The following table gives the amount of water that is used on a farm during the first six months of the year.

\displaystyle \begin{array}{|l|c|} \hline Month & Volume \left(1000\right. of \left.\mathbf{m}^{3}\right) \\ \hline January & 3.00 \\ \hline February & 3.75 \\ \hline March & 3.75 \\ \hline April & 4.00 \\ \hline May & 5.10 \\ \hline June & 5.50 \\ \hline \end{array}

a) Plot the data in the table on a graph.

b) Find the rate of change in the volume of water used between consecutive months.

c) Between which two months is the change in the volume of water used the greatest?

d) Determine the average rate of change in the volume of water used between March and June.

A city’s population (in tens of thousands) is modelled by the function P(t) = 1.2(1.05)^t, where t is the number of years since 2000. Examine the equation for this function and its graph.

a) What can you conclude about the average rate of change in population between consecutive years as time increases?

b) Estimate the instantaneous rate of change in population in 2010.

The height of a football that has been kicked can be modelled by the function

\displaystyle h(t) = -5t^2 + 20t + 1 , where h(t) is the height in metres and t is the time in seconds.

a) What is the average rate of change in height on the interval 0\leq t \leq 2 and on the interval 2 \leq t \leq 4?

b) Use the information given in part a) to find the time for which the instantaneous rate of change in height is 0 m/s. Verify your response.

The movement of a certain glacier can be modelled by d(t) = 0.01t^2 + 0.5t, where d is the distance, in metres, that a stake on the glacier has moved, relative to a fixed position, t days after the first measurement was made. Estimate the rate at which the glacier is moving after 20 days.

Create a graphic organizer, such as a web diagram, mind map, or concept map, for rate of change. Include both the average rate of change and instantaneous rate of change in your graphic organizer.

Create a table to estimate the slope of the tangent to \displaystyle y=x^{3}+1 at \displaystyle P(2,9) . Be sure to approach \displaystyle P from both directions.

Estimate the slope of the tangent line in the graph of this function.

Estimate the slope of the tangent line in the graph of this function.

What does the slope of the tangent represent?

Graph the function f(x)=0.5x^2 + 5x - 15 using your graphing calculator. Estimate the instantaneous rate of change for each value of x.

x = -5

Graph the function f(x)=0.5x^2 + 5x - 15 using your graphing calculator. Estimate the instantaneous rate of change for each value of x.

x =0

Graph the function f(x)=0.5x^2 + 5x - 15 using your graphing calculator. Estimate the instantaneous rate of change for each value of x.

x =-1

Graph the function f(x)=0.5x^2 + 5x - 15 using your graphing calculator. Estimate the instantaneous rate of change for each value of x.

x =3