Problem 36 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \left|\frac{c^{2}-c}{\cos c}\right| ;\left[0, \frac{\pi}{4}\right] $$

Step-by-Step Solution

Answer

The expression is zero at \( c = 0 \), positive for \( 0 < c \leq \frac{\pi}{4} \).

Step 1: Understand the Expression

The given mathematical expression is \( \left| \frac{c^{2}-c}{\cos c} \right| \). It involves a real number \( c \) and we are looking for its values in the interval \( \left[ 0, \frac{\pi}{4} \right] \). The expression inside the absolute values can change sign, which we will explore in subsequent steps.

Step 2: Determine where \( \cos c \) equals zero

First, we need to ensure \( \cos c eq 0 \) because division by zero is undefined. Within the interval \( \left[ 0, \frac{\pi}{4} \right] \), the cosine function does not equal zero. Thus, the expression is defined for this entire interval.

Step 3: Analyze the Quadratic Expression

The quadratic expression in the numerator is \( c^{2} - c \). For values where \( c^{2} - c = 0 \), we solve for \( c \):\[ c(c-1) = 0 \]Thus, \( c = 0 \) or \( c = 1 \). Only \( c = 0 \) lies within the interval \( \left[ 0, \frac{\pi}{4} \right] \).

Step 4: Evaluate and Interpret Solutions

The expression becomes zero when \( c = 0 \). For other values in the interval \( \left(0, \frac{\pi}{4}\right] \), we need to calculate \( \frac{c^{2} - c}{\cos c} \) and apply absolute value. Calculations at different points in the interval confirm that we need to consider the sign of \( c^{2} - c \): negative leads to a positive absolute value.

Step 5: Consider Properties Across the Interval

Since \( c^2 < c \) for values \( c \in (0, 1) \), the fraction within the absolute value is negative, leading to a positive absolute value of the entire expression. For \( c \geq 1 \), the expression is positive, hence its absolute value equals the expression itself. However, these values do not lie inside \([0, \frac{\pi}{4}]\) since \( \frac{\pi}{4} < 1 \).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics, often symbolized by two vertical bars (| |) surrounding a number or expression, such as \( |x| \). It represents the distance of a number from zero on the number line, which means it is always non-negative.

• The absolute value of a positive number is the number itself, \( |5| = 5 \).
• The absolute value of a negative number is its opposite, \( |-3| = 3 \).
• If the number is zero, its absolute value is also zero, \( |0| = 0 \).

In the expression \( \left| \frac{c^{2}-c}{\cos c} \right| \), the absolute value ensures that regardless of whether the fraction \( \frac{c^{2}-c}{\cos c} \) itself is positive or negative, the final result will be positive. This is crucial when calculating functions across intervals, as the actual numeric value might change, but its "magnitude" remains consistent.

Quadratic Expressions

Quadratic expressions are polynomial expressions of degree two, generally represented as \( ax^2 + bx + c \). In the exercise, the quadratic expression \( c^2 - c \) forms the numerator of the given fraction.

• Quadratics can take various forms when graphed, typically as a parabola.
• The zeros of the quadratic \( c^2 - c \) can be found by solving the equation \( c(c-1) = 0 \).
• This results in the solutions \( c = 0 \) and \( c = 1 \).

However, within the interval \( [0, \frac{\pi}{4}] \), only \( c = 0 \) is valid because \( \frac{\pi}{4} \approx 0.785 \), which is less than 1. Quadratic expressions are key to determining the behavior of the function across the interval, indicating where it reaches zero or changes sign.

Trigonometric Functions

Trigonometric functions like sine and cosine are periodic functions that relate angles to sides in right triangles or unit circles. In this exercise, \( \cos c \) is crucial as it's in the denominator of the fraction, \( \frac{c^2 - c}{\cos c} \).

• \( \cos c \) represents the x-coordinate on the unit circle as \( c \) varies as an angle.
• It is crucial to ensure that \( \cos c eq 0 \) because division by zero is undefined.
• Within the interval \( [0, \frac{\pi}{4}] \), \( \cos c \) remains positive and does not reach zero.

Understanding the cosine function aids in determining where the fraction is valid and ensures we can calculate values without encountering mathematical errors.

Inequalities

Inequalities help determine the sign of an expression across a given range. In this exercise, inequalities are used to assess when the quadratic expression \( c^2 - c \) changes sign.

• The inequality \( c^2 - c < 0 \) holds true for \( c \in (0, 1) \).
• For this interval, the fraction \( \frac{c^2 - c}{\cos c} \) is negative, which the absolute value converts to positive.
• Beyond this interval, \( \c >= 1 \), the expression becomes positive, although it isn't relevant since \( \frac{\pi}{4} < 1 \).

Inequalities play a vital role in identifying when expressions are positive or negative, which determines the application of absolute values and affects interpretations of expressions in intervals.