Introduction to Rates of Change Revision Notes for HSC SSCE Mathematics Advanced

Instantaneous Speed and Tangents

Understanding instantaneous speed

Instantaneous speed measures how fast an object is moving at one specific moment in time, rather than over a period. This is different from average speed, which is calculated over an interval.

When we look at a distance-time graph, the instantaneous speed at any point corresponds to the gradient of the tangent to the curve at that point. The tangent represents the exact slope of the curve at a single instant.

Learning objectives:

Relate the instantaneous speed of an object to the gradient of the tangent at a point on its distance-time graph
Estimate the instantaneous speed by calculating average speed over a very small interval
Estimate the instantaneous speed by drawing a tangent to the distance-time graph and calculating its gradient

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The key difference between instantaneous and average speed: instantaneous speed captures motion at a single moment, while average speed describes motion over a time interval. Think of instantaneous speed as your speedometer reading at one exact second, while average speed is your total distance divided by total time for a journey.

What is a tangent?

A tangent is a straight line that touches a curve at exactly one point. At this point of contact, the tangent has the same direction as the curve itself. Think of it as the best straight-line approximation to the curve at that specific point.

The tangent is fundamentally different from other lines that might intersect the curve. It captures the curve's behaviour at one precise location, making it perfect for describing instantaneous rates of change.

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Key property of tangents: A tangent line "just touches" the curve at exactly one point and has the same slope as the curve at that point. This single-point contact makes tangents ideal for measuring instantaneous rates of change.

Gradient of tangent as instantaneous speed

The instantaneous speed of an object at time $t$ is the rate of change of distance at that exact moment. On a distance-time graph, this rate of change is represented by the gradient of the tangent to the curve at time $t$ .

The tangent line touches the curve at a single point, representing the slope of the curve at that instant. This gradient tells us exactly how fast the object is moving at that particular moment.

In the graph above, you can see the difference between:

The secant line (the straight line from $t = 2$ to $t = 4$ ) which gives the average speed over that interval
The tangent line (shown touching the curve at $t = 3$ ) which gives the instantaneous speed at that specific moment

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Secant vs Tangent:

A secant line connects two points on a curve and its gradient gives the average speed between those points
A tangent line touches the curve at exactly one point and its gradient gives the instantaneous speed at that point
As the two points on a secant get closer together, the secant approaches the tangent

For a distance function $d(t)$ , the instantaneous speed at $t = a$ is the limit of the average speed as the time interval becomes smaller and smaller. As we shrink the interval, the secant line approaches the tangent line, and the average speed approaches the instantaneous speed.

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The fundamental concept: Instantaneous speed is what happens when we calculate average speed over an interval that shrinks to zero. Mathematically, this is the limit process that defines the tangent's gradient.

Estimating instantaneous speed from graphs

To estimate instantaneous speed from a distance-time graph, we draw a tangent at the desired time and calculate its gradient. The formula we use is:

$\text{Instantaneous speed} = \frac{\Delta d}{\Delta t}$

where $\Delta d$ represents the change in distance and $\Delta t$ represents the change in time.

In practice, we approximate the tangent by drawing a line that closely follows the curve's slope at the point of interest. We then select two convenient points on this tangent line and calculate the gradient using $\frac{\Delta d}{\Delta t}$ .

chatImportant

Critical principle: The smaller the interval we use, the more accurate our approximation will be. A tiny interval means the secant line is very close to the tangent line, giving us a better estimate of the instantaneous speed.

Worked example 1: Estimating speed using a small interval

lightbulbExample

Worked Example: Using Small Intervals

Problem: A car's distance is modelled by $d(t) = 0.5t^2$ , where $d$ is in metres and $t$ is in seconds. Estimate the instantaneous speed at $t = 3$ .

Strategy: We estimate the instantaneous speed at $t = 3$ by calculating the average speed over a small interval, such as $[3, 3.1]$ .

Solution:

Write the equation: $d(t) = 0.5t^2$

Substitute $t = 3$ : $d(3) = 0.5 \times 3^2$

$= 4.5$

Write the equation again: $d(t) = 0.5t^2$

Substitute $t = 3.1$ : $d(3.1) = 0.5 \times 3.1^2$

$= 4.805$

Write the formula for approximating instantaneous speed: $\text{Approximate instantaneous speed} = \frac{d(t_2) - d(t_1)}{t_2 - t_1}$

Substitute $t_2 = 3.1$ and $t_1 = 3$ : $= \frac{d(3.1) - d(3)}{3.1 - 3}$

Substitute $d(3.1) = 4.805$ and $d(3) = 4.5$ : $= \frac{4.805 - 4.5}{3.1 - 3}$

Evaluate the subtraction: $= \frac{0.305}{0.1}$

$= 3.05$

Answer: The instantaneous speed at $t = 3$ is approximately $3.05$ m/s.

Reflection: The small interval $[3, 3.1]$ ensures the secant line closely approximates the tangent, providing a reasonable estimate. Using an even smaller interval would give an even better approximation.

Key insight: The instantaneous speed at time $t$ is the gradient of the tangent to the distance-time graph at that point, representing the exact speed at that moment.

Worked example 2: Drawing a tangent to estimate speed

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Worked Example: Graphical Method

Problem: The distance-time graph shows a vehicle's journey. Estimate the instantaneous speed at $t = 2$ hours by drawing a tangent and calculating its gradient.

Strategy: We use the tangent line approximated by the secant from $t = 1$ to $t = 3$ passing through $t = 2$ . We calculate the gradient using $\frac{\Delta d}{\Delta t}$ .

Solution:

Calculate change in distance: $\Delta d = 90 - 10$

$= 80$

Calculate change in time: $\Delta t = 3 - 1$

$= 2$

Write the formula: $\text{Instantaneous speed} = \frac{\Delta d}{\Delta t}$

Substitute the values: $= \frac{80}{2}$

Evaluate: $= 40$

Answer: The instantaneous speed at $t = 2$ is approximately $40$ km/h.

Reflection: The secant line from $t = 1$ to $t = 3$ approximates the tangent at $t = 2$ . A smaller interval would refine the estimate further.

Key insight: Instantaneous speed can be estimated from a distance-time graph by calculating the gradient of a tangent at the given time, using points on an approximate tangent line.

Remember!

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Key Points to Remember:

Instantaneous speed is the speed at one exact moment in time, represented by the gradient of the tangent to a distance-time graph at that point
A tangent is a straight line that touches a curve at exactly one point and has the same direction as the curve at that point
The formula for instantaneous speed is $\frac{\Delta d}{\Delta t}$ , where $\Delta d$ is the change in distance and $\Delta t$ is the change in time
To estimate instantaneous speed from a graph, draw a tangent at the desired time and calculate its gradient using two points on the tangent
Using smaller time intervals gives more accurate approximations of instantaneous speed

Introduction to Rates of Change (HSC SSCE Mathematics Advanced): Revision Notes

Instantaneous Speed and Tangents

Understanding instantaneous speed

What is a tangent?

Gradient of tangent as instantaneous speed

Estimating instantaneous speed from graphs

Worked example 1: Estimating speed using a small interval

Worked example 2: Drawing a tangent to estimate speed

Remember!

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