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Optimization in Calculus: Finding Maximum and Minimum Values

Antiderivatives

Derivatives and Antiderivatives

Derivative and Antiderivative of a Function

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Area Under Curves

Areas between curves

Average Value

Average value of a function

Basic Integration Techniques

U-Substitution

U-Substitution: Integrating Radical Functions

Not-So-Obvious U-Substitution

Concept of Limits

Finding limits from graphs

Finding limits algebraically - direct substitution

Finding limits algebraically - when direct substitution is not possible

Limit laws

Evaluating Limits of Functions

limit laws

Evaluating Limits with specific limits given

Introduction to Calculus - Limits

Continuity

Intermediate value theorem

Proving the Existence of Real Roots Using IVT

Intermediate Value Theorem

Application of Intermediate Value Theorem

Proving the Existence of a Real Root Using IVT

Discontinuities of Rational Functions (denominator=0)

Critical Points and Extrema

Critical number & maximum and minimum values

Critical Number & Maximum and Minimum Values

Optimization

Curve Analysis

Curve sketching

Derivative Concept

Estimating Derivatives from a table

Slope and equation of tangent line

Definition of derivative

Definition of Derivative

Derivative using Definition

Vertical Tangent Line of a Function

Applications to definition of derivative

Derivative Rules

Chain rule

Chain Rule

Differentiation of Trigonometric Function

Differentiate Radical Functions with Chain Rule

Differentiate: Trigonometric Functions

Chain rule: Differentiate Trigonometric Functions

Differentiation of Exponential Function

Differentiation of Exponential Functions

Differentiation: Chain Rule with Logarithmic Functions

Differentiation of Logarithmic Function

Product rule

Quotient rule

Differentiation using Quotient Rule

Differentiation using Chain Rule and Quotient Rule

Power rule

Power Rule

Application of the Power Rule

Power rule application

Applying Power Rule

Power Rule with Negative Exponents

Power rule with rational exponents

Power rule and rational exponents

Derivatives of Special Functions

Derivative of inverse trigonometric functions

Derivative of trigonometric functions

Derivative of Trigonometric Functions

Derivative of Composite Trig Functions

Derivative of exponential functions

Derivative of Exponential Functions

Differentiation of Composite Functions

Derivative of power and exponential functions

Derivative of logarithmic functions

Derivative of log with arbitrary base

Derivative of Natural Logarithm Functions

Derivative of Logarithmic Functions

Fundamental Theorem of Calculus

Fundamental theorem of calculus

Fundamental Theorem of Calculus Part II

Fundamental Theorem of Calculus Part I

Implicit Differentiation

Implicit differentiation

Limits at Infinity

Infinite limits - vertical asymptotes

Limits at infinity - horizontal asymptotes

Limit Evaluation

Limits at Infinity of Exponential Functions

Motion and Rates

Position velocity acceleration

Velocity of a Particle

Total Distance Traveled by a Particle

Comparative Analysis of Particle's Velocity and Acceleration

Riemann Sums

Riemann sum

Riemann Sum

Evaluating integrals with a Riemann sum

Evaluating integrals with a Riemann Sum

Riemann Sum with Trapezoid Approximation

Trapezoidal Approximation of Riemann Sums

Tangent Lines

Linear approximation

Use of Linear Approximations in Estimating Square Roots

Linearization of Radical Functions

Linearization of Polynomial Functions

Linearization of Rational Functions

Linearization of Exponential Functions

Linearization of Logarithmic Functions

Linear Approximation of Trigonometric Functions

Parent Assignments

Challenge Zone

Foundation Review

math

Finding two numbers with a given sum that maximize their product

Maximizing area of a rectangular field bordered by a river

Minimizing cost of an open-top rectangular container with fixed volume

Calculating the minimum cost of materials for the container

This lesson explains how optimization problems in calculus turn real-world questions, like maximizing area or minimizing cost, into single-variable functions that can be solved using derivatives, critical numbers, and the second derivative test.

Optimization Calculus | StudyPug

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Overview

Optimization Calculus: Solving Maximum and Minimum Problems

What You'll Learn

Translate a word problem into a constraint equation and an equation for the quantity being optimized

Reduce two variables to one by substituting the constraint into the main equation

Find critical numbers by setting the derivative of the single-variable function equal to zero

Apply the first or second derivative test to confirm whether a critical number gives a maximum or a minimum

Check endpoints of a closed interval since the largest or smallest value can occur there instead of at a critical number

Interpret the final numerical answer in the context of the original problem, such as dimensions, cost, or profit

What You'll Practice

Maximizing products given sum constraints using calculus

Minimizing fencing costs while maximizing enclosed rectangular areas

Finding dimensions that minimize material costs for containers with volume constraints

Setting up and solving optimization equations from word problems

Why This Matters

Before You Start — Make Sure You Can:

This Unit Includes

4 Video lessons

Practice exercises

Learning resources

Skills

Optimization

Derivatives

Critical Points

Word Problems

Maxima and Minima

Cost Minimization

Area Maximization

Volume Constraints