Use derivatives to find tangent lines, optimize functions, and solve related rates.
Derivatives encode slope and rate information. Their applications include finding tangent lines, identifying local maxima and minima (optimization), and solving related rates problems where multiple quantities change simultaneously.
f′(c)=0 or undefined⟹c is a critical point
f′′(c)>0⟹local min at cf′′(c)<0⟹local max at c
Optimization Strategy
1Write the objective function to maximize or minimize.
2Express it in terms of a single variable using constraints.
3Differentiate and set f'(x) = 0 to find critical points.
4Test critical points using the second derivative test or sign analysis.
5Check endpoints if the domain is closed and bounded.
📦Optimization: Maximum Area
Enclose a rectangle with 100 m of fencing. Maximize area A = xy subject to 2x + 2y = 100 → y = 50−x. A(x) = x(50−x) = 50x−x². A'(x) = 50−2x = 0 → x = 25. Maximum area = 25·25 = 625 m².
A(x)=x(50−x)⟹A′(x)=50−2x=0⟹x=25,Amax=625
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Remember This!
Related rates: differentiate both sides of an equation with respect to time t using implicit differentiation. Every variable that changes with time gets a dX/dt factor by the chain rule.
✏️ Try It!
At which x does f(x) = 2x³ − 12x² + 18x − 5 have a local minimum?