Problem 12 - 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. $$ b(x)=\frac{1}{x+3} \text { on }[1,1+h] $$

Step-by-Step Solution

Answer

The average rate of change is \(\frac{-1}{4(h+4)}\).

Step 1: Understanding the Problem

We need to find the average rate of change of the function \(b(x) = \frac{1}{x+3}\) on the interval \([1, 1+h]\). This requires calculating how much the function changes per change in \(x\) over this interval.

Step 2: Definition of Average Rate of Change

The average rate of change of a function \(b(x)\) on the interval \([a, a+h]\) is defined as \(\frac{b(a+h) - b(a)}{h}\). Here, \(a = 1\) and we need to find \(b(1+h)\) and \(b(1)\).

Step 3: Evaluate b(x) at 1

Substitute \(x=1\) into the function to find \(b(1)\): \[b(1) = \frac{1}{1+3} = \frac{1}{4}\].

Step 4: Evaluate b(x) at 1+h

Substitute \(x=1+h\) into the function to find \(b(1+h)\): \[b(1+h) = \frac{1}{(1+h)+3} = \frac{1}{h+4}\].

Step 5: Substitute into the Average Rate of Change Formula

Use the values from Step 3 and Step 4 in the average rate of change formula: \[\frac{b(1+h) - b(1)}{h} = \frac{\frac{1}{h+4} - \frac{1}{4}}{h}\].

Step 6: Simplify the Expression

To simplify \(\frac{\frac{1}{h+4} - \frac{1}{4}}{h}\), first find a common denominator for the terms in the numerator: \[\frac{1}{h+4} - \frac{1}{4} = \frac{4 - (h+4)}{4(h+4)} = \frac{-h}{4(h+4)}\].

Step 7: Combine and Simplify the Expression Further

Substitute the expression from Step 6 into the denominator: \[\frac{-h}{4(h+4) \cdot h} = \frac{-1}{4(h+4)}\].

Step 8: Final Answer

Thus, the average rate of change of \(b(x) = \frac{1}{x+3}\) on the interval \([1, 1+h]\) is \[\frac{-1}{4(h+4)}\].

Key Concepts

Functions in Calculus

Understanding functions is crucial when diving into calculus. A function in calculus is like a rule that assigns an output for each input. This output is derived using a mathematical expression. In our problem, the function is given as \(b(x) = \frac{1}{x+3}\). It involves a simple fraction where the input \(x\) affects the denominator. Functions can represent many real-world situations such as rates, areas, and growth patterns. In calculus, functions help us to evaluate changes and understand behaviors. The rate of change is an essential aspect, providing insight into how the function's values transform as its inputs vary. Thus, exploring functions helps us grasp complex mathematical ideas by approaching them through growth, reduction, or stability, depending on their specific behavior.

Interval Notation

Interval notation is a concise way of describing a set of numbers grouped between endpoints. It uses brackets \([\ ,\ ]\) representing inclusive intervals. In our example, we examine the interval \([1, 1+h]\). This crucially highlights the intended starting and ending points within which our computations are focused.* Square brackets \([\ ,\ ]\) indicate endpoints are included in the interval.* Round brackets \((\ ,\ )\) indicate endpoints are excluded.Using interval notation, we can define the range of \(x\) values considered for a particular problem. It clarifies simple terms like 'between' and outlines specific domains where the function can be applied. This helps ensure calculations hold relevance precisely to our scope of interest, making the mathematics more precise and efficient.

Evaluating Expressions

Evaluating expressions involves calculating their value by substituting variables with numbers. This task becomes central when determining the average rate of change. In this exercise: 1. We substitute \(x = 1\) to find \(b(1)\) resulting in \(b(1) = \frac{1}{4}\).2. We substitute \(x = 1+h\) to evaluate \(b(1+h)\) resulting in \(b(1+h) = \frac{1}{h+4}\).This step ensures we accurately assess the values that a function can take over a specified interval. For deeper understanding, ensure each substitution clearly shows how changes in \(x\) reflect changes in the function, providing direct insight into function characteristics. By practicing these evaluations, complex fractions and formulas become more approachable, allowing more seamless problem solving.