Differentiation & Gradients (Edexcel IGCSE Maths A (Modular): Higher Unit 2): Exam Questions

The curve $y=x^3+2x^2—4x$ is shown on the grid.

By drawing a suitable tangent, find an estimate of the gradient of the curve when x = 1.

A point $D$ lies on the curve.
The x  co-ordinate of $D$ is negative.
The gradient of the tangent at $D$ is 0.

Write down the co-ordinates of $D$.

Ellie runs a race. The graph shows Ellie's speed in the first 10 seconds after the start of the race.

By drawing a suitable tangent, work out the acceleration when $t$ = 9. Give the units of your answer.

Describe what happens to Ellie after 7 seconds.

The diagram shows the graph of $y=\frac{x}{2}+\frac{2}{x}$for $0<\mathrm{x}\le 8.$

Use the graph to solve the equation $\frac{x}{2}+\frac{2}{x}=3$.

By drawing a suitable tangent, find an estimate of the gradient of the graph when x = 1.

Karol runs in a race.

The graph shows her speed, in metres per second, $t$ seconds after the start of the race.

Calculate an estimate for the gradient of the graph when $t=4$ You must show how you get your answer.

Describe fully what your answer to part (a) represents.

Explain why your answer to part (a) is only an estimate.

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

Find: the coordinates where the curve crosses the $x$-axis,

the coordinates where the curve crosses the $y$-axis,

the coordinates of the turning point on the curve,

Sketch the curve showing the points you have found.

A fish bowl is being filled with water.
The graph shows how the diameter of the surface of the water changes with time.

Find an estimate for the gradient at $t$ = 10.

Give an interpretation of the gradient.

The diagram shows the graph of $y=\mathrm{f}\left(x\right)$ for $-4⩽x⩽12$

The point $P$on the curve has $x$ coordinate 2

Use the graph to find an estimate for the gradient of the curve at $P$.

Hence find an equation of the tangent to the curve at $P$. Give your answer in the form $y=mx+c$ .

A particle is moving along a straight line. The fixed point $O$ lies on this line.

The displacement of the particle from $O$ at time $t$ seconds is $s$ metres where

$s=2t^3-9t^2-60t$

Find an expression for the velocity, $v$ $m/s$ of the particle at time $t$ seconds.

Find the time at which the velocity is instantaneously zero.

Part of the curve with equation $y=h\left(x\right)$ is shown on the grid.

Find an estimate for the gradient of the curve at the point where $x=-0.5$
Show your working clearly.

For the curve with equation $y=x^3-7x^2-5x$:
find $\frac{\mathrm{d}y}{\mathrm{d}x}$

find the $x$-coordinates of the two turning points on the curve.

By considering the shape of the curve determine which of your answers to (b) is the $x$-coordinate of a maximum point.

Liquid is leaking out of a container.

The graph shows the depth of the liquid for 60 seconds.

Use the graph to work out an estimate of the rate of decrease of depth at 10 seconds.

You must show your working.

...........................................cm/s

The curve $G$ has equation $y=1-x^3-6x^2-9x$.

Part of the graph of $G$ is shown below.

Write the coordinates of A.

Points B and C are stationary points on $G$.

Find the coordinates of points B and C, stating the nature of the stationary point in each case. 

For which values of $x$ is the gradient of the curve $G$ negative?

For the curve with equation $y=4x+\frac{64}{x}+7$

find $\frac{\mathrm{d}y}{\mathrm{d}x}$

find the coordinates of the stationary points on the curve.

find the exact distance between the two stationary points.

The graph shows information about the speed of a vehicle during the final 50 seconds of a journey.

At the start of the 50 seconds the speed is k metres per second. The distance travelled during the 50 seconds is 1.35 kilometres.

Work out the value of $k$.

$k$ = .......................

A particle is moving along a straight line and passes a fixed point $O$.

The displacement of the particle, from point $O$, at time $t$ seconds is

$s=\frac{1}{3}{t}^3-\frac{5}{2}{t}^2+20t-15$

where $s$ is measured in metres. Initially how far is the particle from $O$?

Find, in terms of $t$, the velocity of the particle.

Find the time at which the particle’s velocity is at its minimum.

For how long is the particle decelerating?

A homeowner wishes to enclose a rectangular part of their garden by building a fence, using an existing wall as one side of the rectangle as shown in the diagram below.

The width of the enclosed rectangle is $w$ metres and its length $l$ metres.

The homeowner has 40 metres of fence to use and would like to use it all in order to maximise the area of the garden to be enclosed. Show that $l=40-2w$

Show that the area of the garden to be enclosed, $A$, is given by $A$= 40$w$ − 2$w^2$

Find $\frac{\mathrm{d}A}{\mathrm{d}W}$

Find the value of $w$ that maximises $A$

Find the dimensions of the rectangle that produce the maximum area that can be enclosed using all of the fence.

Also find the maximum area.

The diagram shows a cuboid of volume $V{\mathrm{cm}}^3$

Show that $V=15+16x-x^2-2x^3$

There is a value of $x$ for which the volume of the cuboid is a maximum.

Find this value of $x$.
Show your working clearly.
Give your answer correct to 3 significant figures.

A particle $P$ is moving along a straight line.
The fixed point $O$ lies on this line.
At time $t$ seconds where $t⩾0$, the displacement, $s$ metres, of $P$ from $O$ is given by

$s=t^3+5t^2–8t+10$

Find the displacement of $P$ from $O$ when $P$ is instantaneously at rest.

Give your answer in the form $\frac{a}{b}$ where $a$ and $b$ are integers.

A cuboid with a square cross section is to be made from rods as shown in the diagram. The shorter rods making the square are of length xx cm and the longer rods

are of length $y$ cm.

Explain why 12 rods in total will be needed to make the cuboid, and state how many of each length will be required. 

The total length of the rods is to be fixed at 36 cm.

The total length of the rods is to be fixed at 36 cm.

Find $y$ in terms of $x$

Show that the volume of the cuboid, $V$ cm³ is $V$ = 9$x^2$ − 2$x^3$.

Find the value of $x$ that maximises the volume.

Find the maximum volume.

A particle is moving along a straight line that passes through the fixed point O.
The displacement, $s$ metres, of the particle from O at time $t$ seconds is given by

$s=2t^3-5t^2+6t-5$

Find the value of $t$ when the acceleration of the particle is 5 m/s²

A curve, $C$, has equation $y=2x^2+8k^{2}x-3$ where $k$ is a constant.

Show that when $k$ = 0, the turning point on $C$ has coordinates (0, -3).

Show that when $\mathit{k}$ ≠ 0, the turning point on $C$ must have a negative $\mathit{x}$-coordinate.

When $k$≠ 0 determine whether or not the $y$-coordinate of the turning point is negative.

Part of the graph with equation $y=2x^4-16x^2+3$is shown below.

The graph has three stationary points, indicated on the graph by points $P$, $Q$ and$R$.
Find the area of the triangle $P$$QR$.

The diagram shows a cuboid with a square cross-section.

The sides of the square face are $x$cm and the length of the cuboid is $y$cm.

The cuboid is to have a fixed surface area, $A$, of 25 cm².

Show that the volume of the cuboid, $V$ cm³ is given by

$V=\frac{25}{4}x-\frac{1}{2}{x}^3$

Show that the value of $x$ that maximises the volume of the cuboid is $\frac{5\sqrt{6}}{6}$

Find the maximum volume of the cuboid, correct to 3 significant figures.

A particle $P$ moves along a straight line that passes through the fixed point $O$

The displacement, $x$ metres, of$P$from $O$ at time t seconds, where $t⩾0$, is given by

$x=4t^3-27t+8$

The direction of motion of  $P$ reverses when $P$ is at the point $A$ on the line.

The acceleration of $P$ at the instant when $P$ is at $A$ is $a\mathrm{m}/{\mathrm{s}}^2$. Find the value of  $a$.

a = ..................................... 

Two particles, $P$ and $Q$, move along a straight line.

The fixed point $O$ lies on this line.

The displacement of $P$ from $O$ at time $t$ seconds is $s$ metres, where

$s=t^3–4t^2+5tfort>1$

The displacement of $Q$ from $O$ at time $t$ seconds is $x$ metres, where

$x=t^2–4t+4fort>1$

Find the range of values of $t$ where $t>1$ for which both particles are moving in the same direction along the straight line.

The point A is the only stationary point on the curve with equation $y=k{x}^2+\frac{16}{x}$  where $k$is a constant.