2022 Live Review 2 | AP Calculus AB | Working with Applications of the Derivative

Mark Corelli and Verge Cornelius, teachers from different high schools, greet their audience and introduce day two of a calculus workshop. They recap the previous day's focus on applications of derivatives and prepare to delve into topics such as related rates, tangent line approximation, curve sketching, and extreme values. They also mention that a significant portion will be dedicated to theorems like the extreme value theorem and mean value theorem on day five. The speakers encourage students to engage with the material and prompt them to set up their calculators for a warm-up problem involving the velocity of an object.

The instructors present a problem involving the calculation of an object's velocity at a specific time using a given height function. They demonstrate both manual and calculator-assisted methods to find the derivative and evaluate it at the given time, emphasizing the utility of calculators in such problems. They also discuss the importance of understanding the context of derivatives in physics, such as velocity and acceleration, and the significance of accurate calculation up to three decimal places as per AP exam standards.

Mark's velocity while riding a bike is described by a function of time. The problem at hand is to find the derivative (velocity) at a specific time, which represents acceleration. The instructors use the concept of the derivative to explain the physical phenomenon and emphasize the need to include units and context when presenting the answer. They also touch upon the concept of speed, which is the absolute value of velocity, and how it differs from velocity when considering direction.

Using a table of position values for Verge at different times, the instructors illustrate how to approximate velocity at a time not explicitly given in the table. They explain the concept of using the average rate of change to estimate the instantaneous rate of change, specifically focusing on the interval closest to the time of interest. The problem highlights the application of derivatives in real-world scenarios and the importance of selecting the most relevant data points for accurate approximations.

The instructors compare the velocities of Mark and Verge at a specific time, using the previously calculated values. They clarify the difference between speed and velocity and how to interpret the derivative in terms of increasing or decreasing values. The comparison serves as a practical example of how to apply calculus concepts to compare rates of change in different scenarios.

The instructors analyze the behavior of a function by examining its first and second derivatives. They use the sign of the first derivative to determine whether the function is increasing or decreasing and the second derivative to assess concavity. The analysis includes identifying intervals of decrease and points of inflection, which are essential for understanding the function's overall behavior and shape.

The session concludes with a problem involving the natural logarithm function, where the goal is to find the maximum value of the function. The instructors demonstrate how to take the first and second derivatives, set them to zero to find critical points, and then use these points to analyze the function's behavior. They emphasize the importance of testing these points to confirm whether they correspond to a maximum or minimum value.

In the final part, the instructors briefly mention the next day's focus on limits and continuity, which will cover a range of topics from the entire course. They summarize the key takeaways from the day, including the importance of differentiating to find rates, understanding the implications of the first and second derivatives, and the significance of limits in calculus. They also encourage students to practice and look forward to the next session.