $16{\text{ cm}}^3{\text{ s}}^{-1}$ without bursting.

Given that the radius of the balloon is determined by the function $r\left(t\right)=\frac{t}{\pi }+\frac{1}{2}$, $t\ge 0$,

show that the maximum time the balloon can be inflated for, without bursting, is $\frac{3}{2}\pi$ seconds.

An ice lolly which is in the shape of a cylinder of radius $r$ cm and length $8r$ cm is melting at a constant rate of $0.4{\text{ cm}}^3{\text{ s}}^{-1}$.

Assuming that the ice lolly remains in the shape of a cylinder (mathematically similar to the original cylinder) whilst it melts, find the rate at which its surface area is decreasing at the point when its radius is 0.3 cm.

The volume of liquid in a hemispherical bowl is given by the formula

$V=\frac{1}{3}\pi h^2\left(3R-h\right)$

where $R$ is the radius of the bowl and $h$ is the depth of liquid (that is, the height between the bottom of the bowl and the level of the liquid).

In a particular case, a bowl is leaking liquid through a small hole in the bottom at a rate directly proportional to the depth of liquid.

Show that the depth of liquid in the bowl is decreasing by

$\frac{k}{\pi \left(2R-h\right)}$

where $k$ is a constant.

The function $\text{f}$ is defined by $\text{f}\left(x\right)=4x+\frac{3}{x}$ for $x\in \mathrm{ℝ}$, $x\ne 0$.

Find the set of values of $x$ for which $\text{f}$ is a decreasing function.

The function $\text{f}$ is defined by $\text{f}\left(x\right)=\sqrt{x}-\frac{7}{\sqrt{x}}$ for $x>0$.

Show that $\text{f}$ is an increasing function.

A curve is described by the equation $y=\text{f}\left(x\right)$, where $\text{f}\left(x\right)=7-2x^2+\sqrt{x}$, $x⩾0$.

Find $\text{f}\text{'}\left(x\right)$ and $\text{f}\text{'}\text{'}\left(x\right)$.

$P$ is the stationary point on the curve.

Find the coordinates of $P$ and determine its nature.

A plant pot in the shape of a square-based pyramid, stood on its vertex, is being filled with soil at a rate of $72{\text{ cm}}^3{\text{ s}}^{-1}$.

The plant pot has a height of 1 m and a base length of 40 cm.

Find the rate at which the depth of soil is increasing at the moment when the depth is 60 cm.

(The volume of a pyramid is a third of the area of the base times the height.)

A curve has equation $y=5-(x-3{)}^2$.

$A$ is the point on the curve with $x$-coordinate 0, and $B$ is the point on the curve with $x$-coordinate 6.

$C$ is the point of intersection of the tangents to the curve at $A$ and $B$.

Find the coordinates of point $C$.

Calculate the area of triangle $ABC$.

A curve is described by the equation $y=\text{f}\left(x\right)$, where

$\text{f}\left(x\right)=\frac{1}{\sqrt{x}},x>0$

$P$ is the point on the curve such that the normal to the curve at $P$ also passes through the origin.

Find the coordinates of point $P$. Give your answer in the form $\left(2^a,2^b\right)$, where $a$ and $b$ are rational numbers to be found.

Write down the equation of the normal to the curve at $P$.

Show that an equation of the tangent to the curve at $P$ is

$\left(2^{\frac{1}{3}}\right)x+\left(2^{\frac{5}{6}}\right)y=3$

The diagram below shows the part of the curve with equation $y=3-\frac{1}{4}{x}^2$ for which $y>0$. The marked point $P(x,y)$ lies on the curve. $O$ is the origin.

Show that $O{P}^2=9-\frac{1}{2}{x}^2+\frac{1}{16}{x}^4$.

Find the minimum distance from $O$ to the curve, using calculus to prove that your answer is indeed a minimum.

The top of a patio table is to be made in the shape of a sector of a circle with radius $r$ and central angle $\theta$, where $\theta$ is strictly between 0° and 360°.

Although $r$ and $\theta$ may be varied, it is necessary that the table have a fixed area of $A{\text{ m}}^2$.

Explain why $r>\sqrt{\frac{A}{\pi }}$.

Show that the perimeter, $P$, of the table top is given by the formula

$P=2r+\frac{2A}{r}$

Show that the minimum possible value for $P$ is equal to the perimeter of a square with area $A$. Be sure to prove that your value is a minimum.

An expanding spherical air bubble has radius $r$ cm at time $t$ seconds, determined by the function

$r\left(t\right)=0.3+0.1t^2$

The bubble will burst if the rate of expansion of its volume exceeds $4t{\text{ cm}}^3{\text{ s}}^{-1}$.

Find, to one decimal place, the length of time the bubble expands for.

A small conical pot, stood on its base, is being filled with salt via a small hole at its vertex. The cone has a height of 6 cm and a radius of 2 cm.

Salt is being poured into the pot at a constant rate of $0.3{\text{ cm}}^3{\text{ s}}^{-1}$.

Find, to three significant figures, the rate of change in depth of the salt at the instant when the pot is half full by volume.

A large block of ice used by sculptors is in the shape of a cuboid with dimensions $x$ m by $2x$ m by $5x$ m. The block melts uniformly, with its surface area decreasing at a constant rate of $k$${\text{m}}^2{\text{ s}}^{-1}$. You may assume that as the block melts, the shape remains mathematically similar to the original cuboid.

Show that the rate of melting, by volume, is given by

$\frac{15kx}{34}{\text{ m}}^3{\text{ s}}^{-1}$

In the case when $k=0.2$, the block of ice remains solid enough to be sculpted as long as the rate of melting, by volume, does not exceed $0.05{\text{ m}}^3{\text{ s}}^{-1}$.

Find the value of $x$ for the largest block of ice that can be used for ice sculpting under such conditions, giving your answer as a fraction in its lowest terms.

The volume of liquid in a hemispherical bowl is given by the formula

$V=\frac{1}{3}\pi h^2\left(3R-h\right)$

where $R$ is the radius of the bowl and $h$ is the depth of liquid, that is, the height between the bottom of the bowl and the level of the liquid.

In a particular case, liquid is leaking through a small hole in the bottom of a bowl at a rate directly proportional to the depth of liquid.

When the bowl is full, the rate of volume loss is equal to $\pi$.

Show that the rate of change of the depth of the liquid is inversely proportional to $R\left(h-2R\right)$.