Monday Night Calculus: Derivatives of Trigonometric Functions

The video begins with Curtis Brown introducing the Monday Night Calculus webinar, welcoming viewers to the session focused on derivatives of trigonometric functions. He introduces Steve and Tom, who will lead the discussion, and expresses gratitude to Mark Corali for promoting the event and supporting AP Calculus teachers. The segment also acknowledges international viewers from Canada and emphasizes the importance of understanding trigonometric functions in radian measure rather than degrees, highlighting a common mistake made by students on the AP Calculus exam.

Steve starts the main topic by discussing the derivatives of the sine function using an inquiry-based learning approach. He shares his experience of teaching this concept in China and emphasizes the graphical, numerical, and analytical methods of understanding derivatives. Steve uses the top graph to illustrate tangent lines and slopes to estimate the derivative of sine function. He then transitions to a more rigorous analytical approach, using the definition of the derivative and trigonometric identities to derive the result that the derivative of sine function is cosine function. The segment also addresses common errors and encourages participants to think about the graphical representation of the derivative.

The discussion continues with an in-depth exploration of the derivatives of trigonometric functions using limit laws and trigonometric identities. The focus is on the sine function, where Steve uses the definition of the derivative and manipulates the expression to isolate terms. He combines terms, factors out sine and cosine, and uses the limit laws to find that the derivative of sine is cosine. The segment also poses questions to the audience, encouraging them to consider different ways to estimate and prove the limit involved in the derivative calculation. Steve also mentions the use of L'Hopital's rule and Taylor series in estimating limits.

Steve presents an example of applying the product rule to find the derivative of a function that involves both sine and a polynomial. He emphasizes the importance of verifying the analytical results through graphical evidence. Steve uses technology to graph the function and its derivative, showing that the derivative is positive when the function is increasing and negative when the function is decreasing. This visual confirmation helps students understand the relationship between the function and its derivative. The segment also discusses the importance of knowing the derivatives of trigonometric functions for AP Calculus exams and provides a problem for the audience to practice finding the derivative of the tangent function using the quotient rule.

The focus shifts to utilizing technology for calculating derivatives and tangent lines. Steve demonstrates how to use a graphing calculator to find the equation of a tangent line to a function at a specific point. He emphasizes the importance of understanding the quotient rule and the use of numerical answers in AP Calculus exams. The segment also includes a discussion on the use of CAS systems and the benefits of defining functions and values for later use in calculations. Steve provides a problem involving the secant function and demonstrates how to find the equation of the tangent line using both symbolic and numerical methods.

The video concludes with a discussion on solving parametric motion problems. Steve presents a problem involving a particle moving along a horizontal line and asks viewers to find the first time the particle is at rest. He explains how to translate the problem into mathematical terms and uses the product rule to find the derivative. Steve also discusses the importance of verifying results using graphical and numerical methods. He uses a graphing calculator to graph the function and demonstrate the motion of the particle, highlighting the value of technology in visualizing and understanding calculus concepts.

Tom takes over to explore the use of technology in understanding and visualizing trigonometric functions and their derivatives. He uses a graphing calculator to plot the sine function and its derivative, the cosine function, emphasizing the visual connection between the two. Tom discusses the ability to change tick marks to multiples of pi for better understanding and the dynamic updating of functions for different derivative estimations. He also highlights the power of technology in quickly visualizing and understanding the concepts of derivatives and local linearity.

Curtis wraps up the session by thanking Steve and Tom for their contributions and the insights they've shared on calculus. He encourages teachers to involve their students in the Monday Night Calculus sessions and to use the provided resources and problem sets for their classes. Curtis also mentions the availability of the session links and problem sets in the chat and looks forward to the next topic on local linearity.