The Instantaneous Rate Lab
How fast is something changing at a single instant? Draw a tangent, measure its gradient, and use the same rate-of-change thinking to grow and shrink money with compound multipliers.
Get the method right under pressure
Free interactive practice on the steps that lose marks under exam pressure.
Start revising freeWhat you'll cover
How fast, right now?
Average speed over a journey is easy. But how fast are you going at one instant? The instantaneous rate of change answers that. On a curved graph it is the steepness at a single point, and the same idea powers compound growth in money.
Rate and tangent words
Four terms run through this module:
Average or instantaneous?
The gradient of a straight line on a curve tells you a rate, but which one depends on the line.
Gradient of a tangent
A tangent to a curve passes through the points (2, 10) and (6, 30). Work out its gradient, using gradient = rise / run.
Match each graph to what the gradient means
- A distance-time graph
- A velocity-time graph
- A tangent at a point
- A chord between two points
- The gradient is the speed
- The gradient is the acceleration
- The instantaneous rate at that point
- The average rate between the points
Draw, pick, divide
To find the instantaneous rate at a point on a curve: draw the tangent at that point, pick two points far apart on it, then work out rise / run. At GCSE this is done by drawing and measuring, not by calculus.
Plot the distance
On this distance-time graph the distance follows d = 2t^2, where t is time in seconds and d is distance in metres. The points at t = 0, 1, 2 and 3 are already marked. Work out d when t = 4 and plot that point.
Compound growth
2000 pounds is invested at 5% compound interest for 2 years. Using the multiplier 1.05, find the amount after 2 years, in pounds.
Why compound wins
Why does compound interest give more than simple interest over time?
- Interest is earned on interest already added
- The interest rate is always higher
- The money is taxed less
- The bank adds a fixed bonus
Rates of change
Tap the TWO statements that are TRUE.
- The instantaneous rate of change is the gradient of the tangent at a point.
- Compound interest earns interest on interest already added.
- A chord gives the instantaneous rate at a single point.
- At GCSE you find the instantaneous rate using calculus.
- Simple and compound interest always give the same amount.
The method in order
Put the steps for finding an instantaneous rate from a curve into order.
- Draw the tangent to the curve at the point
- Pick two points far apart on the tangent
- Work out the rise and the run between them
- Divide the rise by the run to get the gradient
Rates in words
The instantaneous rate of change at a point is the gradient of the _____ drawn there. To find it, you work out the rise divided by the _____. On a distance-time graph this gradient is the _____. When money grows by 4% a year it is multiplied by 1.04 each year, which is called _____ interest. To find a value after a percentage decrease, you multiply by a number less than _____.
Pick the right tool
Decide what each situation needs.
- You want the exact speed of a car at one instant on a curved distance-time graph. What do you draw?
- Savings of 1000 pounds grow by 3% each year. How do you find the amount after 5 years?
- A laptop worth 500 pounds loses 20% of its value each year. Which multiplier finds its value after each year?
Explain the rate
Explain how to find the instantaneous rate of change at a point on a curve, and explain how it is different from the average rate of change.
- Explain what the instantaneous rate of change means
- Describe how to find it using a tangent and rise / run
- Explain what a chord gives instead
- Give one real context, such as speed or money