AP Calculus AB: Lesson 2.2 The Derivative Function

Michelle Crummel introduces the concept of the derivative function, focusing on the function f(x) = 4x - x^2. She outlines an activity to compute the derivative values at specific points using the limit definition. The goal is to observe patterns in the derivative values and conjecture a general formula for f'(a) based on the observed values. The process begins with finding f'(0) using the limit as h approaches 0, simplifying the expression to isolate the derivative, and finding that f'(0) equals 4.

The script continues with the calculation of derivative values at points f'(1), f'(2), and f'(3). Through simplification and factoring, it is found that f'(1) = 2, f'(2) = 0, and f'(3) = -2. A pattern is observed where the derivative decreases by 2 as the input value increases by 1. Based on this, conjectures are made for f'(4) and f'(5), and a general formula f'(a) = -2a + 4 is proposed, illustrating the relationship between input and output values.

The explanation shifts to the definition of the derivative function, f'(x), as the limit of (f(x+h) - f(x))/h as h approaches 0. This definition provides the slope of the tangent line at any arbitrary point x on the graph of f. The script differentiates between finding the slope at a specific point (f'(a)) and the slope at any point (f'(x)). The process of finding the derivative for the given function f(x) = 4x - x^2 is demonstrated, resulting in the derivative function f'(x) = 4 - 2x.

The script explains how to graph the derivative function by plotting points that correspond to the slopes of tangent lines at various x-coordinates of the original function f(x). It provides a method to sketch the graph of the derivative without the derivative equation, using the original function's graph to estimate slopes and plot points. The process involves identifying points where the slope is zero, and using symmetry or estimation to plot additional points, ultimately connecting them to form the derivative graph.

The video script discusses techniques for estimating the graph of the derivative function from the graph of the original function. It emphasizes identifying points where the derivative is zero, as these correspond to horizontal tangent lines on the original graph. The process involves visual estimation of slopes at various points on the graph and plotting these on the derivative graph, providing a rough sketch of the derivative's behavior.

The script presents several examples of finding the derivative function for different types of functions, including constant functions, linear functions, quadratic functions, and others. Each example demonstrates the process of applying the limit definition to find the derivative, simplifying expressions, and factoring out the variable h to isolate the derivative. The results include derivatives such as 0 for a constant function, 1 for a linear function with a slope of 1, 2z for a quadratic function z^2, and -1/t^2 for the function 1/t.

The final part of the script focuses on finding the derivative of the square root function. The process involves multiplying by the conjugate to simplify the expression and eliminate the h in the denominator. The limit is then taken as h approaches 0, resulting in the derivative of the square root function being 1/(2√y), with the original variable x replaced by y to maintain consistency with the function notation.