Existence Theorems in Calculus: Free Course

About this course

Existence theorems give conclusions only when their hypotheses are satisfied. This course compares the intermediate value, extreme value and mean value theorems through graphs, tables and examples. Study the complete public source playlist from Khan Academy in English, with all 12 recordings in the on-site classroom. Original lesson titles, source order and repeated entries are preserved.

Course at a glance

What you’ll learn

• State the role of continuity and differentiability in existence theorems.
• Distinguish IVT, EVT and MVT conclusions.
• Check theorem conditions in examples.
• Use a theorem to justify an existence claim.

Skills you’ll gain

• Intermediate value theorem
• Extreme value theorem
• Mean value theorem
• Mathematical justification

Course Overview

The intermediate value theorem concerns values attained between two function outputs under continuity assumptions. The extreme value theorem addresses maximum and minimum values on an appropriate closed interval. The mean value theorem connects an average rate across an interval with a derivative at some interior point. Worked examples involving polynomials and square-root functions illustrate both use and condition checking. Graphs and tables help organize the available information, but isolated tabulated values alone do not establish every needed hypothesis. Distinguish a guarantee that a point exists from a method that actually locates all such points.

How to study the course

• Videos 1–3: this part of the source list includes “Mean value theorem” through “Existence theorems intro”. 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 4–6: this part of the source list includes “Mean value theorem application” through “Mean value theorem example: polynomial”. 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 7–9: this part of the source list includes “Intermediate value theorem example” through “Mean value theorem example: square root function”. 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 10–12: this part of the source list includes “Extreme value theorem” through “Conditions for IVT and EVT: table”. 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

Choose a continuous polynomial on a closed interval and formulate one claim for each theorem. State the hypotheses separately from the conclusion. Then modify the example to introduce a discontinuity or a corner and identify which guarantees may no longer apply. Explain why failure of a hypothesis does not by itself prove that the desired point is absent.

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.