Lecture Note: §4 Applications of Derivatives

Last Update: 2025-10-28

In this chapter, we learn what derivatives tell us about the shape of a graph of a function and, in particular, how they help us locate maximum and minimum values of functions. One of the most important applications of the derivative is its use as a tool for finding the optimal (best) solutions to problems.

§4.1 Extreme Values of Functions

• Definition of Absolute and Local Extreme Value
• The Extreme Value Theorem
  • If \(f\) is continuous on a closed interval \([a,b]\), then \(f\) attains both an absolute maximum value \(M\) and an absolute minimum value \(m\) in \([a, b]\).
  • Intuitively plausible
• Finding Extrema
  • The First Derivative Theorem for Local Extreme Values
  • The critical point of \(f\) (\(f’\) is zero or undefined).
• Finding the Absolute Extrema of a Continuous Function \(f\) on a Finite Closed Interval
  1. Find all critical points of \(f\) on the interval.
  2. Evaluate \(f\) at all critical points and endpoints.
  3. Take the largest and smallest of these values.

§4.2 The Mean Value Theorem

• Suppose that \(y = f (x)\) is continuous over the closed interval \([a, b]\) and differentiable at every point of its interior \((a, b)\).
  • Rolle’s Theorem. If \(f (a) = f (b)\), then there is at least one number \(c\) in \((a, b)\) at which \(f’(c) = 0\).
  • The Mean Value Theorem. There is at least one number \(c\) in \((a, b)\) at which \(\dfrac{f(b) - f(a)}{b-a} = f’(c)\) or, equivalently, \(f(b) - f(a) = f’(c)(b-a)\).

§4.3 Monotonic Functions and the First Derivative Test

• Monotonicity
  • \(f’(x) > 0\), then \(f\) is increasing;
  • \(f’(x) < 0\), then \(f\) is decreasing.
• First Derivative Test for Local Extrema.

§4.4 Concavity and Curve Sketching

• Concavity
  • If the graph of \(f\)) lies above all of its tangents on an interval \(I \), then \( f \) is called concave up on \(I \). If the graph of \( f \)) lies below all of its tangents on \( I \), then \( f \) is called concave down on \(I\).
  • concave up: \(f’\) is increasing, \(f^{\prime\prime} >0\)
  • concave down: \(f’\) is decreasing, \(f^{\prime\prime} <0\)
  • point of inflection of \(f\)
• Second Derivative Test for Local Extrema
• Procedure for Graphing \(y = f(x)\)
  • What does \(f\) says about \(f\)?
  • What does \(f’\) says about \(f\)?
  • What does \(f^{\prime\prime}\) says about \(f\)?

§4.5 Applied Optimization

• Solving Applied Optimization Problems

§4.6 Newton’s Method

Solving \(f(x) = 0\).

• Newton’s Method
  1. Guess the initial value \(x_0\)
  2. \(x_{n+1} = x_n - \dfrac{f(x_n)}{f’(x_n)}\), if \(f’(x_n) \neq 0\).

§4.7 Antiderivatives

• Antiderivative.
• Indefinite Integral \(\int f(x) dx\)
  • The collection of all antiderivatives of \(f\) is called the indefinite integral of \(f\) with respect to \(x\).
• Initial Value Problem.