Optimisation & Rates of Change (Grade 12) — Worksheet & Memo

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Grade 12 · Calculus

Optimisation & Rates 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
A rectangle has a perimeter of 40 m. Determine its maximum possible area.
(2)
A = x(20 − x), A′ = 20 − 2x = 0 → x = 10 → 100 m²✓✓ (2)
2
The height of a ball is s(t) = 20t − 5t². Determine its velocity after 1 second.
(2)
v(t) = 20 − 10t → v(1) = 10 m·s⁻¹✓✓ (2)
3
For the same ball, determine when it reaches its maximum height.
(2)
v = 0 → 20 − 10t = 0 → t = 2 s✓✓ (2)
4
For the same ball, determine its maximum height.
(2)
s(2) = 40 − 20 = 20 m✓✓ (2)
5
For the same ball, determine its velocity when it strikes the ground.
(2)
s = 0 at t = 4 → v(4) = −20 m·s⁻¹✓✓ (2)
6
Determine the acceleration of the ball.
(2)
a(t) = v′(t) = −10 m·s⁻² (constant)✓✓ (2)
7
Why must you use the constraint before differentiating?
(2)
Because you cannot differentiate a formula containing two variables — the constraint eliminates one.✓✓ (2)
8
What does a negative velocity tell you?
(2)
The object is moving in the opposite direction (downwards, here).✓✓ (2)

TOTAL: 16 marks