Problem 15 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

find the slope of the secant line passing through points \(P\) and \(Q\), where \(P\) and \(Q\) are points of the graph of \(f(x)\) with the indicated \(x\) -coordinates. \(f(x)=x^{2}+3 x+1 ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=b\) and \(x=b+h(h \neq 0)\), respectively.

Step-by-Step Solution

Answer

The slope of the secant line passing through points \(P\) and \(Q\) is \(2b + h + 3\).

Step 1: Find the Y Coordinates of P and Q

First, plug in \(x = b\) and \(x = b + h\) into the function \(f(x) = x^{2} + 3x + 1\) to find their corresponding \(y\) coordinates. That is,\nfor point \(P\): \(f(b) = b^{2} + 3b + 1\), \nand for point \(Q\): \(f(b + h) = (b + h)^{2} + 3(b + h) + 1\).

Step 2: Simplify the expression for Q's Y coordinate

Simplify \(f(b + h)\) expression: \((b + h)^{2} + 3(b + h) + 1 = b^{2} + 2bh + h^{2} + 3b + 3h + 1\).

Step 3: Calculate the Slope of the Secant Line

Now, use the slope formula \(\frac{y_{2} - y_{1}}{x_{2} - x_{1}}\), which in this case becomes \(\frac{f(b + h) - f(b)}{(b + h) - b} = \frac{(b^{2} + 2bh + h^{2} + 3b + 3h + 1) - (b^{2} + 3b + 1)}{h}\).

Step 4: Simplify the Slope

Simplify the numerator in the slope: \(b^{2} + 2bh + h^{2} + 3b + 3h + 1 - b^{2} - 3b - 1 = 2bh + h^{2} + 3h\). Thus, the slope of the secant line is \(\frac{2bh + h^{2} + 3h}{h}\).

Step 5: Finalize the Slope

Factor out \(h\) in the numerator to simplify the slope: \(h(2b + h + 3)\). Hence, the slope of the secant line becomes \(\frac{h(2b + h + 3)}{h} = 2b + h + 3.\)

Key Concepts

Difference Quotient

Understanding the difference quotient is essential when exploring how functions change at certain points. In calculus, the difference quotient is given by the formula \[\begin{equation}\frac{f(x+h)-f(x)}{h}\end{equation}\]to calculate the average rate of change of the function over the interval from x to x + h, provided h is not zero.

An intuitive way to think about it is momentarily substituting a function's curve with a straight line between two points, which can give us an average slope between those points. When you apply the difference quotient to our exercise, you lay the groundwork for understanding the slope of the secant line—a crucial step before diving into derivatives and the concept of the instantaneous rate of change.

Rate of Change

The rate of change is a measure of how a quantity changes as another quantity changes. In the context of functions, this often refers to how a function's output (y-value) changes with respect to changes in its input (x-value). The average rate of change between two points is represented by the slope of the secant line that passes through these points.

For example, in our exercise, the slope of the secant line between points P and Q gives us the average rate of change of the function f(x) over the interval from b to b+h. This concept is fundamental in calculus because it paves the way for understanding how functions behave locally, leading to the concept of derivatives, which are the function's instantaneous rate of change.

Secant Line in Calculus

Now let's take a closer look at the secant line itself within calculus. A secant line is a straight line that intersects two points on the curve of a function. To find this line's slope—which reflects the average rate of change of the function between these two points—you use the very same difference quotient we discussed earlier.

In our case, identifying the secant line required finding points P and Q on the graph of the function. We then used their coordinates to calculate the slope. Unlike a tangent line, which touches the curve at just one point and represents the instantaneous rate of change, the secant line bridges two points. As h approaches zero, the secant line approaches the tangent line, offering a bridge from average to instantaneous rates of change.

Functions and Graphs

The relationship between functions and graphs provides a visual understanding of concepts like rate of change and the secant line. Each function can be represented by a graph in the coordinate system, with the horizontal axis typically representing the input (x-values) and the vertical axis representing the output (f(x) or y-values).

In our exercise with the function f(x), graphing it would show us a parabola, and the slope of the secant line connecting P and Q represents how steeply this parabola ascends or descends between these points. This graphical perspective is crucial for visual learners and reinforces abstract concepts by grounding them in visual representations. It serves as an invaluable tool for understanding and predicting the behavior of functions, both in math and real-world applications.