Problem 5 - College Algebra with Corequisite Support

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. \(f(x)=4 x^{2}-7\) on \([1, \quad b]\)

Step-by-Step Solution

Answer

Average rate of change is \(4(b + 1)\).

Step 1: Understand the Formula for Average Rate of Change

The average rate of change of a function between two points is calculated using the formula: \( \frac{f(b) - f(a)}{b - a} \). This is similar to finding the slope of the line connecting two points \((a, f(a))\) and \((b, f(b))\) on the graph of the function.

Step 2: Evaluate the Function at Given Points

First, we need to find the values of the function \( f(x) = 4x^2 - 7 \) at the endpoints of the interval \([1, b]\).- Calculate \( f(1) = 4(1)^2 - 7 \):\[ f(1) = 4 - 7 = -3 \]- Now calculate \( f(b) = 4b^2 - 7 \). We will leave this expression in terms of \( b \) since we do not have a specific numeric value for \( b \).

Step 3: Substitute into Average Rate of Change Formula

Substitute \( f(1) \) and \( f(b) \) into the average rate of change formula:\[\frac{f(b) - f(1)}{b - 1} = \frac{(4b^2 - 7) - (-3)}{b - 1}\]Simplify the expression in the numerator:\[ 4b^2 - 7 + 3 = 4b^2 - 4 \].

Step 4: Simplify the Expression

The expression for the average rate of change is:\[\frac{4b^2 - 4}{b - 1}\]This can be factored in the numerator as:\[ 4(b^2 - 1) = 4(b - 1)(b + 1) \]So the expression becomes:\[\frac{4(b - 1)(b + 1)}{b - 1}\]We can cancel \(b - 1\) from the numerator and denominator:\[ 4(b + 1) \]

Step 5: Final Result

So, the average rate of change of \( f(x) = 4x^2 - 7 \) on \([1, b]\) is: \[ 4(b + 1) \].

Key Concepts

Quadratic Functions

Quadratic functions are an essential part of algebra and calculus, often appearing in many areas of mathematics. At their core, a quadratic function is defined as any function that can be written in the form:
\( f(x) = ax^2 + bx + c \), where \( a \), \( b \), and \( c \) are constants, with \( a eq 0 \).
In our exercise, the function \( f(x) = 4x^2 - 7 \) is a quadratic function. The main characteristic of quadratic functions is their U-shaped graph known as a parabola. The exact shape and position of the parabola depend on the coefficients \( a \), \( b \), and \( c \).

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), it opens downwards.
• The vertex of the parabola is the highest or lowest point, depending on the direction it opens.

Understanding the basics of quadratic functions can help when calculating the average rate of change between points on their curve. This concept is analogous to finding how steep a hill is between two points rather than the whole hill.

Interval Notation

Interval notation is a crucial concept in mathematics, particularly when dealing with functions on certain domains. It is a way of specifying a particular range of x-values.
This notation is designed to help us easily communicate which values are included or excluded from a given set. In interval notation, a pair of numbers are used within brackets:

• Square brackets [ ] mean the endpoint is included.
• Parentheses ( ) mean the endpoint is not included.

In the given exercise, the interval \([1, b]\) includes '1' but leaves 'b' open, meaning it can be reached but not necessarily included depending on further context.When calculating the average rate of change, understanding interval notation is essential because it tells us exactly between which points we need to evaluate the function's behavior. It clearly marks what's taken into account and what's not. This helps avoid mistakes and confusion in calculations, providing clarity.

Function Evaluation

Function evaluation is a fundamental skill in algebra used often in calculus problems like finding average rates of change. It involves calculating the output of a function for given input values.
For example, in our task, we're dealing with \( f(x) = 4x^2 - 7 \).When evaluating this function for a specific x-value, such as \( x = 1 \), simply substitute '1' into every \( x \) in the function:
\( f(1) = 4(1)^2 - 7 = -3 \).Evaluating at \( x = b \) requires a slightly different approach since 'b' is a variable:

• We substitute \( b \) into the function instead of a number.
• This keeps the function in terms of \( b \), allowing for algebraic simplification.

Once function evaluations are completed at the interval endpoints, these results are plugged into the average rate of change formula.
Function evaluation is the process that ensures you're working with accurate values warranted by the problem setup or user-given conditions.