Problem 32 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \left|\frac{4 c}{c+4}\right| ;[0,1] $$

Step-by-Step Solution

Answer

The expression is \( \frac{4c}{c+4} \) on [0,1], with a range from 0 to \(\frac{4}{5}\).

Step 1: Analyze the Expression Inside the Absolute Value

First, consider the expression \( \frac{4c}{c+4} \). We need to examine how this expression behaves numerically over the interval \([0,1]\).

Step 2: Determine Expression Sign in \([0,1]\)

Check if the expression \( \frac{4c}{c+4} \) changes sign over the interval \([0,1]\). Since \(4c\) is always non-negative and \(c+4\) is positive due to \(c\) being non-negative, \(\frac{4c}{c+4}\) is always non-negative over the interval.

Step 3: Simplify Absolute Value Expression

Due to \( \frac{4c}{c+4} \) being non-negative, the absolute value does not change the expression. Therefore, \( \left| \frac{4c}{c+4} \right| = \frac{4c}{c+4} \) for \(c\) in \([0,1]\).

Step 4: Evaluate Expression at Interval Endpoints

Now, evaluate \( \frac{4c}{c+4} \) at the endpoints of the interval: \(c=0\), \( \frac{4 \times 0}{0+4} = 0 \), and \(c=1\), \( \frac{4 \times 1}{1+4} = \frac{4}{5} \).

Step 5: Conclusion on Range of the Function

Over the interval \([0,1]\), the expression \( \left| \frac{4c}{c+4} \right| \) takes values ranging from 0 to \(\frac{4}{5}\).

Key Concepts

Absolute Value

The absolute value of a number is its distance from zero on the number line, regardless of direction. In mathematical terms, the absolute value of a number \( x \), denoted as \( |x| \), is:

• \( x \), if \( x \geq 0 \)
• -\( x \), if \( x < 0 \)

The absolute value is used to express non-negative quantity. In the context of the given problem, we evaluated \( \left| \frac{4c}{c+4} \right| \). Since the expression was shown to be non-negative over the interval \([0, 1]\), the absolute value did not alter the expression's form. This highlights how absolute value functions ensure that any expression inside its bars becomes non-negative.

Expression Evaluation

Expression evaluation involves calculating the value of an algebraic expression. Let's consider the expression \( \frac{4c}{c+4} \). This can be evaluated based on the value given for the variable \( c \). Through substitution and simplification, the numerical value of the expression is determined.

When evaluating expressions, it's crucial to pay attention to:

• The domain of the expression, meaning the values \( c \) can take without the denominator becoming zero.
• Whether the expression is defined over that domain interval.

For instance, substituting the values \( c=0 \) and \( c=1 \) into \( \frac{4c}{c+4} \), gave us the specific outputs needed for analyzing the interval \([0,1]\). This establishes the foundation for the following concept.

Interval Analysis

Interval analysis refers to examining how an expression behaves over a specified range (or interval) of input values. The problem examined the behavior of \( \frac{4c}{c+4} \) over the interval \([0,1]\). Understanding how an expression behaves within an interval is crucial when determining the function's output range or when solving inequality problems.

In this exercise, we needed to verify if the expression remained non-negative and continuous over the chosen interval, which dictates no sudden jumps or undefined points. The key steps involved in interval analysis are:

• Checking sign changes within the interval to identify behavior consistency.
• Evaluating endpoints separately to ensure they are included in the function's range.

By confirming that \( \frac{4c}{c+4} \) didn't change sign and stayed positive, we safely said it was consistent over \([0,1]\).

Function Range

The range of a function is the set of all possible output values it can produce for given inputs. After determining how the function behaves over a specific interval (from interval analysis), finding its range becomes straightforward.

In the problem, we analyzed \( \left| \frac{4c}{c+4} \right| \) to find it behaved from 0 to \( \frac{4}{5} \) over \([0, 1]\). The endpoints \( c=0 \) and \( c=1 \) were used to establish these boundaries:

• At \( c = 0: \ rac{4 \times 0}{0+4} = 0 \)
• At \( c = 1: \ rac{4 \times 1}{1+4} = \frac{4}{5} \)

By following these outputs, the range becomes the interval \([0, \frac{4}{5}]\). The comprehension of this concept is vital as it ensures that all potential outputs given the constraints are covered, providing a complete picture of the function's behavior.