Applications of Derivatives

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📚 Part 1: Multiple Choice (Choose the best answer)

1. Given a position function s(t) = t³ − 6t² + 9t (t in seconds, s in metres). What is the velocity v(2)?

−6 m/s

−3 m/s

0 m/s

3 m/s

2. For the same s(t), at which times is the object at rest (v(t) = 0)?

t = 0 and t = 3 s

t = 1 s and t = 3 s

t = 2 s only

t = 1 s and t = 2 s

3. A rectangle has fixed perimeter P = 40 m. Let one side be x. For A(x) = x(20 − x), the value of x that gives maximum area is:

x = 5 m

x = 10 m

x = 15 m

x = 20 m

4. For f(x) = x⁴ − 4x³ + 10, determine the concavity at x = 1:

Concave up

Concave down

Point of inflection

f''(1) = 0 (flat)

✏️ Part 2: Short Answers & Worked Problems

5. Differentiate s(t) = t³ − 6t² + 9t to find v(t) and a(t). Then calculate v(0) and a(2). Show working.

6. For A(x) = x(20 − x): find A'(x), the critical point, and use the second derivative to classify it (max/min). Show working.

7. Let f(x) = x³ − 3x² + 4. Find the critical point(s), determine whether each is a local max or min, and state intervals where f is increasing or decreasing (briefly justify using derivatives).

8. Explain in one or two sentences: In a motion context, what does the second derivative (acceleration) tell you about the object's motion? Relate sign to speeding up or slowing down.

9. Extension (change a parameter): If the rectangle perimeter is increased to P = 60 m, what side length x maximises area? Show a one-line justification.