Applications of Derivatives, Extreme Values of Functions, and the Mean Value Theorem

Applications of Derivatives

• Rate measure: \(dy/dx\) = \(\Delta y/\Delta x\) = \(f'(x)\)
• Errors and approximations.
• Increasing and decreasing functions.
• Maxima and minima.
• Tangent lines and normal lines.

Roles theorem

• If \(f\) is continuous on \([a,b]\) and differentiable on \((a,b)\), and \(f(a) = f(b)\), then there exists a number \(c\) in \((a,b)\) such that \(f'(c) = 0\) (the slope of the tangent line is zero).