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Identifying Rates

Identifying Rates. Rates describe how much one quantity changes with respect to another. Can you think of any Rates that you have seen or heard in your everyday lives?. Examples. Try these 2 Examples. Example 2. Example 1.

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Identifying Rates

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  1. Identifying Rates

  2. Rates describe how much one quantity changes with respect to another. Can you think of any Rates that you have seen or heard in your everyday lives?

  3. Examples

  4. Try these 2 Examples Example 2 Example 1 • A car travels 259 kilometres using 35 litres of petrol. Express this rate in km/L. • Which of the following represent a rate? • a20 m/s • b75 cents per packet • c$13

  5. Answer 2 Answer 1

  6. Constant rate of change

  7. When the rate of change of one quantity with respect to another does not alter, the rate is constant. • if petrol is $1.60 per litre, then every litre of petrol purchased at this rate always costs $1.60. This means 10 litres of petrol would cost $16.00 and 100 litres of petrol would cost $160.00. Calculating the gradient from the graph

  8. Example

  9. Solution

  10. Example 2

  11. Solution

  12. Variable rates

  13. If a rate is not constant (is changing), then it must be a variable rate.

  14. Example

  15. Solution

  16. What is an average rate? If a rate is variable, it is sometimes useful to know the average rate of change over a specified interval.

  17. Example 1

  18. Solution

  19. Example 2

  20. Solution

  21. Example 3

  22. Solution

  23. Solution

  24. Instantaneous Rates

  25. What is an instantaneous Rate? • If a rate is variable, it is often useful to know the rate of change at any given time or point, that is, the instantaneous rate of change. • For example, a police radar gun is designed to give an instantaneous reading of a vehicle's speed. This enables the police to make an immediate decision as to whether a car is breaking the speed limit or not.

  26. Calculating Instantaneous Rates: drawing a tangent to the curve at the point in question calculating the gradient of the tangent over an appropriate interval (that is, between two points whose coordinates are easily identified).
Note: The gradient of the curve at a point, P, is defined as the gradient of the tangent at that point.

  27. Example 1 aUse the following graph to find the gradient of the tangent at the point where L = 10. bHence, find the instantaneous rate of change of weight, W, with respect to length, L, when L = 10.

  28. Solution

  29. Example 3

  30. Solution

  31. Soultion

  32. Rates of change of polynomials

  33. Rates of change of polynomials • We have seen that instantaneous rates of change can be found from a graph by finding the gradient of the tangent drawn through the point in question. The following method uses a series of approximations to find the gradient.

  34. Example

  35. Example

  36. Solution

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