Tangent and normal equations; problem solving
Answers are for parents and teachers.
1
Write the tangent and normal equations of f(x) = x³ − 3x² + 2 at x₀ = 1.
f(1) = 0, f′(x) = 3x² − 6x, f′(1) = −3. Tangent: y = −3(x − 1) = −3x + 3. Normal: y = (1/3)(x − 1).
2
Write the equation of the tangent to y = x² − 5x + 6 that is parallel to y = 3x − 2.
2x − 5 = 3 ⇒ x₀ = 4, f(4) = 2. Tangent: y − 2 = 3(x − 4), i.e. y = 3x − 10.
3
If s(t) = 2t³ − 15t² + 24t (m), when does the point stop and during which time interval does it move backwards?
v = 6t² − 30t + 24 = 6(t − 1)(t − 4). It stops at t = 1 s and 4 s; for 1 < t < 4, v < 0 — it moves backwards.
4
For f(x) = x² + 3x compute Δf and Δf/Δx at x₀ = 2, Δx = 0.05 and compare with f′(2).
f(2.05) = 4.2025 + 6.15 = 10.3525; f(2) = 10; Δf = 0.3525; Δf/Δx = 7.05; f′(2) = 7: close.
5
What angle does the tangent to y = ln x at x₀ = 1 make with the Ox axis?
f′(1) = 1 ⇒ tan α = 1 ⇒ α = 45°.