Estimating Gradients

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 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  = 3.

Interpret the value of the gradient.

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.

The diagram shows part of the graph of 

By drawing a suitable straight line, use your graph to find estimates for the solutions of 

is the point on the graph of  where 

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

The curve 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.

Part of the curve with equation is shown on the grid.

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

and are points on a curve.

is    is

Work out the instantaneous rate of change of with respect to at point .

The average rate of change of with respect to between points and is worked out.

Which statement is correct?
Tick one box.

	It is positive.
	It is zero.
	It is negative.
	You cannot tell if it is positive or negative.

A container is filled with water in 5 seconds.
The graph shows the depth of water, cm, at time 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 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 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 = 300.

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

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

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

The diagram shows part of the graph of  

By drawing a suitable straight line, use your graph to find estimates for the solutions of 

is the point on the graph of where 

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

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 = 0 and = 10.

Find an estimate for the gradient at = 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  .

	-3	-2.5	-2	-1.5	-1	0	1	1.5	2	2.5	3
	-19	-9.1		0.1	1	-1	-3	-2.1	1	7.1

 
Complete the table of values.
  

Draw the graph of    for  .

A straight line through (0, –17) is a tangent to the graph of  .

[1]

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

Find the equation of the tangent.
Give your answer in the form  .
 

 = ................................................ [3]