Applications of Differentiation (Edexcel A Level Maths: Pure): Exam Questions

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=x^3+x^2-5xx\in \mathrm{ℝ}$

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

Using algebra, solve the equation $3x^2+2x-5=0$.

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

Show that the point $\left(1,2\right)$ is a local maximum on the curve with equation

$y=-2x^3+3x^2+1$

A curve $C$ has equation

$y=3x^3+6x^2-5x+1x\in \mathrm{ℝ}$

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

Verify that $C$ has a stationary point at $x=\frac{1}{3}$ and determine its nature, giving a reason for your answer.

A curve $C$ has equation

$y=3x-\frac{1}{2}{x}^{2}x\in \mathrm{ℝ}$

The point $P$ lies on $C$ and has $x$-coordinate $5$.

Find the gradient of $C$ at $P$.

Find the equation of the normal to $C$ at $P$, giving your answer in the form $ax+by+c=0$, where $a$, $b$ and $c$ are integers to be found.

A curve $C$ has equation

$y=2x^2-\frac{2}{3}{x}^3-\frac{5}{3}x\in \mathrm{ℝ}$

Show that the point $(2,1)$ is a local maximum point on $C$.

A curve has equation

$y=\frac{2}{3}{x}^3-\frac{7}{2}{x}^2-4x+5$

Find $\frac{\mathrm{d}y}{\mathrm{d}x}$ writing your answer in simplest form.

Hence find the range of values of $x$ for which $y$ is decreasing.

A curve $C$ has equation

$y=x^2-2x-24\sqrt{x},x>0$

Find

(i) $\frac{\mathrm{d}y}{\mathrm{d}x}$

(ii) $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$

Verify that $C$ has a stationary point when $x=4$.

Determine the nature of this stationary point, giving a reason for your answer.

Figure 1 shows part of the curve with equation $y=3x^2-2$

The point $P\left(2,10\right)$ lies on the curve.

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

The point $Q$ with $x$ coordinate $2+h$ also lies on the curve.

Find the gradient of the line $PQ$, giving your answer in terms of $h$ in simplest form.

Explain briefly the relationship between part (b) and the answer to part (a).

In this question your must show all stages of your working.

Solutions relying on calculator technology are not acceptable.

The curve $C$ has equation

$\begin{array}{ccc}y=\frac{1}{3}{x}^2-2\sqrt{x}+3& & x\ge 0\end{array}$

The point $P$ lies on $C$ and has $x$ coordinate $4$

The line $l$ is tangent to $C$ at $P$.

Show that $l$ has equation

$13x-6y-26=0$

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 an 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 an equation of the normal to $C$ at $P$ is

$x+2y=3$

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

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

A curve has the 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 local minimum 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 this must be the 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.

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=7x^2-2x(x^2+5)x\in \mathrm{ℝ}$

Show that $\text{f}$ is a decreasing function for all $x\in \mathrm{ℝ}$.

Given that

$y=\frac{4}{x}-\sqrt[3]{\frac{27}{x}}$

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

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

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=4x+\frac{3}{x}x\in \mathrm{ℝ}, x\ne 0$

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

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=\sqrt{x}-\frac{7}{\sqrt{x}}x>0$

Show that $\text{f}$ is an increasing function for all $x>0$.

The curve $C$ has equation $y=\mathrm{f}\left(x\right)$ where

$\mathrm{f}\left(x\right)=a{x}^3+15x^2-39x+b$

and $a$ and $b$ are constants.

Given

• the point $\left(2,10\right)$ lies on $C$

• the gradient of the curve at $\left(2,10\right)$ is $-3$

(i) show that the value of $a$ is $-2$

(ii) find the value of $b$.

Hence show that $C$ has no stationary points.

Factorise completely $9x-x^3$

The curve $C$ has equation

$y=9x-x^3$

Sketch $C$ showing the coordinates of the points at which the curve cuts the $x$-axis.

The line $l$ has equation $y=k$ where $k$ is a constant.

Given that $C$ and $l$ intersect at $3$ distinct points, find the range of values for $k$, writing your answer in set notation.

Solutions relying on calculator technology are not acceptable.

The curve $C$ has equation

$\begin{array}{ccc}y=5x^4-24x^3+42x^2-32x+11& & x\in \mathrm{ℝ}\end{array}$

Find

(i) $\frac{\mathrm{d}y}{\mathrm{d}x}$

(ii) $\frac{{\mathrm{d}}^{2}y}{\mathrm{d}{x}^2}$

(i) Verify that $C$ has a stationary point at $x=1$

(ii) Show that this stationary point is a point of inflection, giving reasons for your answer.

A company makes drinks containers out of metal.

The containers are modelled as closed cylinders with base radius $r$ cm and height $h$ cm and the capacity of each container is $355$ cm³

The metal used

• for the circular base and the curved sides costs $0.04$ pence/cm²

• for the circular top costs $0.09$ pence/cm²

Both metals used are of negligible thickness.

Show that the total cost, $C$ pence, of the metal for one container is given by

$C=0.13\pi r^2+\frac{28.4}{r}$

Use calculus to find the value of $r$ for which $C$ is a minimum, giving your answer to 3 significant figures.

Using $\frac{{\mathrm{d}}^{2}C}{\mathrm{d}{r}^2}$ prove that the cost is minimised for the value of $r$ found in part (b).

Hence find the minimum value of $C$, giving your answer to the nearest integer.

A company makes toys for children.

Figure 5 shows the design for a solid toy that looks like a piece of cheese.

The toy is modelled so that

• face $ABC$ is a sector of a circle with radius $r$ cm and centre $A$

• angle $BAC=0.8$ radians

• faces $ABC$ and $DEF$ are congruent

• edges $AD,CF$ and $BE$ are perpendicular to faces $ABC$ and $DEF$

• edges $AD,CF$ and $BE$ have length $h$ cm

Given that the volume of the toy is $240{\mathrm{cm}}^3$ show that the surface area of the toy, $S{\mathrm{cm}}^2$, is given by

$S=0.8r^2+\frac{1680}{r}$

making your method clear.

Using algebraic differentiation, find the value of $r$ for which $S$ has a stationary point.

Prove, by further differentiation, that this value of $r$ gives the minimum surface area of the toy.

The function $\text{f}$ is defined by

$\text{f}\left(x\right)=x^3-5x^2+3x-2x\in \mathrm{ℝ}$

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

The curve $C$ has equation

$y=3x^2-6x+\sqrt{2x}$

The point $P(2,2)$ lies on $C$.

Find an equation of the tangent to $C$ at $P$, giving your answer in the form $y=mx+c$, where $m$ and $c$ are constants to be found.

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 curve $C$ has equation

$y=x(x+6{)}^2+4(3x+11)$

Find the coordinates of the stationary point of $C$ and determine its nature.

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

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

The 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.

The curve $C$ has equation

$y=7-2x^2+\sqrt{x},x\ge 0$

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

The curve $C$ has a stationary point at $P$.

Find the coordinates of $P$ and determine its nature, justifying your answer.

[A sphere of radius $r$ has volume $\frac{4}{3}\pi r^3$ and surface area $4\pi r^2$]

A manufacturer produces a storage tank.

The tank is modelled in the shape of a hollow circular cylinder closed at one end with a hemispherical shell at the other end as shown in Figure 9.

The walls of the tank are assumed to have negligible thickness.

The cylinder has radius $r$ metres and height $h$ metres and the hemisphere has radius $r$ metres.

The volume of the tank is 6 m³.

Show that, according to the model, the surface area of the tank, in m², is given by

$\frac{12}{r}+\frac{5}{3}\pi r^2$

The manufacturer needs to minimise the surface area of the tank.

Use calculus to find the radius of the tank for which the surface area is a minimum.

Calculate the minimum surface area of the tank, giving your answer to the nearest integer.

The curve $C$ has equation

$y=5-(x-3{)}^2$

The points $A$ and $B$ lie on $C$ and have $x$-coordinates $0$ and $6$ respectively.

The tangents to $C$ at $A$ and $B$ intersect at the point $D$.

Find the coordinates of $D$.

Find the exact area of triangle $ABD$.

A curve $C$ has equation $y=\text{f}\left(x\right)$, where

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

The point $P$ lies on $C$ such that the normal to $C$ at $P$ passes through the origin $O$.

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

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

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

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

In this question you must show all stages of your working.

Solutions relying entirely on calculator technology are not acceptable.

Figure 1 shows a sketch of part of the curve $C$ with equation

$y=3-\frac{1}{4}{x}^2,y>0$

The point $P(x,y)$ lies on $C$ and $O$ is the origin.

Show that the square of the distance from $O$ to $P$ is given by

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

Using calculus, find the exact minimum distance from $O$ to $C$. You must justify that your answer is a minimum.

Figure 2 shows the design for a patio table top.

The table top is modelled as a sector of a circle with radius $r$ metres and central angle $\theta$ radians, where $0<\theta <2\pi$.

The area of the table top is fixed at $A$ m².

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

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

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

Using calculus, show that the minimum possible value for $P$ is equal to the perimeter of a square of area $A$. Justify that your value is a minimum.