Problem 14 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

For Problems 14 through 16, 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. \(f(x)=\frac{3}{x}+2 x ;\) the \(x\) -coordinates of \(P\) and \(Q\) are \(x=3\) and \(x=3+k(k \neq 0)\), respectively.

Step-by-Step Solution

Answer

The slope of the secant line passing through points \(P\) and \(Q\) is found by simplifying the expression \([f(3+k) - f(3)] / k\), where \(f(x) = 3/x + 2x\) and \(k \neq 0\).

Step 1: Understand and Calculate the Coordinates of P and Q

First, plug in the x coordinates into the function to get the y coordinates of points P and Q. For point P, when \(x=3\), calculate \(f(3)\) in the equation. For point Q, when \(x = 3+k\), calculate \(f(3+k)\) to get the coordinates of Q.

Step 2: Find the Secant Line Slope

The slope of the line passing through points \(P\) and \(Q\) is given by \([f(3+k) - f(3)] / [3+k -3]\), which simplifies to \([f(3+k) - f(3)] / k\). Subsitute the coordinate values from the previous step to get the slope.

Step 3: Simplify and Evaluate

Simplify the expression to get the slope of the secant line.

Key Concepts

Slope of a Line

Understanding the slope of a line is crucial in mathematics, as it measures the steepness, or angle, of a line. The slope is characterized by the ratio of the vertical change (rise) to the horizontal change (run) between any two points on the line. In mathematical terms, if you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line, the slope \( m \) is calculated as \( m = \frac{y_2 - y_1}{x_2 - x_1} \).

When applying this to a secant line — a line cutting across a curve at two points — this formula helps determine the average rate of change of the function between those two points. In calculus, this average rate of change can lead to an understanding of instantaneous rates of change when the secant line becomes a tangent line, with both points becoming infinitesimally close.

In the provided exercise, the function \( f(x) = \frac{3}{x} + 2x \) involves a more complex application, incorporating function evaluation within the slope calculation, with given points P and Q having specific \( x \) coordinates. Here, the slope formula is adapted to accommodate the function values at these points.

Secant Lines in Calculus

Secant lines serve an important role in calculus, often acting as a bridge to more advanced concepts like derivatives. A secant line intersects a curve at two distinct points and can be used to approximate the slope of the curve at one point. The term comes from the Latin word 'secare', meaning 'to cut,' reflecting the line's action on a curve.

When you're dealing with functions and their graphs, finding the slope of a secant line involves substituting the \( x \) values of the two points into the function to obtain their corresponding \( y \) values. This procedure is vividly demonstrated in the exercise: you calculate the coordinates of points P and Q on the graph of \( f(x) \) and then use these coordinates to determine the slope of the secant line that cuts through these points.

As the distance between the two points along the \( x \) axis (denoted by \( k \) in the exercise) gets smaller, the secant line closely approximates the tangent line at point P. In the limit, where \( k \) approaches zero, the concept of the derivative comes into play, solidifying the secant line's importance in the study of calculus.

Function Evaluation

Function evaluation is a foundational skill in calculus that involves finding the output values (\( y \) values) of a function given specific input values (\( x \) values). This process is essential for determining the exact coordinates of points on a graph.

In the context of our exercise, to evaluate the function \( f(x) = \frac{3}{x} + 2x \) at \( x = 3 \) and \( x = 3+k \) requires substituting these values into the function. For example, for point P at \( x=3 \) you would compute \( f(3) = \frac{3}{3} + 2(3) \) to find the \( y \) coordinate. Evaluating at \( x = 3+k \) for point Q is a bit more involved as it includes an expression instead of a number.

The accuracy of function evaluation directly affects the precision of the slope calculation for the secant line. Missteps in this initial step can lead to errors in finding the secant line slope. Therefore, this step lays the groundwork for mastering the key aspects of calculus—focusing on both the computational and conceptual understanding of functions and their graphical representations.