Calculus 1 Exam 2 Review Problems and Solutions (including Implicit Differentiation)

Calculus Exam Review Problem Types: 1) Differentiability & Linear Approximation, 2) Derivative Rules & Implicit Differentiation, 3) Derivatives of Elementary Functions, 4) Physics & Optimization, 5) Mean Value Theorem. Infinite Powers: https://amzn.to/37PBMjb.

The derivative rules include the Product Rule, Quotient Rule, and Chain Rule. Elementary functions include power, polynomial, exponential, logarithms, trigonometric, inverse trigonometric (arcsine, arctangent), and general inverse functions. Simple indefinite integrals are also done. The topics related to the Mean Value Theorem are: the Racetrack Principle, Increasing Function Theorem, Decreasing Function Theorem, and Constant Function Theorem. For a cubic function, critical points are found and classified as local maximum points or local minimum points with the First Derivative Test and the Second Derivative Test.

Links and resources
===============================
🔴 Subscribe to Bill Kinney Math: https://www.youtube.com/user/billkinneymath?sub_confirmation=1
🔴 Subscribe to my Math Blog, Infinity is Really Big: https://infinityisreallybig.com/
🔴 Follow me on Twitter: https://twitter.com/billkinneymath
🔴 Follow me on Instagram: https://www.instagram.com/billkinneymath/
🔴 You can support me by buying “Infinite Powers, How Calculus Reveals the Secrets of the Universe”, by Steven Strogatz, or anything else you want to buy, starting from this link: https://amzn.to/3eXEmuA.
🔴 Check out my artist son Tyler Kinney’s website: https://www.tylertkinney.co/

(0:00) Types of problems
(0:29) Continuity, differentiability, the Mean Value Theorem, and related concepts.
(7:28) Find a complicated derivative (differentiation problem with product rule, quotient rule, and chain rule).
(11:02) Derivative of an inverse function
(14:25) Abstract Product Rule and Chain Rule
(17:52) Population rate of change interpretation of a derivative
(21:58) Cooling coffee rate of change interpretation of a derivative
(25:14) Physics free fall problem (find the maximum height)
(29:26) Basic antiderivatives (indefinite integrals, including a differential equation initial value problem for the third problem)
(32:59) Derive the Quotient Rule using the Product Rule and Chain Rule
(35:42) Find and interpret a partial derivative (for the period of a mass on a spring in terms of the mass m and spring stiffness constant k). Also do a linear approximation.
(42:09) Implicit differentiation problem (find the derivative of an implicitly defined function)
(49:24) Find the local extrema of a cubic function (optimization using critical points, the First Derivative Test, and Second Derivative Test)
(56:59) Linear approximation (local linearization) to approximate a fourth root
(1:02:10) Newton’s Method to approximate a fourth root
(1:06:06) Use infinitesimals to derive the derivative of x^2 (use the differential dx)
(1:08:24) Inequality proof using the Racetrack Principle
(1:13:54) Prove a function is constant with the Constant Function Theorem

AMAZON ASSOCIATE
As an Amazon Associate I earn from qualifying purchases.