Calculus 1 Final Exam Review Part 1 | Behind the Scenes with Professor V | How I Write Exams

Professor V introduces the video by explaining the plan to review a Calculus 1 final exam. The focus will be on working through a comprehensive study guide to provide insights into the topics covered, the skill set required, and the types of questions that might be on the final exam. The video aims to help students identify areas for further study and reinforce their understanding of calculus concepts.

The video delves into the concept of finding limits, emphasizing the importance of mastering different types of limits, particularly those that do not involve L'Hopital's rule. The professor demonstrates how to evaluate limits by direct substitution and factoring, especially in cases where the limit appears as an indeterminate form. The process of factoring is highlighted as a crucial skill for solving such problems.

Absolute value functions and their piecewise nature are discussed, with the professor explaining how to rewrite absolute value functions as piecewise functions. The video emphasizes the importance of understanding where the function's domain splits and how to evaluate limits at points where the domain changes. The concept of one-sided limits is also introduced, showing how to determine if a limit exists from the left or right side of a critical point.

The Squeeze theorem, also known as the sandwich theorem, is introduced as a method for finding limits when direct substitution is not possible. The professor provides a visual example to illustrate how the theorem works, emphasizing its usefulness when dealing with trigonometric functions. The theorem's conditions and application are explained in detail, showing how it can be used to confirm the existence of a limit.

The video discusses the use of graphical analysis to determine limits, particularly for piecewise functions. The professor demonstrates how to sketch the graph of a function and use it to find limits as x approaches specific values. The concept of one-sided limits is revisited, with examples showing how to find the limit as x approaches a value from the right or left side.

Continuity is defined with three conditions that must be met for a function to be continuous at a point. The professor explains these conditions and how they relate to the function's behavior. The Intermediate Value Theorem is introduced, which states that if a function is continuous on a closed interval and the function values at the endpoints are different, then the function will take on any value between those two values at some point within the interval.

The video addresses the concept of limits as x approaches positive or negative infinity. The professor explains how to evaluate these limits by looking at the highest power in the numerator and denominator and provides examples of how to handle indeterminate forms. The process of dividing by the highest power of x is demonstrated, along with how to handle cases where the limit does not exist.

The video presents a problem involving finding a value that makes a piecewise function continuous at a specific point. The professor outlines the three conditions for continuity and demonstrates how to apply them to find the required value. The process involves finding the limit as x approaches the point of discontinuity and ensuring that the function value matches this limit.

The video concludes with a review of differentiation rules, including the chain rule, product rule, quotient rule, and various applications. The professor provides examples of how to apply these rules to find derivatives of functions, including more complex examples involving the quotient and chain rules. The importance of factoring and understanding the underlying concepts is emphasized.

Professor V wraps up the video by discussing strategies for approaching the final exam, emphasizing the importance of not cutting corners and providing thorough evidence of understanding. The video mentions the availability of playlists for calculus and pre-calculus review, and encourages students to subscribe for more content. The professor signs off, promising a follow-up video to continue the review of calculus concepts.