Calculus Study Guide: Absolute Extrema, Optimization, Indefinite Integrals, and Applications

4.5 Absolute Maxima and Minima

Definitions and Concepts

Absolute extrema (maximum and minimum) are the highest and lowest values of a function on a given interval. These can occur at critical numbers or endpoints of the interval.

• Absolute Maximum: The largest value of f(x) on an interval [a, b].

• Absolute Minimum: The smallest value of f(x) on an interval [a, b].

• Critical Number: A value c in the domain of f where f'(c) = 0 or f'(c) does not exist.

Procedure: Finding Absolute Extrema on a Closed Interval

1. Find the critical numbers of f in the interval [a, b].

2. Evaluate f at each critical number and at the endpoints a and b.

3. The largest value is the absolute maximum; the smallest is the absolute minimum.

First Derivative Test for Local Extrema

• If f'(x) changes from positive to negative at c, f has a local maximum at c.

• If f'(x) changes from negative to positive at c, f has a local minimum at c.

Second Derivative Test for Absolute Extrema

• If f''(c) > 0, f has a local minimum at c.

• If f''(c) < 0, f has a local maximum at c.

Example

Find the absolute extrema of f(x) = 5ln(x) - x on [1,3]:

• Find critical numbers by solving f'(x) = 5/x - 1 = 0 ⇒ x = 5 (not in [1,3]).

• Evaluate f at endpoints: f(1) = 5ln(1) - 1 = -1, f(3) = 5ln(3) - 3 ≈ 2.493.

• Absolute minimum at x = 1, absolute maximum at x = 3.

4.6 Optimization

Optimization Problems

Optimization involves finding the maximum or minimum value of a function that models a real-world situation. Typical problems include maximizing area, profit, or minimizing cost, distance, etc.

1. Read the problem carefully and create equations for the quantities involved.

2. Express the quantity to be optimized as a function of one variable.

3. Find the domain of the function.

4. Find critical numbers and test for maxima or minima.

5. Answer the original question, interpreting the result in context.

Example

A company manufactures and sells smartphones. The weekly price-demand and cost equations are:

• p = 500 - 0.04x (price-demand)

• C(x) = 26000 + 120x (cost)

To maximize weekly profit, express profit as P(x) = R(x) - C(x), where R(x) = x 0.04x. Find the value of x that maximizes P(x) by setting P'(x) = 0 and solving for x.

5.1 Indefinite Integrals

Definition

The indefinite integral of f(x) is a function F(x) such that F'(x) = f(x). It is written as:

$\int f(x)\,dx = F(x) + C$

where C is the constant of integration.

Basic Integration Rules

• $\int x^n\,dx = \frac{x^{n+1}}{n+1} + C$, for $n \neq -1$

• $\int e^x\,dx = e^x + C$

• $\int a\,dx = ax + C$

Properties

• $\int [f(x) + g(x)]\,dx = \int f(x)\,dx + \int g(x)\,dx$

• $\int kf(x)\,dx = k\int f(x)\,dx$

Applications

• Finding the original function from its derivative (antiderivative).

• Solving basic differential equations.

5.3 Fundamental Theorem of Calculus

Statement

• If f is continuous on [a, b] and F is any antiderivative of f, then:

$\int_a^b f(x)\,dx = F(b) - F(a)$

Average Value of a Function

The average value of a function f over [a, b] is:

$f_{avg} = \frac{1}{b-a}\int_a^b f(x)\,dx$

Review Questions and Applications

• Find absolute extrema for various functions on closed intervals.

• Apply the first and second derivative tests to determine local extrema.

• Solve optimization problems involving area, cost, and revenue.

• Evaluate indefinite and definite integrals using basic rules.

• Find particular antiderivatives given initial conditions.

• Solve simple first-order differential equations.

• Apply calculus to real-world scenarios such as maximizing profit, minimizing cost, and modeling growth or decay.

Sample Table: Comparison of Extrema Tests

Test	How to Use	What It Determines
First Derivative Test	Check sign changes of f'(x) around critical points	Local maxima/minima
Second Derivative Test	Evaluate f''(x) at critical points	Concavity; local maxima/minima if f''(x) ≠ 0
Endpoint Test	Evaluate f(x) at endpoints of interval	Absolute maxima/minima

Additional info:

• Some questions involve interpreting graphs to find extrema.

• Applications include business (profit, cost, revenue), geometry (area, fencing), and economics (marginal cost).

• Students are expected to justify answers using calculus concepts and show all steps.