Problem 18 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Let \(g\) be a function that is locally linear. We know that \(g^{\prime}(a)\) is the slope of the tangent line to the graph of \(g\) at point \(A=(a, g(a))\). Let \(Q=(t, g(t))\) be an arbitrary point on the graph of \(g, Q\) distinct from \(A\). (a) Write a difference quotient (i.e., an expression of the form \(\frac{?-?}{2-?}\), the quotient of two differences) that gives the slope of the secant line through points \(A\) and \(Q\). (b) Take the appropriate limit of the difference quotient in part (a) to arrive at an expression for \(g^{\prime}(a)\).

Step-by-Step Solution

Answer

a) The difference quotient is \(\frac{g(t)-g(a)}{t-a}\).\n b) The expression for \(g^{\prime}(a)\) is \(\lim_{t \to a} \frac{g(t) - g(a)}{t - a}\).

Step 1: Setting up the Difference Quotient

Form a difference quotient which is the formula to get the slope of the secant line through points \(A\) and \(Q\). This is given by \(\frac{g(t) - g(a)}{t - a}\). This formula represents the change in the function \(g\) over the change in its input.

Step 2: Limit of the Difference Quotient

To find \(\(g^{\prime}(a)\)\), the derivative of \(g\) at \(a\), we take the limit as \(t\) goes to \(a\) of this difference quotient. This will give the slope of the tangent line at \(A\), which is \(g^{\prime}(a)\). So, applying the limit, the derivative of \(g\) at \(a\) is: \(\lim_{t \to a} \frac{g(t) - g(a)}{t - a}\). This is the formal definition of the derivative.

Key Concepts

Difference Quotient

The difference quotient is a fundamental tool in calculus that measures the average rate of change of a function over a given interval. It is usually represented in the form \(\frac{f(x+h) - f(x)}{h}\), where \(h\) is the change in the input value of the function \(f\). In simple terms, it compares the difference in the function's output values to the difference in its input values.

For the problem at hand, the difference quotient is expressed as \(\frac{g(t) - g(a)}{t - a}\), which calculates the average rate of change of the function \(g\) from \(a\) to \(t\). This quotient is crucial as it lays the foundation for understanding the behavior of the function as the input value \(t\) approaches \(a\). It is a stepping stone towards finding the precise instantaneous rate of change or the function's derivative at a specific point.

Tangent Line Slope

The slope of a tangent line to a curve at a given point is a measure of how steep the line is at that precise location. It reflects the rate of change of the function at that point and is equivalent to the value of the derivative at that same point.

In the context of the exercise, the slope of the tangent line \(g^\prime(a)\) is what we are ultimately trying to find. By taking the limit of the difference quotient as \(t\) approaches \(a\), we are honing in on the infinitesimally small interval around \(a\) to determine the function's rate of change at \(a\) itself. This action essentially transitions the average slope of the secant line into the precise slope of the tangent line.

Secant Line Slope

Conversely, the slope of a secant line is the average rate of change of the function across two points. In the problem provided, these two points are \(A=(a, g(a))\) and \(Q=(t, g(t))\). The secant line slope can be visualized as the slope of the straight line that connects two points on the curve of the function \(g\).

The formula \(\frac{g(t) - g(a)}{t - a}\) represents the slope of this secant line. As \(t\) varies, the slope of the secant line provides an average rate of change between \(A\) and \(Q\). This concept is important because as \(t\) gets closer to \(a\), the secant line starts to resemble the tangent line at \(A\), leading us towards the derivative.

Limit of a Function

The limit of a function is a fundamental concept in calculus that describes the behavior of a function as the input approaches a certain value. The limit does not necessarily represent the function's output at that value, but rather where the function's value is headed or the trend it displays.

Applying the concept of limits to the difference quotient \(\frac{g(t) - g(a)}{t - a}\) as \(t\) approaches \(a\) leads to understanding the derivative \(g^\prime(a)\). In this exercise, taking the limit of the difference quotient allows us to move from the average rate of change over an interval to the instantaneous rate of change at a point - which provides us with \(g^\prime(a)\), the exact slope of the tangent line at \(a\). Understanding the calculation of limits is crucial as they are the gateway to finding derivatives, which have numerous applications across various fields of study.