Applications of the derivative: problem solving
Answers are for parents and teachers.
1
For f(x) = x³ − 6x² + 5 find the intervals of monotonicity, the extrema and the greatest and least values on [−1, 5].
f′ = 3x(x − 4): increasing on (−∞, 0), (4, ∞); decreasing on (0, 4). Max f(0) = 5, min f(4) = −27. On [−1, 5]: f(−1) = −2, f(5) = −20; greatest 5, least −27.
2
A body moves upwards by h(t) = 12t − 4t² (m). Find the velocity, the acceleration, the greatest height and the flight time.
v = 12 − 8t, a = −8 m/s². v = 0 ⇒ t = 1.5 s; h(1.5) = 18 − 9 = 9 m. h = 0 ⇒ t = 3 s.
3
Find the abscissa at which the tangents to y = x² − 4x + 5 and y = −x² + 2x + 1 are parallel.
2x − 4 = −2x + 2 ⇒ x = 1.5.
4
Using the sign of its derivative, find where y = ln(x² + 1) increases and decreases.
y′ = 2x/(x² + 1): negative on (−∞, 0) — decreasing; positive on (0, ∞) — increasing; minimum y = 0 at x = 0.
5
For a rectangular plot with perimeter 40 m, which sides make its diagonal smallest? Take d²(x) = x² + (20 − x)².
(d²)′ = 4x − 40 = 0 ⇒ x = 10: a 10 × 10 square; d² = 200, d = 10√2 ≈ 14.1 m.