The Derivative at a Point (Calculus Problems and Solutions)

Use the limit definition of the derivative to find the derivative f(x)=x^2 at x=2, f(x)=1/x^2=x^-2 at x=-1, and f(x)=sqrt(x)=x^1/2 (square root) at x = 4 (use a difference quotient in each case). Interpret the derivative as the slope of a tangent line and rate of change. Use linear approximation to estimate a function value. Estimate the derivative of atmospheric pressure with respect to altitude. 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 #derivatives #calculusderivatives

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(0:00) Use a limit to find f'(2) when f(x)=x^3 (limit definition of the derivative, the limit of a difference quotient).
(6:24) Use a limit to find f'(-1) when f(x)=1/x^2
(9:54) Use a limit to find f'(4) when f(x)=x^1/2=sqrt(x)
(12:49) Find the tangent line at x=10 when f(10)=5 and f'(10)=2
(17:57) Tangent line approximation of f(x)
(21:08) Fill in the blanks about f and f’ when given a tangent line
(22:41) Geometric interpretations of function values and derivative values
(28:39) Atmospheric pressure application (interpret the derivative)

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