The derivative: geometric and physical meaning
Answers are for parents and teachers.
1
From the definition find the derivative of f(x) = x² + 4x and compute f′(3).
10
2
A point moves by s(t) = 2t³ − t (m). Using (t³)′ = 3t², find the velocities at t = 1 s and t = 3 s.
v(t) = 6t² − 1; v(1) = 5 m/s and v(3) = 53 m/s.
3
Find the derivative of f(x) = 2/x (x ≠ 0) from the definition. What can you say about its sign?
f(x + h) − f(x) = −2h/((x + h)x), hence f′(x) = −2/x². The derivative is always negative.
4
At which points of y = x³ is the tangent parallel to the line y = 12x − 5? ((x³)′ = 3x².)
3x² = 12 ⇒ x = ±2; the points are (2, 8) and (−2, −8).
5
The volume of water in a tank is V(t) = 5t² + 20 litres (t in minutes, (t²)′ = 2t). Give the rate of change of the volume at t = 4 min, with its unit.
V′(t) = 10t, V′(4) = 40 L/min.