Problem 1 - Thomas Calculus

Question

Find the average rate of change of the function over the given interval or intervals. \(f(x)=x^{3}+1\) a. \([2,3] \quad\) b. \([-1,1]\)

Step-by-Step Solution

Answer

For \([2,3]\), the average rate is 19. For \([-1,1]\), it is 1.

Step 1: Understand the Problem

We are asked to find the average rate of change of the function \( f(x) = x^3 + 1 \) over two intervals: \([2, 3]\) and \([-1, 1]\). This involves understanding how much the function's output value changes per unit change in input over the specified intervals.

Step 2: Determine the Formula for Average Rate of Change

The average rate of change of a function \( f(x) \) over an interval \([a, b]\) is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \). This formula essentially calculates the slope of the secant line between the points \((a, f(a))\) and \((b, f(b))\).

Step 3: Calculate \(f(x)\) for Interval [2, 3]

Evaluate the function at the endpoints of the interval \([2, 3]\). First, calculate \( f(2) \) and \( f(3) \).\[ f(2) = 2^3 + 1 = 8 + 1 = 9 \]\[ f(3) = 3^3 + 1 = 27 + 1 = 28 \]

Step 4: Compute Average Rate of Change for Interval [2, 3]

Substitute the values obtained for \( f(2) \) and \( f(3) \) into the formula for the average rate of change:\[ \text{Average Rate of Change} = \frac{f(3) - f(2)}{3 - 2} = \frac{28 - 9}{1} = 19 \]

Step 5: Calculate \(f(x)\) for Interval [-1, 1]

Evaluate the function at the endpoints of the interval \([-1, 1]\). First, calculate \( f(-1) \) and \( f(1) \).\[ f(-1) = (-1)^3 + 1 = -1 + 1 = 0 \]\[ f(1) = 1^3 + 1 = 1 + 1 = 2 \]

Step 6: Compute Average Rate of Change for Interval [-1, 1]

Substitute the values obtained for \( f(-1) \) and \( f(1) \) into the formula for the average rate of change:\[ \text{Average Rate of Change} = \frac{f(1) - f(-1)}{1 - (-1)} = \frac{2 - 0}{2} = 1 \]

Key Concepts

Secant Line

When exploring the average rate of change of a function, we often refer to something called the "secant line". Visualize a function's graph as a curvy line on a plane. A secant line in geometry is a straight line that intersects a curve at two distinct points. For our purpose, these two points are defined by the endpoints of the interval we are analyzing.

• In the context of the average rate of change, the secant line connects the points \((a, f(a))\) and \((b, f(b))\) on the graph of the function.
• The slope of this line gives us the average rate of change over the interval \([a, b]\).

Thus, the average rate of change is the same as the slope of this secant line. Finding this slope helps us understand how the function behaves between these points by showing us the ratio of change in output value to change in input value.

Function Evaluation

To compute the average rate of change for a function like \(f(x) = x^3 + 1\), it is crucial first to evaluate the function at specified points. Function evaluation simply means plugging particular values into the function and solving for the output.

• For instance, to find \( f(2)\) for the function \(f(x) = x^3 + 1\), substitute 2 into the function: \(f(2) = 2^3 + 1\), which will give us 9.
• This process is repeated for each endpoint of the interval, which gives us two output values, one for each endpoint.

These evaluations define the points that form the ends of our secant line on the graph. Without these evaluations, we cannot determine the starting and ending points needed to calculate the average rate of change. Function evaluation sets up the stage for all other calculations in the process.

Interval Analysis

When calculating average rates of change, the concept of interval analysis comes into play. Intervals \([a, b]\) describe a range of x-values over which we want to measure change in the function.

• An interval offers two critical points: the start point \(a\) and the endpoint \(b\), creating a section of the x-axis to examine.
• In the context of the problem, each interval is used to determine how the output of a function changes as we move from \(a\) to \(b\).

For example, when analyzing the interval \([2, 3]\), evaluating the function at 2 and then at 3 reveals how the function has changed within this interval. Interval analysis helps focus our calculations and provides a framework to analyze changes systematically. It is crucial for understanding segments of functions, rather than their entire scope at once.