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

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 $x\in \mathrm{ℝ}$.

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.

Given that

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

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

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

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

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

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.

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.

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 $x>0$.

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 $C$.

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.

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.

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.