Intro to differential calculus

Introduction to Differential Calculus

Using rates of change

​What is differential Calculus?

Differential calculus focuses on rates of change. A rate of change describes the speed at which one variable changes with respect to another.
Think - Pair - Share
what situations can you think of that knowing the rate of change could be useful?

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Review - Constant rate of change

​WHAT IS CONTINUOUS RATE OF CHANGE?

Review - Constant rate of change

In calculating and interpreting the gradient of linear functions we have encountered rates of change previously. Let's review rates of change for linear functions.

​This graph of distance against time forms a straight line, as it would for any two variables where the rate of change is constant. The rate of change, in this case the speed, is given by the change in distance divided by the change in time. This can be calculated using any two points from the table or graph.

Review - Constant rate of change

The rate of change of a linear function is given by the gradient of the line. The units for a rate of change will be the dependent variable units divided by the independent variable units. 

In calculating and interpreting the gradient of linear functions we have encountered rates of change previously. Let's review rates of change for linear functions.

Multiple Choice

The graph shows the progress of two competitors in a BMX race.

Who is travelling faster?

• Uther

• Maximilian

Fill in the Blanks

Multiple Choice

Does the function that passes through the following points:

{(−2,−5),(1,−20),(2,−25),(7,−50),(9,−60)}

have a constant or a variable rate of change?

• Constant

• Variable

​

In reality, rate of change varies, rather than stays constant.

​Average Rate of Change

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We can still calculate the change in distance divided by the change in time between two points for a journey with a variable speed, this calculation will now give us the average rate of change.

This line that intersects a curve at two distinct points is called a secant.

The gradient of the secant is the average rate of change.

​Average Rate of Change

​

To calculate the average rate of change between two points,

we calculate the change in the dependent variable divided by the change in the independent variable.

This is equivalent to calculating the gradient of the secant passing through points A and B.

​Average Rate of Change

​

In calculus function notation is commonly used.
The equivalent statement for the average rate of change between x=a and x=b is:

Average rate of change equals

​Average Rate of Change

Open Ended

The volume of a lake over a 5 week period has been recorded.

What is the average rate of change in the first week?

Open Ended

The volume of a lake over a 5 week period has been recorded.

What is the average rate of change over the whole 5 week period?

Open Ended

The volume of a lake over a 5 week period has been recorded.

What was the average rate of change of volume in the last 3 weeks?

Open Ended

A population of rabbits is growing according to the function: P(t)=200×1.08t, where t is time in months.

(a) Find the average rate of change in the population between 2 and 4 months.

(b) Find the average rate of change in population between 4 and 7 months.

Instantaneous rates of Change

Instantaneous rate of change is the rate of change at a particular point  

Instantaneous rates of Change

Take for example a trip from Esperance to Perth, the average speed for the journey, which is the total distance travelled divided by the time taken. If I drove from Esperance to Perth in 9hrs, and the distance travelled is 714km, then my average speed is 714/9 km/h = 79.3km/h. 

​However, it is impossible to drive this route with a constant speed. Instead, it is more likely I went 110km/h on the highway and 50km/h through the small towns.
The specific points is the instantaneous rate of change.

Instantaneous rates of Change

But how do we calculate instantaneous rates of change?

If we think about the formula for average rates of change, for our example on the previous slide we have: 
Average speed=change distance divided by change in time​. However, in any particular instant, no distance has been travelled and no time has elapsed so we need a new method.

​An object is moving according to the distance-time graph shown below. We want to estimate for the instantaneous rate of change at point A.

​Think: How can we estimate the speed at this point?

​

​imagine if at point A the object then continued from this point at the speed it had at point A rather than continuing along the curve. The object would follow a straight line, if we extend this line in both directions, we obtain the line shown.

​

A ​line that touches a curve and has a gradient matching the rate of change of the curve at the point of contact is called a tangent. Therefore, we can estimate the instantaneous rate of change at A by calculating the gradient of the tangent.

An alternative to sketching a tangent at point A is finding the average rate of change between A and a point, B, which is very close to A.

​Predict - what do you think is going to happen to the value of our rate of change as we move point B closer to A?

Your task

Use a spreadsheet to estimate the instantaneous rate of change at x = 3 for the following functions. You will do this by selecting an appropriate point B and decreasing it so it approaches point A.

1 - f(x) = 3x^2

2 - f(x)=x^3-3x^2+2x-1
3 - f(x) = 4^x

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​This is an example of how you should set up your spreadsheet and the formulas you should use.

Open Ended

What were your estimates of the instantaneous rates of change?

What did you notice as you decreased the difference between points A and B?

Let's Practice!

Math Response

Consider the graph given by the function f(x) = 5(2)^x and the tangent line at x = 1 is given by y = 7x + 3

What is the instantaneous rate of change?

Type answer here

Correct Answer: 7

Deg^°

Rad

Math Response

Consider this function

use your preferred method to estimate the instantaneous rate of change at x = 5

Type answer here

Correct Answer: 2

Deg^°

Rad

Independent Practice

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Textbook work
​Cambridge Mathematical Methods Units 1&2 for Western Australia
Exercise 17B page 540

Open Ended

What is constant rate of change?

Open Ended

How can you calculate the average rate of change?

Be specific and use appropriate terminology

Open Ended

What is the name of the line that you draw and use to find the instantaneous rate of change?

Poll

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