Dave4Math / Courses / Applications of the Derivative

Applications of the Derivative

Learn powerful methods for apply derivatives to the real world. With this course, you'll master applications of the derivative and become skilled at curve sketching.


January 11, 2021

Created by Dave

This course begins Summer 2021.
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Have you ever felt like you were being challenged in a math course, but yet you didn’t quite feel like you understood the basics? It would help if you had time and practice to understand the basics. Introducing Dave’s online math course: Applications of the Derivative. Dave really challenges confident students while encouraging those less so. His lessons are clear and engaging. You can become confident. 

Who This Course is For

  • Students in college who want to learn more mathematics.
  • Anyone interested in the applications of the derivative.
  • Students who are enrolled in calculus 1 and want to improve their grade.
  • Anyone interested in getting prepared for calculus 2.
  • Anyone wanting to learn about extrema of functions, the Mean Value Theorem, the first and second derivative tests, and curve sketching.


There is no required textbook, though you will need an up-to-date web browser, paper, and pen. A hand-held scientific calculator of your choice is required.

What You’ll Learn in Applications of the Derivative

  • Absolute Extrema of Functions
  • Relative Extrema of Functions
  • Finding the Extreme Values
  • An Optimization Problem
  • Rolle’s Theorem
  • Mean Value Theorem
  • Consequences of the Mean Value Theorem
  • Determining the Number of Zeros
  • Increasing and Decreasing Functions
  • The First Derivative Test
  • Finding the Relative Extrema of a Function
  • Concavity
  • Inflection Points
  • The Second Derivative Test
  • Determining the Shape of a Graph
  • Infinite Limits
  • Vertical Asymptotes
  • Limits at Infinity
  • Horizontal Asymptotes
  • Interest Compounded Continuously
  • Infinite Limits at Infinity
  • Precise Definitions
  • The Graph of a Function
  • Guide to Curve Sketching
  • Slant Asymptotes
  • Formulating Optimization Problems
  • The Indeterminate Forms Type I
  • l’Hôpital’s Rule
  • The Indeterminate Forms Type II
  • The Indeterminate Forms Type III
  • What is Newton’s Method?
  • Applying Newton’s Method
  • When Newton’s Method Does Not Work

Course Description

This course begins with a thorough investigation into the extrema of functions. We study both absolute extrema and relative extrema by working through several examples. Next, we analyze Rolle’s Theorem and the Mean Value Theorem. We both motivate the theorem and provide proof. Some consequences of the Mean Value Theorem and several examples are detailed.

After that, we begin a thorough investigation into sketching the graphs of functions. Increasing and decreasing functions and the first derivative test are examined first. Following this, we study concavity, inflection points, and the second derivative test.

We then continue our investigation to the graphs of functions by studying limits involving infinity and different types of asymptotes. We cap these lessons off with a thorough guide to curve sketching, including slant asymptotes, vertical tangent, and cusps.

Now that a thorough analysis of the graph of a function is accomplished, we begin studying optimization problems. We establish guidelines for solving optimization problems and work through several examples to demonstrate how to use these guidelines.

The exciting conclusion to this course is to examine indeterminate forms. We begin by classifying seven indeterminate forms into three classes and explain the strategy behind each type. The primary tool being l’Hopital’s Rule along with algebraic manipulation.

In the end, we explore Newton’s method, how to apply it, and when it fails.

Recommended Prerequisites for Applications of the Derivative

I recommend the prerequisite course Theory of Derivatives. If you’re not sure if this course is for you, checkout the course contents below or find out more by checking out my free calculus 1 articles.

Course Content

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David A. Smith

Mathematics Educator

I sincerely believe that the potential in every student can be unlocked. As an accomplished and dedicated instructor, I have firm confidence that I can provide significant value to your studies.

I relish teaching and use it as a channel for my creativity as I seek to deliver real-life and relevant connections between the mathematics that I teach and my students. Promoting an active learning environment and excitement about mathematics is my dream come true.

David Smith (Dave) has a B.S. and M.S. in Mathematics and has enjoyed teaching precalculus, calculus, linear algebra, and number theory at both the junior college and university levels for over 20 years. David is the founder and CEO of Dave4Math.