Question

\(g(x)=x^{2}+1\) (a) Find the average rate of change from -1 to 2 . (b) Find an equation of the secant line containing \((-1, g(-1))\) and \((2, g(2))\).

Short answer

(a) The average rate of change from -1 to 2 is 1. (b) The equation of the secant line is y = x + 3.

Step-by-step solution

- Evaluate the Function at Given Points

First, compute the values of the function at the endpoints of the interval: Evaluate at \(-1\): \[g(-1) = (-1)^{2} + 1 = 1 + 1 = 2\]Evaluate at \(2\): \[g(2) = 2^{2} + 1 = 4 + 1 = 5\]

- Calculate the Average Rate of Change

The average rate of change of the function over an interval is given by the formula: \[ \text{Average Rate of Change} = \frac{g(x_2) - g(x_1)}{x_2 - x_1} \]Substitute the given points, \(-1\) and \(2\), as \(x_1\) and \(x_2\) respectively:\[ \text{Average Rate of Change} = \frac{g(2) - g(-1)}{2 - (-1)} = \frac{5 - 2}{2 - (-1)} = \frac{3}{3} = 1 \]

- Find the Points for the Secant Line

We already have the points; they are \(-1, g(-1)\) and \(2, g(2)\): \[ (-1, 2) \quad \text{and} \quad (2, 5) \]

- Determine the Slope of the Secant Line

The slope (\(m\)) of the secant line is the same as the average rate of change: \[ m = 1 \]

- Write the Equation of the Secant Line

Using the point-slope form of a line equation \(y - y_1 = m(x - x_1)\), we can write the equation of the secant line. Select one of the points, for example, \(-1, 2\): \[ y - 2 = 1(x - (-1)) \]Simplify: \[ y - 2 = x + 1 \]Thus, the equation of the secant line is: \[ y = x + 3 \]

Key concepts

Secant Line

A secant line is a straight line that intersects a curve at two or more points. It's quite useful in calculus and other areas of mathematics for approximating the slope of a curve. In our exercise, the secant line intersects the curve of the function at the points \( -1, g(-1) \) and \( 2, g(2) \). The primary goal is to find the equation of this line.
This line can be helpful in studying the behavior of the function between those two points.

Slope

The slope of a line measures its steepness. It's calculated by the difference in the y-coordinates divided by the difference in the x-coordinates between two points. For our secant line, we find the slope using the points \((-1, 2) \) and \( (2, 5) \): \[ m = \frac{5 - 2}{2 - (-1)} = \frac{3}{3} = 1. \] This slope tells us how much the y-value of the function changes for each unit change in the x-value.
In other words, for each step we move to the right on the x-axis, the value on the y-axis increases by 1.

Function Evaluation

Function evaluation involves calculating the output of a function for specific values of the input. In our problem, we evaluate the function \( g(x) = x^2 + 1 \) at the points -1 and 2:
- For \(-1\): \ g(-1) = (-1)^2 + 1 = 2 \
- For \(2\): \ g(2) = 2^2 + 1 = 5 \
This step is crucial because we need these values to determine the slope and the coordinates of the secant line. It shows how the function behaves at specific points on the x-axis.

Point-Slope Form

The point-slope form of a line equation is incredibly useful when you know a point on the line and the slope. It's given by \ y - y_1 = m(x - x_1) \, where \( (x_1, y_1) \) is a point on the line and \( m \) is the slope.
From our exercise, we know the slope \( m = 1 \) and a point \(-1, 2\). Plugging these into the formula gives us: \[ y - 2 = 1(x - (-1)) \] Simplifying further, we get \[ y - 2 = x + 1 \] or \[ y = x + 3. \] This final equation represents the secant line connecting the points on the curve.