Differentiation & Gradients (Edexcel IGCSE Maths): Exam Questions

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 $k$ ≠ 0, the turning point on $C$ must have a negative $x$-coordinate.

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

The distance-time graph shows information about part of a car journey.

Use the graph to estimate the speed of the car at time 5 seconds.

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 $P\quad QR$.

The graph shows information about the velocity of a parachutist after jumping from a plane.

By drawing a suitable tangent, find an estimate of the gradient of the curve after 3 seconds.

Interpret the value of the gradient.

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.

The graph shows the temperature of a fish tank over the first 6 hours after a heater is added.

By drawing a suitable tangent, find an estimate of the gradient of the curve when $h$ = 3.

Interpret the value of the gradient.

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 $t \text{greater} - \text{than} or \text{slanted} \text{equal} to 0$, 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 $am/s^{2}$. Find the value of $a$.

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

The diagram shows parts of the graphs of y=f(x) and y=g(x)

Write down the value of x where the gradient of the curve y=g(x) is zero.

Calculate an estimate for the gradient of the curve y = f(x) at the point on the curve where x = 4.

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 diagram shows part of the graph of $y=x^{2}---2x+3$

By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^{2}---3x---1=0$

$P$ is the point on the graph of $y=x^{2}---2x+3$ where $x=2$

Calculate an estimate for the gradient of the graph at the point $P$.

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

Given that the coordinates of $A$ are $( \frac{2}{3} , a )$

find the value of $a$. Show your working clearly.

$a=$ .................................................

The curve $y=x^{3}---3x---2$is shown on the grid.

Write down the co-ordinates of the points where the gradient of the curve is zero.

Write down the range of values of x when the gradient of the curve is negative.

Find an estimate of the gradient of the curve when *x *= 2.

The curve $C$ has equation $y=ax^{3}+bx^{2}--12x+6$ where $a$ and $b$ are constants.

The point $A$ with coordinates (2, –6) lies on $C$.

The gradient of the curve at $A$ is 16.

Find the $y$ coordinate of the point on the curve whose $x$ coordinate is 3. Show clear algebraic working.

Part of the curve with equation $y=f(x)$ is shown on the grid.

Find an estimate for the gradient of the curve at the point where $x=2$ Show your working clearly.

A particle $P$ is moving along a straight line.

The fixed point $O$ lies on the line.

At time t seconds $( t \text{greater} - \text{than} or \text{slanted} \text{equal} to 0 )$, the displacement of $P$ from $O$ is s metres where

$s=t^{3}--9t^{2}+33t--6$

Find the minimum speed of $P$.

...................................................... m/s

$A$ and $B$ are points on a curve.

$A$ is $(2,7)$ $B$ is $(12,0)$

Work out the instantaneous rate of change of $y$ with respect to $x$ at point $A$.

The average rate of change of $y$with respect to $x$ between points $A$ and $B$ is worked out.

Which statement is correct? Tick one box.

$ABCED$ is a five-sided shape.

$ABCD$ is a rectangle. $CED$ is an equilateral triangle.

$AB=x\mathrm{cm}BC=y\mathrm{cm}$

The perimeter of $ABCED$is 100 cm. The area of $ABCED$is $R$ cm²

Show that $R = \frac{x}{4} ( 200 - [ 6 - \sqrt{3} ] x )$

(i) Find the value of $x$ for which $R$ has its maximum value.

Give your answer in the form $\frac{p}{q-\sqrt{3}}$ where $p$ and $q$ are integers.

$x=$ ....................................................... [2]

(ii) Explain why the maximum value of $R$ is given by this value of $x$.

A container is filled with water in 5 seconds.

The graph shows the depth of water, $d$ cm, at time $t$ seconds.

The water flows into the container at a constant rate.

Which diagram represents the container? Circle the correct letter.

Use the graph to estimate the rate at which the depth of water is increasing at 3 seconds.

You must show your working.

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

A particle moves along a straight line. The fixed point $O$ lies on this line. The displacement of the particle from $O$ at time $t$ seconds , $t \text{greater} - \text{than} or \text{slanted} \text{equal} to 0$ , is $s$ metres where

$s=t^{3}+4t^{2}-5t+7$

At time $T$ seconds the velocity of $P$ is $Vm/s$ where $V \text{greater} - \text{than} or \text{slanted} \text{equal} to - 5$

Find an expression for $T$ in terms of $V$.

Give your expression in the form $\frac{-4+\sqrt{k+mV}}{3}$ where $k\quad m$ and are integers to be found.

$T=...............$

A ball is thrown from a point 6 metres above the ground.

The graph shows the height of the ball above the ground, in metres.

Estimate the speed of the ball, in m/s, after 1 second. You must show your working.

...............................m/s

The graph shows the distance travelled by a particle over 8 seconds.

Estimate the speed of the particle at 5 seconds.

................................................... m/s

The graph shows the speed, v metres per second (m/s), of a car at time t seconds.

Use the graph to estimate the acceleration at* t *= 7.

...................................................m/s²

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 $D$.

Use differentiation to find $\frac{dy}{dx}$ for the following:

$y=x^{4}$

$y=2x^{-3}$

$y=\frac{4}{x}$

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.

Use differentiation to find $\frac{dy}{dx}$ for the following:

$y=4x^{3}+2x$

$y=-5x^{-2}$

$y=\frac{1}{3x}$

The diagram shows the graph of $y=\frac{x}{2}+\frac{2}{x}$for $0<x\leq8.$

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.

Use differentiation to find $\frac{dy}{dx}$ for the following:

$y=2x^{3}-6x^{2}+3x-4$

$-\frac{5}{3x^{4}}$

$\frac{2}{3}x^{2}+\frac{1}{5}x-\frac{3}{2x}$

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.

For the curve with equation $y=2x^{2}-6x-11$:

find $\frac{dy}{dx}$

Find the coordinates of the point on the curve where the gradient is 2.

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.

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 curve has equation $y=x^{3}+\frac{7}{2}x^{2}-2x+9$

Find $\frac{dy}{dx}$

Find the gradient of the curve at the point where:

(i) $x=-3$

(ii) $x=\frac{2}{3}$

What can you say about the tangents to the curves at these two points?

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.

The diagram shows the graph of $y=f(x)$ for $- 4 \text{less} - \text{than} or \text{slanted} \text{equal} to x \text{less} - \text{than} or \text{slanted} \text{equal} to 12$

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 $P$ passes the fixed point $O$ whilst moving along a straight line.

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

$s=6t^{3}-12t^{2}+7t$

Find expressions for the velocity, $vm/s$, and the acceleration, $am/s^{2}$ of the particle at time $t$ seconds.

Find the time at which the acceleration is 3 $m/s^{2}$.

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

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.

Part of the curve with equation $y=h(x)$ 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.

The curve $C$ has equation $y=5x^{3}--x^{2}--6x+4$.

Find $\frac{dy}{dx}$.

$\frac{dy}{dx}$ = ..............................................

There are two points on the curve $C$ at which the gradient of the curve is $2$.

Find the $x$ coordinate of each of these two points. Show clear algebraic working.

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?

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

$y=x^{3}--6x^{2}--15x$.

Find $\frac{dy}{dx}$.

$\frac{dy}{dx}=$....................................

The curve with equation $y=x^{3}--6x^{2}--15x$ has two stationary points.

Work out the coordinates of these 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 $k$.

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

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

find $\frac{dy}{dx}$

find the coordinates of the stationary points on the curve.

find the exact distance between the two stationary points.

The curve $C$ has equation $y=\frac{1}{3}x^{3}-9x+1$.

Find $\frac{dy}{dx}$.

Find the range of values of $x$ for which $C$ has a negative gradient.

The velocity-time graph shows the first 40 seconds of a car in a race.

Work out the average acceleration for the first 40 seconds. Give the units of your answer.

Estimate the time during the 40 seconds when the instantaneous acceleration = the average acceleration.

You must show your working on the graph.

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?

Calculate the gradient of $y=24+5x-x^{2}$ at $x=-1.5$.

The graph gives information about the variation in the temperature of an amount of water that is left to cool from 80° C.

Work out an estimate for the rate of decrease of temperature at $t$= 300.

Work out the average rate of decrease of the temperature of the water between $t$ = 0 and $t$ = 800.

The instantaneous rate of decrease of the temperature of the water at time $T$ seconds is equal to the average rate of decrease of the temperature of the water between $t$ = 0 and $t$ = 800.

Find an estimate for the value of $T$. You must show how you got your answer.

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 $\frac{dA}{dW}$

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.

Differentiate $6+4x-x^{2}$.

Find the coordinates of the turning point of the graph of $y=6+4x-x^{2}$ .

( ...................... , ...................... )

$y=x^{3}-6x^{2}-15x$

Find $\frac{dy}{dx}$

The curve with equation $y=x^{3}-6x^{2}-15x$ has two stationary points.

Work out the coordinates of these two stationary points.

The diagram shows part of the graph of $y=x^{2}---2x+3$

By drawing a suitable straight line, use your graph to find estimates for the solutions of $x^{2}---3x---1=0$

$P$ is the point on the graph of $y=x^{2}---2x+3$where $x=2$

Calculate an estimate for the gradient of the graph at the point $P$.

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.

The curve C has equation $y=4x^{3}+x^{2}-20x$

Find $\frac{dy}{dx}$

Find the $x$coordinates of the points on C where the gradient is 4 Show clear algebraic working.

Clare emptied a tank and recorded the depth of water each minute.

Plot the graph of depth against time.

Work out the average rate of decrease of the depth of the water in Clare's tank between $t$ = 0 and $t$ = 10.

Find an estimate for the gradient at $t$ = 7.

Give an interpretation of part (c).

A particle $P$ is moving along a straight line. The fixed point $O$ lies on this line. At time $t$ seconds where $t \text{greater} - \text{than} or \text{slanted} \text{equal} to 0$, 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 curve has equation $y=4x^{3}-8x+5$

Find the $x$ coordinates of the two points on the curve where the gradient is $\frac{1}{3}$

Olympic medallist Usain runs in a race. The graph shows his speed, in metres per second (m/s), during the first 10 seconds of the race.

Use the graph to find how long it took Usain to reach his top speed.

Work out an estimate for Usain's acceleration at 2 seconds. Give the units of your answer.

Calculate the difference in Usain's acceleration between 2 and 6 seconds.

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.