Application problems combining velocity, maximum and minimum values, and approximation.
Example ① A sphere whose radius is increasing. When the radius is r(t) = t cm, find the rate of change of volume at t=3。
Solution:V = (4/3)πr³。dV/dt = 4πr²·dr/dt = 4π·9·1 = 36π cm³/s
Solution:V = (4/3)πr³。dV/dt = 4πr²·dr/dt = 4π·9·1 = 36π cm³/s
Key Point
- Calculate the rate of change with respect to time using differentiation of composite functions (the chain rule)
Practice Problems
- One side of a square, a(t)=2t, is increasing. What is the rate of change of its area at t=1?
Solutions and Explanations
- Solution:8
S=a²=4t²、dS/dt=8t、at t=1, it is 8。