The Second Derivative (Calculus Problems and Solutions)

Given the graph of f(x), sketch the graphs of the first derivative f'(x) and the second derivative f”(x). The sign of the second derivative tells you where the graph of f(x) is concave up and concave down (concavity). This can be applied to rising or falling prices and the physics of motion (position, velocity, and acceleration). 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

#calculus #RateOfChange #RateOfChangeOfQuantites

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(0:00) Given the graph of f(x), graph f'(x) and f”(x) (graph the first derivative (slope of f(x)) and graph the second derivative (concavity of f(x)))
(8:16) Stock price as a function of time: sign of first and second derivatives
(11:24) Identify signs of f'(x) and f”(x), or if approximately zero from a graph of f(x)
(14:36) From position graph and marked points, find where position, velocity, and acceleration are positive, negative, increasing, or decreasing.
(20:43) Given graph of f'(x), where is f(x) greatest and least? Where is f'(x) greatest and least? Where is f”(x) greatest and least?

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