Analyzing Functions with Derivatives: Free Course

About this course

Derivatives describe the behavior of a function, not just a symbolic calculation. This course uses first and second derivatives to justify increasing intervals, extrema, concavity, graph shape and optimization. Study the complete public source playlist from Khan Academy in English, with all 47 recordings available in the on-site classroom. Lessons retain their original titles and source order, including worked examples, introductions and repeated entries.

Course at a glance

What you’ll learn

• Check hypotheses when using existence theorems.
• Find and classify critical points with derivative evidence.
• Analyze concavity and verify inflection points.
• Translate constrained situations into optimization problems.

Skills you’ll gain

• Function analysis
• Extrema
• Concavity
• Optimization

Course Overview

Mean-value and extreme-value theorem examples establish the importance of hypotheses before conclusions. Critical points lead into increasing and decreasing intervals and the first-derivative test. Relative extrema are distinguished from absolute extrema, including the need to examine endpoints on a closed interval. Concavity and inflection-point lessons then add information from the second derivative, with explicit examples of mistakes caused by treating every candidate as a confirmed inflection point. Graphical connections among a function and its derivatives support a more complete picture of behavior. Optimization problems apply that reasoning to quantities such as volume, profit, material cost and motion. For these applications, building the function and defining its allowed domain come before differentiation. The repeated critical-point recording is retained exactly where the source includes it.

How to study the course

• Videos 1–11: this part of the source list includes “Mean value theorem” through “Finding decreasing interval given the function”. Identify the function, variable and conditions. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 12–23: this part of the source list includes “Finding increasing interval given the derivative” through “Inflection points (graphical)”. Connect the method to the meaning of the problem. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 24–35: this part of the source list includes “Analyzing concavity (algebraic)” through “Justification using second derivative: inflection point”. Rework a demonstration with a changed parameter. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.
• Videos 36–47: this part of the source list includes “Justification using second derivative: maximum point” through “Horizontal tangent to implicit curve”. Check the result and explain its interpretation. Keep a brief working note, revisit an unclear explanation and test one example before moving to another part of the material.

Who is this for?

This course suits learners with algebra and trigonometry preparation who want the specific calculus topic in the title. It also provides a recorded reference for revisiting a topic after an earlier lesson or study attempt. Choose examples appropriate to your background and allow time to work through the method instead of judging understanding only by how familiar the terminology sounds.

Before you start

Comfort manipulating algebraic expressions and interpreting functions is important. Review the prerequisite calculus stated by a lesson before using an advanced technique.

Put the lessons into practice

Analyze a function on a closed interval: list candidates, check endpoints, build sign charts and sketch a graph consistent with both derivatives. Then write a simple box-volume or cost optimization problem. State the feasible domain and explain why the selected value is optimal within it. Include one rejected candidate and the reason it does not give the required conclusion.

This optional study activity is provided by WikiFree for independent practice. Keep your working notes and use the relevant classroom recording to check a step, definition or interpretation. It is not an instructor-marked assignment.

Certificate / Assessment

Watching the public videos and using WikiFree completion tracking does not award a certificate, academic credit, an examination voucher or instructor assessment. Use your practice output to review what you can explain and repeat independently. The recorded lessons provide study material; completion tracking records your progress through that material.

Frequently asked questions

Provider: Khan Academy. Videos play from the Khan Academy channel on YouTube.