Estimating Gradients (Edexcel IGCSE Maths A: Higher): 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 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 $y=mx+c$ .

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.

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 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.

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.

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 $P$.

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).

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.

The table shows some values of  $y=x^3-3x-1$.

  Complete the table of values.   

Draw the graph of  $y=x^3-3x-1$  for  $-3\le x\le 3$.

A straight line through (0, –17) is a tangent to the graph of  $y=x^3-3x-1$.

(i) On the grid, draw this tangent.    

[1]

(ii) Find the co-ordinates of the point where the tangent meets your graph.   

(................ , ................) [1]

(iii) Find the equation of the tangent. Give your answer in the form  $y=mx+c$.  

$y$ = ................................................ [3]