Unlocking Maxima and Minima: Mastering the Difference Quotient

PreAlgebra Grades High School 4:57 Video
Print Lesson Plan Watch Video

Lesson Description

Explore the power of the difference quotient to find relative maximums and minimums of polynomial functions. This lesson provides a step-by-step guide to calculating derivatives and using them to analyze graph behavior.

Video Resource

Finding Relative Maximums and Minimums Using Difference Quotient

Mario's Math Tutoring

Duration: 4:57
Watch on YouTube

Key Concepts

  • Difference Quotient
  • Derivative as a Limit
  • Relative Maxima and Minima
  • Horizontal Tangent Lines
  • Polynomial Graphing

Learning Objectives

  • Calculate the derivative of a polynomial function using the difference quotient.
  • Determine the x-values where a function has relative maximums or minimums by setting the derivative equal to zero.
  • Find the coordinates of relative maximums and minimums by substituting x-values into the original function.
  • Sketch an accurate graph of a polynomial function using information about its zeros, end behavior, and relative extrema.

Educator Instructions

  • Introduction (5 mins)
    Briefly review the concept of slope and tangent lines. Introduce the idea of finding turning points (relative maxima and minima) on a graph.
  • The Difference Quotient (10 mins)
    Present the formula for the derivative using the difference quotient: lim (h->0) [f(x+h) - f(x)] / h. Explain that this formula calculates the slope of the tangent line to the graph of f(x) at a given point.
  • Example Problem (20 mins)
    Work through the example from the video: f(x) = 2x^3 - 6x. Step-by-step, show how to: 1. Substitute (x+h) into the function. 2. Expand and simplify the expression. 3. Apply the difference quotient formula. 4. Take the limit as h approaches zero to find the derivative.
  • Finding Maxima and Minima (10 mins)
    Explain that relative maximums and minimums occur where the tangent line is horizontal (slope = 0). Set the derivative equal to zero and solve for x. These x-values are the locations of the turning points.
  • Graphing (10 mins)
    Substitute the x-values of the turning points back into the original function to find the y-coordinates. Plot these points. Find the zeros of the function. Use end behavior to sketch an accurate graph.
  • Practice (10 mins)
    Provide students with similar problems to practice finding the relative maximums and minimums of different polynomial functions using the difference quotient.

Interactive Exercises

  • Group Calculation
    Divide students into groups and assign each group a different polynomial function. Have them work together to find the derivative using the difference quotient and then identify the relative maximums and minimums. Each group presents their solution.
  • Graphing Challenge
    Provide students with a set of polynomial functions and have them sketch the graphs by hand, using the information they gather from the derivative and the function itself. Then, they check their graphs using a graphing calculator.

Discussion Questions

  • Why does setting the derivative equal to zero help us find relative maximums and minimums?
  • How does the difference quotient relate to the concept of a tangent line?
  • What are some limitations of using the difference quotient (compared to other differentiation techniques)?
  • How does understanding the end behavior and zeros of a function help to graph it accurately?

Skills Developed

  • Algebraic Manipulation
  • Limit Calculation
  • Problem-Solving
  • Analytical Thinking
  • Graphing Skills

Multiple Choice Questions

Question 1:

The difference quotient is used to find the:

Correct Answer: Slope of the tangent line

Question 2:

At a relative maximum or minimum, the derivative of a function is:

Correct Answer: Zero

Question 3:

The formula for the derivative using the difference quotient is:

Correct Answer: lim (h->0) [f(x+h) - f(x)] / h

Question 4:

After finding the x-value where the derivative is zero, you substitute it back into the ________ to find the y-coordinate of the extremum.

Correct Answer: Original Function

Question 5:

What does end behavior describe?

Correct Answer: How the graph behaves as x approaches positive or negative infinity

Question 6:

Which of the following is a turning point on a graph?

Correct Answer: Relative Maximum

Question 7:

What is the primary purpose of finding the derivative of a function?

Correct Answer: To find the slope of the tangent line at any point

Question 8:

Which concept is essential for evaluating the difference quotient as h approaches 0?

Correct Answer: Limits

Question 9:

If the derivative of a function is positive at a point, what does this indicate about the function at that point?

Correct Answer: The function is increasing

Question 10:

What does Pascal's triangle help expand?

Correct Answer: Binomials

Fill in the Blank Questions

Question 1:

The derivative of a function represents the slope of the __________ line.

Correct Answer: tangent

Question 2:

Relative maxima and minima are also known as __________ points.

Correct Answer: turning

Question 3:

To find the x-values of relative extrema, set the __________ equal to zero.

Correct Answer: derivative

Question 4:

The end behavior of a polynomial function is determined by its __________ term.

Correct Answer: leading

Question 5:

The difference quotient is an application of the __________ of a function.

Correct Answer: limit

Question 6:

Using Pascal's Triangle aids in expanding __________ expressions, which can simplify finding the derivative.

Correct Answer: binomial

Question 7:

The process of finding the derivative is called __________.

Correct Answer: differentiation

Question 8:

Points where the derivative is zero or undefined are also referred to as __________ points.

Correct Answer: critical

Question 9:

The __________ of a polynomial function describes how it behaves as x approaches infinity or negative infinity.

Correct Answer: end behavior

Question 10:

Graphically, a relative maximum is the highest point in a __________ region of the graph.

Correct Answer: local

Educational Standards

PC.F.1.1:Interpret characteristics of a function defined by an expression in the context of the situation. PC.F.1.2:Sketch the graph of a function that models a relationship between two quantities, identifying key features. PC.F.3.1:Predict solutions involving functions that are quadratic, polynomial of higher order, rational, exponential, and logarithmic.

Teaching Materials

Download ready-to-use materials for this lesson:

Student Worksheet Classroom Slides Assessment Quiz
Add to Google Classroom

User Actions

Sign in to save this lesson plan to your favorites.

Sign In

Share This Lesson

Related Lesson Plans