Mean Value Theorem and its Corollaries

When does a function satisfy the hypotheses and conclusions of the Mean Value Theorem? What are some corollaries of the Mean Value Theorem? How are these corollaries proved? Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

🔴 Calculus 1, Lesson 1.1: Linear Functions (3Blue1Brown SoME2): https://www.youtube.com/watch?v=lmCjnxzIals

🔴 Calculus 1, Lesson 1.2: Exponential Functions: https://www.youtube.com/watch?v=47ZqW19AjD4

🔴 Calculus 1, Lesson 1.3: Quadratic Functions, Transformations of Functions, and Function Composition: https://www.youtube.com/watch?v=1T58CnSME4g

🔴 Calculus 1, Lesson 1.4: Logarithmic Functions (Intro to Horizontal Compressions & Inverse Functions): https://www.youtube.com/watch?v=oS-gvIoJf60

🔴 Calculus 1, Lesson 1.5: Trigonometric Functions and Inverse Trigonometric Functions: https://www.youtube.com/watch?v=UFYjlnGUenE

🔴 AP Calculus AB Exam Review: Practice Exam Problems & Solutions (Multiple Choice, No Calculator): https://www.youtube.com/watch?v=2AG_Dt3x7q0

🔴 AP Calculus AB Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=RLgUwP44V_M

🔴 AP Calculus BC Exam Review: Practice Exam (30 Question Multiple Choice, No Calculator): https://www.youtube.com/watch?v=1Q7ALcwT97A

🔴 AP Calculus BC Exam Review: Free Response Practice Exam Problems & Solutions: https://www.youtube.com/watch?v=ZWisJEDP44w

🔴 Calculus 1 Exam 1 Review Problems and Solutions (Precalculus, Limits, Continuity, Derivatives): https://www.youtube.com/watch?v=UjmvUwvd7c4

🔴 Calculus 1 Course, Lecture 1: The Big Ideas (Rates & Areas, the Infinity Principle & Circular Area): https://www.youtube.com/watch?v=2V_ehJC5QUY

🔴 Calculus 1 Course Lecture Playlist: https://www.youtube.com/playlist?list=PLmU0FIlJY-MkFrKtM0S0NLu-TxZa5jBIx

The corollaries include the Increasing Function Theorem, Constant Function Theorem, and the Racetrack Principle.

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(0:00) Mean Value Theorem statement
(3:28) f(x)=x^3 over [a,b]=[-2,3]. Are the hypotheses and conclusion of the MVT satisfied? If the conclusion is satisfied, find the value(s) of c.
(7:22) f(x)=sin^-1(x) over [a,b]=[-1,1] (the arcsine or inverse sine function). Are the hypotheses and conclusion of the MVT satisfied? If the conclusion is satisfied, find the value(s) of c.
(11:31) f(x)=|x|=abs(x) over [a,b]=[-1,1] (the absolute value function). Are the hypotheses and conclusion of the MVT satisfied? If the conclusion is satisfied, find the value(s) of c.
(13:34) f(x)=x^(1/3)=cuberoot(x) over [a,b]=[-1,8] (cube root function). Are the hypotheses and conclusion of the MVT satisfied? If the conclusion is satisfied, find the value(s) of c.
(16:46) Find a tangent line using the Mean Value Theorem
(19:13) Prove the Increasing Function Theorem by using the Mean Value Theorem
(26:24) Prove f(x)=x^3+x is an increasing function over the real number line R
(29:27) Use Constant Function Theorem to prove a general solution of the differential equation y’=y is y=Ce^x
(36:46) Use the Racetrack Principle to prove ln(x) is less than or equal to x – 1 for all x greater than or equal to 1

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