Local Linearization (Tangent Line Approximation)

In Calculus, the local linearization to a differentiable function f near a point x=a is a linear approximation given by L(x)=f(a)+f'(a)*(x-a). We compute this for many examples, including to approximating a square root and pendulum period, and we also consider the error function E(x)=f(x)-L(x). 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

#linearization #tangentline #approximation

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(0:00) Idea of local linearization (and formula)
(1:38) f(x)=ln(1+x) and a = 0 example
(3:40) f(x)=sqrt(1+3x) and a = 0 example
(5:36) f(x)=1/x and a = 4 example
(8:10) f(x)=sin(x) and a = π/6 example
(9:43) Estimate sqrt(4.2)=4.2^(.5) without a calculator (use linear approximation)
(15:25) (1+x)^(1/3) is approximately 1+x/3 near x = 0
(17:34) Estimate magnitude of the error with technology (TI graphing calculator)
(21:48) Approximate solution to transcendental algebra equation
(25:10) Period of a pendulum word problem
(36:53) Error function and its graphical behavior

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