AP Calculus AB: Lesson 2.1: The Difference Quotient

In this lesson, Michelle Kremmel introduces the concept of derivatives as the limit of a difference quotient, which is the instantaneous rate of change of a function. The average rate of change is reviewed first, defined as the change in the function's value over an interval divided by the change in the independent variable. This concept is related to the slope of a line and is generalized from the idea of average velocity. The lesson sets the stage for understanding derivatives by using the average rate of change formula and visualizing it on a graph, where the slope of the secant line between two points on the function represents the average rate of change.

The lesson continues by formally defining the derivative as the limit of the average rate of change as the interval width approaches zero, thus becoming the instantaneous rate of change. This is illustrated with the difference quotient, which is the formula for the derivative. The derivative is denoted as f'(a) and represents the slope of the tangent line to the function at a specific point x=a. The importance of the limit in this definition is emphasized, as it distinguishes the derivative from the average rate of change. The derivative's existence, indicated by the limit, is also discussed, noting that it may not always exist at certain points.

This section delves deeper into the derivative's interpretation as the slope of the curve or the tangent line at a point. The concept of differentiability is introduced, stating that if the derivative exists at x=a, the function is differentiable at that point. The derivative's role in determining the instantaneous rate of change is reiterated, and the difference between average and instantaneous rates of change is highlighted. The vocabulary of differentiation is also introduced, including terms like 'differentiable,' 'differentiate,' and 'differentiation.'

The lesson explains that the derivative, or the instantaneous rate of change, has units of the dependent variable per unit of the independent variable. This is illustrated with an example where the cost function is measured in dollars and the quantity in bracelets, resulting in the derivative being measured in dollars per bracelet. The derivative's geometric interpretation as the slope of the tangent line to the curve at a point is also discussed, emphasizing the difference between the constant slope of a line and the varying slope of a curve.

The point-slope form of the equation of a line is introduced as the method for writing the equation of the tangent line to a function at a specific point. The importance of memorizing this equation is stressed, as it is central to calculus. The relationship between the derivative, the slope of the tangent line, and the curve's slope at a point is reiterated. The lesson also poses potential questions that might be asked regarding the tangent line, such as finding the slope of the tangent line at a specific x-value or approximating function values using tangent lines.

The script uses a series of graphs to visually demonstrate the concept of the derivative. As the interval width (h) approaches zero, the secant lines between points on the function become closer and closer to the tangent line at a point, illustrating the transition from the average rate of change to the instantaneous rate of change. The importance of the limit in the definition of the derivative is emphasized through this visual progression.

An example is provided to demonstrate the calculation of the derivative using the difference quotient. The function f(x) = x - x^2 is used to find the derivative at x=2. The process involves algebraic manipulation to simplify the expression and then taking the limit as h approaches zero, resulting in the derivative value. The significance of the limit in obtaining the instantaneous rate of change is highlighted.

The properties of linear functions are discussed, emphasizing that the slope is constant across the entire function. This is contrasted with the varying slope of a curve at different points. The average rate of change for a linear function is shown to be constant, regardless of the interval chosen, and the derivative of a linear function is computed to demonstrate that it is constant, as expected.

The derivative is applied to a real-world problem involving the volume of a balloon changing as it is filled with air. The volume function V(r) = (4/3)πr^3 is given, and the average rate of change of volume with respect to the radius is calculated for different intervals. The derivative is used to find the instantaneous rate of change at specific points, highlighting the application of derivatives in understanding how quantities change in real situations.

The lesson concludes with an example of estimating the value of a function at a point using the tangent line. Given a function with a known value and derivative at x=2, the tangent line equation is used to estimate the function value at x=2.5. This demonstrates the practical use of derivatives in approximating function values and understanding the local behavior of functions.