Unit 4: Introductory Differential Calculus
4.1 Rates of Change
Rate of Change – a comparison between two quantities of different kinds
Units in rates of change are invariably in the form <UNIT> per <UNIT>
words/minute, km/h, $/h
Ex 1 Consider a trip from Adelaide to Melbourne
|
Place |
Time taken (min) |
Distance Travelled (km) |
|
Adelaide tollbooth |
0 |
0 |
|
Tailem Bend |
63 |
98 |
|
Bordertown |
157 |
237 |
|
Nhill |
204 |
324 |
|
Horsham |
261 |
431 |
|
Ararat |
317 |
527 |
|
Midland H/W Junction |
386 |
616 |
|
Melbourne |
534 |
729 |
Average Rate of Change – the gradient of the line segment joining the two points. A line that touches a graph in two points is called a secant.
(a) Calculate the average speed from Bordertown to Nhill.
The same process works for curves.
Ex 2 is on the projector
Estimate the average rate of change for the period from week 3 to week 6.
Instantaneous Rate of Change
From the last example:
What is the rate of change of the number of rats in the previous example at the beginning of week 4?
To do this: we're going to create a secant between two points and examine what happens as the distance between them shrinks.
Ex 3 Another example, with a cubic
p. 602#1-5 p. 604# 1-4 p. 605# 1-2 p. 607# 1-2