Differentiable and Non-Differentiable Functions

Where is a function differentiable or non-differentiable, based on its graph? How is it related to continuity? We work through interesting examples such as x*sin(1/x) to see what happens as we zoom in to the graph looking for local linearity. A proof that differentiability implies continuity is done at the end. Infinite Powers, How Calculus Reveals the Secrets of the Universe: https://amzn.to/37PBMjb.

A function f is differentiable at a point c if the derivative f'(c) exists. This means the function f is locally linear. Its tangent line approximation does a good job (local linearization).

If a function f is differentiable at c, then f is continuous at c. Therefore, if f is not continuous at c, then f is not differentiable at c.

Calculus problems and solutions. Real Analysis problems and solutions.

🔴 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

#calculus #differentiability #differentiability_of_functions

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(0:00) Where is f not differentiable (and not continuous) graphically?
(4:29) f(x)=|x|=abs(x) is not differentiable at x = 0 (the absolute value function is not differentiable at x=0)
(7:15) f(x)=sqrt(x^2+.0001) is differentiable at x = 0 (zoom in to see the local linearity)
(10:54) f(x)=sin(1/x) when x is not zero and f(0)=0 is not continuous and not differentiable at x=0
(14:24) f(x)=x*sin(1/x) when x is not zero and f(0)=0 is continuous but not differentiable at x=0
(16:39) f(x)=x^2*sin(1/x) when x is not zero and f(0)=0 is continuous and is differentiable at x=0
(18:57) Real life non-differentiable function (acceleration g due to gravity as a function of the distance from the center of the earth)
(25:17) Computation of g (as a “constant” acceleration due to gravity near the surface of the earth using the universal gravitation constant G, the mass of the earth M, and the radius of the earth R)
(27:11) Proof that differentiability implies continuity (and therefore non-continuity implies non-differentiability)

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