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Chapter 3: Problem 2

Explain why the slope of a secant line can be interpreted as an average rate of change.

Short Answer

Expert verified
Answer: The slope of a secant line and the average rate of change share the same formula: \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\), where \(x_1\) and \(x_2\) are two different input values in the function, and \(f(x_1)\) and \(f(x_2)\) are their corresponding output values. This identical formula demonstrates that the slope of a secant line, which is a straight line passing through two points on a curve, represents the average rate of change between those two points in the function.

Step by step solution

01

Define secant line

A secant line is a straight line that passes through two points on a curve in a graphical representation of a function. In general, the secant line represents the average rate of change between those two points.
02

Describe average rate of change

An average rate of change is the change in the dependent variable divided by the change in the independent variable between two points in a function. It gives us an idea of how the function is changing on average over a particular interval. Mathematically, the average rate of change is defined as: Average rate of change = \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\) where \(x_1\) and \(x_2\) are two different input values in the function, and \(f(x_1)\) and \(f(x_2)\) are their corresponding output values.
03

Calculate the slope of a secant line

We can calculate the slope of a secant line using the formula for the slope of a line: Slope of secant line = \(\frac{y_2-y_1}{x_2-x_1}\) Here, (\(x_1\), \(y_1\)) and (\(x_2\), \(y_2\)) are the coordinates of the two points on the curve where the secant line intersects. Since the points lie on the curve described by the function, their coordinates are \((x_1, f(x_1))\) and \((x_2, f(x_2))\).
04

Compare the formulas

Now, let's compare the formulas for the slope of a secant line and the average rate of change: Slope of secant line = \(\frac{y_2-y_1}{x_2-x_1} = \frac{f(x_2)-f(x_1)}{x_2-x_1}\) Average rate of change = \(\frac{f(x_2)-f(x_1)}{x_2-x_1}\) As we can see, the formulas for the slope of a secant line and the average rate of change are identical. This demonstrates that the slope of a secant line can be interpreted as an average rate of change between the two points it intersects on the curve.

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