Estimating Gradients & Areas under Graphs Practice Questions

Work out the average rate at which the volume of water in the pool increases between $t = 2$ and $t = 8$.

Give the units with your answer.

Explain how the table shows that water is flowing into the pool at an increasing rate.

Write down what the area under this graph represents.

Write down what the gradient of this graph represents. Give the units.

Is Jay's estimate an underestimate or an overestimate of the actual distance travelled? Give a reason for your answer.

$A$ is the point on the curve where $x = 2$ and $B$ is the point on the curve where $x = 4$.

Work out the gradient of the chord $AB$.

By drawing a suitable tangent, calculate an estimate for the gradient of the curve at the point $B$.

Work out the average speed of the ball over the first 6 seconds.

Use the graph to estimate the speed of the ball at time 4 seconds.

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

Work out an estimate for the distance the drone travels in the first 16 seconds.

Use 4 strips of equal width.

Is your answer to part (a) an underestimate or an overestimate of the actual distance travelled? Give a reason for your answer.

Work out the average rate of change of the value of the van over the 4 years.

Give the units with your answer.

The value of the van is plotted against time and a smooth curve is drawn through the points.

Use the table to explain how the steepness of the curve changes as $t$ increases, and what this means for the value of the van.

Work out the average rate of change of the temperature of the coffee between $t = 0$ and $t = 10$.

Give the units with your answer.

A smooth curve is drawn through the points. The tangent to the curve at $t = 15$ passes through the points $(10, 48)$ and $(20, 40)$.

Use this tangent to find the rate at which the temperature of the coffee is falling at $t = 15$.

Work out an estimate for the area of the region bounded by the curve $C$, the $x$-axis, the line $x = 1$ and the line $x = 4$.

Use 3 strips of equal width.

Explain whether your answer to part (a) is an underestimate or an overestimate of the actual area.

The actual area of the region is $16.64$, correct to 4 significant figures.

Work out the percentage error in your estimate from part (a).

Work out the average speed of the train between $t = 3$ and $t = 5$.

Work out the gradient of the chord joining the point on the graph of $s$ against $t$ where $t = 3$ to the point where $t = 3.1$.

The gradient of the chord from $t = 3$ to $t = 3.01$ is $15.02$.

Using this information and your answers to parts (a) and (b), write down the speed of the train at $t = 3$. Give a reason for your answer.