Problem 11 - College Algebra

Question

For the following exercises, find the average rate of change of each function on the interval specified for real numbers \(b\) or \(h\) in simplest form. $$ a(t)=\frac{1}{t+4} \text { on }[9,9+h] $$

Step-by-Step Solution

Answer

The average rate of change is \(-\frac{1}{13(13+h)}\)."

Step 1: Understanding the Problem

We are asked to find the average rate of change of the function \( a(t) = \frac{1}{t+4} \) over the interval \([9, 9 + h]\). This involves understanding that we need to evaluate the change in the function's output over this interval.

Step 2: Apply the Average Rate of Change Formula

The average rate of change of a function \( f \) from \( x = a \) to \( x = b \) is given by the formula \( \frac{f(b) - f(a)}{b - a} \). Here, \( a = 9 \) and \( b = 9+h \).

Step 3: Evaluate the Function at the Endpoints

Calculate \( a(9) = \frac{1}{9+4} = \frac{1}{13} \) and \( a(9+h) = \frac{1}{(9+h) + 4} = \frac{1}{13+h} \). These values will be used in the rate of change formula.

Step 4: Substitute into the Formula

Substitute these function values into the average rate of change formula: \( \frac{a(9+h) - a(9)}{(9+h) - 9} = \frac{\frac{1}{13+h} - \frac{1}{13}}{h} \).

Step 5: Simplify the Expression

To simplify the expression, find a common denominator for \(\frac{1}{13+h} - \frac{1}{13}\). This results in: \( \frac{13 - (13+h)}{(13+h) \cdot 13} \). The numerator simplifies to \(-h\), so the expression becomes \( \frac{-h}{(13+h) \cdot 13} \).

Step 6: Simplify the Final Expression

Simplify further by dividing both the numerator and denominator by \( h \), resulting in: \( \frac{-1}{13(13+h)} \). Thus, the average rate of change is \( -\frac{1}{13(13+h)} \).

Key Concepts

Interval Notation

Interval notation is a method used to describe a set of numbers along an interval on the real number line. It simplifies how we represent subsets of the real numbers. When you see an interval written like

• \([9, 9+h]\), this represents all numbers between 9 and 9 plus the variable \(h\), including both endpoints.

We use square brackets, [ and ], to indicate that the endpoint is included in the interval, which refers to a closed interval.
In contrast, using parentheses, () or (9, 9+h) would mean that the endpoints are not included, refered to as an open interval.
You might also find mixed notations like [9, 9+h), where one endpoint is included and the other is not. Understanding these symbols helps you determine whether endpoints are part of the interval, vital in functions and calculus.

Rational Functions

Rational functions are mathematical expressions representing the ratio of two polynomials. For example, the function

• \(a(t) = \frac{1}{t+4}\) is a simple rational function.

The numerator is 1, and the denominator is \(t+4\). These expressions can be actionable for finding average rates of change, as in the problem we've solved.
The behavior of rational functions can be complex, with more extensive denominators possibly having asymptotes, holes, or specific restrictions where the function is not defined.
As the denominator approaches zero, the function's value tends to infinity or negative infinity, leading to vertical asymptotes. Understanding rational functions is crucial when manipulating algebraic expressions, especially in calculus and real-world modeling.

Simplifying Expressions

Simplifying expressions involves reducing expressions to their most concise form without changing their value or meaning. This process is critical when working with formulas or solving equations. For example, in finding the average rate of change, we encountered an expression:

• \(\frac{\frac{1}{13+h} - \frac{1}{13}}{h}\).

To simplify, we looked for a common denominator to combine the fractions \(13(13+h)\) and simplify the numerator,

• resulting in \(\frac{-h}{(13+h) \cdot 13}\).

By dividing the entire expression by \(h\), we further simplified to

• \(\frac{-1}{13(13+h)}\).

The simplification needs to be done step-by-step, ensuring mathematical rules, like order of operations and factoring, are respected. Simplifying makes comparisons, evaluations, or graphing functions more manageable and insightful.