Average Gradient & Average Rate of Change (Grade 12) — Worksheet & Memo

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Grade 12 · Functions

Average Gradient & Average Rate of Change (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 16

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Determine the average gradient of f(x) = x² + 1 between x = 1 and x = 3.
(2)
10 − 23 − 1 = 4✓✓ (2)
2
Determine the average gradient of f(x) = x² + 1 between x = −1 and x = 2.
(2)
5 − 22 + 1 = 1✓✓ (2)
3
Write down the formula for average gradient between x = a and x = b.
(2)
f(b) − f(a)b − a✓✓ (2)
4
Is the average gradient the same as the gradient at a point? Explain.
(2)
No — it is the gradient of the straight line joining two points; the gradient at a point is the gradient of the tangent there.✓✓ (2)
5
Determine the average gradient of f(x) = x² between x = 2 and x = 2,001 (three decimals).
(2)
4,001… which tends to 4 as the interval shrinks✓✓ (2)
6
What does the average gradient become as the two points move together?
(2)
The derivative — the gradient of the tangent at that point.✓✓ (2)
7
A car travels 240 km in 3 hours. What is its average rate of change of distance?
(2)
2403 = 80 km/h✓✓ (2)
8
Determine the average gradient of f(x) = 2x + 5 between any two points.
(2)
2 — for a straight line the average gradient is the same everywhere.✓✓ (2)

TOTAL: 16 marks