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

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Grade 12 · Functions and Graphs

Average Gradient & Average Rate of Change (Grade 12)

MEMORANDUM

Name:

Class:

Date:

Mark

/ 14

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 − 2)/(3 − 1) = 4✓✓ (2)
2
Determine the average gradient of f(x) = x² + 1 between x = −1 and x = 2.
(2)
(5 − 2)/(2 + 1) = 1✓✓ (2)
3
Write down the formula for average gradient between x = a and x = b.
(1)
(f(b) − f(a))/(b − a)✓ (1)
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).
(3)
4,001… which tends to 4 as the interval shrinks✓✓✓ (3)
6
What does the average gradient become as the two points move together?
(1)
The derivative — the gradient of the tangent at that point.✓ (1)
7
A car travels 240 km in 3 hours. What is its average rate of change of distance?
(1)
240/3 = 80 km/h✓ (1)
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: 14 marks