Fall 2023 MNC: Up close and personal with differentiability and local linearity

The video begins with Curtis Brown introducing the Monday Night Calculus session on October 2nd, 2023. He expresses excitement for the attendees and introduces Steve Kokaska and Tom Dick, two calculus experts who will be sharing their knowledge. Curtis mentions that he did not do his homework and will be managing the chat during the session. The attendees are eager to start, with some mentioning their preference for the calculus session over watching the Manning brothers. Curtis also discusses the importance of differentiability and local linearity in calculus.

The session continues with a focus on differentiability and local linearity. The speaker explains the strict definition of the derivative of a function at a point and introduces the alternate definition involving a limit. The discussion includes the importance of experience in choosing the appropriate definition for finding the derivative function. The speaker emphasizes that a function is differentiable at a point if its graph is smooth, meaning there are no sharp corners or abrupt changes in direction. The concept of local linearity is introduced, explaining that a differentiable function's graph appears as a straight line near a certain point, making the tangent line a good approximation.

The video segment involves a demonstration using a calculator to graph functions and understand the concept of tangent lines. Tom shares his screen and uses a TI-84 calculator to illustrate the graph of a function. Curtis and Tom discuss the appearance of the graph and how it can be identified as a straight line, specifically suggesting it might be y equals 1.3x. They discuss the importance of zooming in on the graph to understand the tangent line and the significance of differentiability. A trick for teachers is shared on how to use the Trace function without showing the expression on the screen.

The discussion moves to analyzing two specific functions, f(x) as the absolute value of x and g(x) as the square root of (x squared + 0.01) minus 0.01. The domain of both functions is identified as all real numbers. The speaker instructs the audience to graph both functions and observe their similarities. A table of values for both functions is presented, highlighting the closeness of their values except for a difference of about 0.01. The speaker emphasizes the importance of understanding the symmetry of functions and how it affects the graph and values.

The focus shifts to finding the derivative of function G. The speaker explains the process of using the definition of a limit to find the derivative at zero. The calculation involves simplifying the expression and evaluating the limit to get the general derivative expression. The derivative of g at zero is found to be zero, indicating a tangent line with a slope of zero. The speaker emphasizes the importance of understanding the definition of the derivative and the ability to manipulate limits to find derivatives.

The session continues with an examination of differentiability through the graphs of two functions. The speaker and Tom discuss the points at which the functions are not differentiable, noting the lack of continuity and the presence of sharp corners. They emphasize that for a function to be differentiable, it must be continuous. The speaker also discusses the importance of recognizing where a graph is differentiable and where it is not, which is a key skill for understanding calculus concepts.

The video segment involves working with functions involving absolute values. The speaker rewrites the function f(x) as x times the absolute value of x and analyzes its differentiability at zero. The derivative is found using the definition of the derivative and the properties of absolute values. The function is found to be differentiable at zero with a derivative of zero. The speaker also discusses the concept of piecewise-defined functions and their derivatives, emphasizing the importance of understanding the behavior of functions near zero.

The speaker reflects on the relationship between tangent lines and the behavior of functions. They discuss the importance of understanding the slope of the tangent line and how it relates to the increasing and decreasing nature of the function. A question from the chat about the placement of the equality sign in the function's domain is addressed, highlighting the importance of clarity in mathematical notation. The speaker also introduces the concept of the sign function and its role in differentiating the behavior of the function based on the sign of the argument.

The video concludes with a comparison of tangent lines for three different functions at the point where x equals zero. The speaker calculates the equation of the tangent line for each function and finds that they all have the same slope and pass through the point (0, 1). They discuss the importance of understanding the tangent line at a specific point versus the tangent function as a whole. The speaker and Tom use a graphing calculator to visualize the functions and their tangent lines, noting that all functions approach the tangent line as they get closer to the point of tangency. The session ends with a discussion on the best linear approximation for each function based on how closely they follow the tangent line over a range of x values.

The video addresses several complex calculus problems, including the volume of a solid of revolution and the identification of local maxima and minima. The speaker emphasizes the importance of proper notation when using integration by parts and the expectation that BC students be familiar with this concept. The video also discusses the use of sign charts for determining local maxima and minima, noting that while they are useful for organization, a written justification is required for the AP exam. The session concludes with a discussion on identifying and classifying points of discontinuity as removable or non-removable.