Using derivatives to solve real-world problems
If f'(x) > 0 on an interval, then f(x) is:
Positive derivative indicates function is increasing
Find critical points of f(x) = x³ - 3x
Critical points where f'(x)=0: 3x²-3=0, so x²=1, x=±1
A ball is thrown upward with height h(t) = -16t² + 64t. When does it reach maximum height?
Maximum when h'(t)=0: -32t+64=0, so t=2
Find equation of tangent line to y = x² at (2,4)
Slope f'(2)=4, point (2,4): y-4=4(x-2) → y=4x-4
For optimization, we find maximum/minimum by:
Critical points (where f'(x)=0 or undefined) are candidates for extrema