Estimating Gradients & Areas under Graphs

Karol runs in a race.

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

Calculate an estimate for the gradient of the graph when 
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.

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.

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  = 10.

Give an interpretation 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.

The speed-time graph shows 20 seconds of a car journey.

Harry wants to estimate the distance the car travels in this time.

He uses a triangle and a trapezium, as shown, to estimate the area under the graph.

Complete Harry’s method to estimate the distance the car travels.

.......................m

For this journey, which of these is true for Harry’s method?
Tick one box.

     It works out an overestimate of the distance

       It works out an underestimate of the distance

     It could work out an overestimate or an underestimate of the distance

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 .

Leo runs for 12 seconds.

The graph shows his speed.

Show that the distance he runs is less than 67.5 metres.

Work out his average acceleration for the first 9 seconds.

State the units of your answer.

A car moves from rest.

The graph gives information about the speed, metres per second, of the car seconds after it starts to move.

Calculate an estimate of the gradient of the graph at = 15

Work out an estimate for the distance the car travels in the first 20 seconds of its journey.
Use 4 strips of equal width.

The diagram shows the graph of  for 

The point on the curve has  coordinate 2

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

Hence find an equation of the tangent to the curve at .
Give your answer in the form  .