Calculus 1 Question- linear approximation

Question

The volume of a sphere with radius r is V(r) = (4/3)(π)(r^3). Use a linear approximation for V (r) at r = 2 to approximate the volume of a sphere with radius r = 1.94. What is the error in this approximation?

Answer 1

ΔV = V(1.94) - V(2) ≈ dV = 4πr²dr; V(1.94) ≈ V(2) + 4πr²dr;

V(2) = 4/3π·2³ = 32π/3; dr = Δr = 1.94 - 2 = - 0.06;

Then V(1.94) = 32π/3 - 4π·2²·(- 0.06) = 32π/3 - 0.96π = π(32/3 - 24/25) = 728π/75 = 50.824