Problem 35 - Calculus Early Transcendentals: Pearson New International Edition

Question

$$ \left|\frac{c^{2}+\sin c}{10 \ln c}\right| ;[2,4] $$

Step-by-Step Solution

Answer

The maximum value of the expression in [2, 4] is approximately 1.100.

Step 1: Evaluate the Expression Inside the Absolute Value

The expression inside the absolute value is \( \frac{c^2 + \sin c}{10 \ln c} \). To evaluate this, plug in values of \( c \) within the closed interval \([2, 4]\).

Step 2: Substitute \( c = 2 \)

Substitute \( c = 2 \) into the expression:\[\frac{2^2 + \sin(2)}{10 \ln(2)} = \frac{4 + \sin(2)}{10 \ln(2)}\]

Step 3: Compute \( \sin(2) \) and \( \ln(2) \)

Calculate \( \sin(2) \approx 0.909 \) and \( \ln(2) \approx 0.693 \).

Step 4: Calculate the Expression for \( c = 2 \)

Plug the approximate values back into the expression:\[\frac{4 + 0.909}{10 \times 0.693} \approx \frac{4.909}{6.93} \approx 0.709\]

Step 5: Substitute \( c = 4 \)

Substitute \( c = 4 \) into the expression:\[\frac{4^2 + \sin(4)}{10 \ln(4)} = \frac{16 + \sin(4)}{10 \ln(4)}\]

Step 6: Compute \( \sin(4) \) and \( \ln(4) \)

Calculate \( \sin(4) \approx -0.757 \) and \( \ln(4) \approx 1.386 \).

Step 7: Calculate the Expression for \( c = 4 \)

Plug the approximate values back into the expression:\[\frac{16 - 0.757}{10 \times 1.386} \approx \frac{15.243}{13.86} \approx 1.100\]

Step 8: Compare Values within the Interval

The expression's values for \( c = 2 \) and \( c = 4 \) are approximately \( 0.709 \) and \( 1.100 \) respectively. The maximum value within the interval will be \( 1.100 \).

Step 9: Find the Absolute Maximum Value

The absolute maximum value for the expression inside its absolute value, across the interval \([2,4]\), is approximately \( |1.100| = 1.100 \).

Key Concepts

Absolute Value

Absolute value is a fundamental concept in mathematics that indicates the distance of a number from zero on the number line. It is always a non-negative value. Absolute value is represented by vertical bars, such as in \(|x|\), which means the absolute value of \(x\).
Key points about absolute value include:

• The absolute value of a positive number is the number itself.
• The absolute value of a negative number is the opposite of the number, making it positive.
• The absolute value of zero is zero.

When solving expressions that involve absolute value, such as \(\left|\frac{c^2 + \sin c}{10 \ln c}\right|\), it means we're interested in the non-negative value of the result. In the steps provided, once the maximum value of the evaluated expression was found to be 1.100, the absolute value is directly 1.100, since it is already a positive number.

Logarithms

Logarithms play a critical role in mathematics, especially in calculus. A logarithm answers the question: "To what exponent must the base be raised, to yield a given number?" The common base used in natural logarithms is \(e\), approximately 2.718.
When you see \(\ln(x)\), it denotes the natural logarithm of \(x\). This is crucial in calculus because natural logarithms have properties that make differentiation and integration more manageable.
In the expression \(\frac{c^2 + \sin c}{10 \ln c}\), the logarithm part \(\ln(c)\) is essential because it acts as a divisor, impacting the overall size of the fraction. Computing \(\ln(2)\) and \(\ln(4)\) gave us approximately 0.693 and 1.386 respectively, which were then used to evaluate the expression at different values of \(c\). This helped understand how changing \(c\) affects the output of the logarithmic function, thus impacting the overall result.

Trigonometric Functions

Trigonometric functions are another cornerstone of calculus and are used to relate the angles of triangles to the lengths of their sides. They include sine, cosine, tangent, and their reciprocals. In our example exercise, the sine function, \(\sin c\), is crucial.
Here's what you should know about sine and how it was applied:

• \(\sin(x)\) represents the sine of an angle \(x\), a trigonometric function that outputs the y-coordinate of a point on the unit circle.
• The sine function has a periodic nature, which means it repeats its values in a predictable cycle.
• It ranges from -1 to 1, making it a bounded function.

For the values within the closed interval [2, 4], \(\sin(c)\) was calculated as approximately 0.909 for \(c=2\) and -0.757 for \(c=4\). These values were essential in determining the overall values of the expressions \(4 + \sin(2)\) and \(16 + \sin(4)\), affecting the numerator and thus the entire fraction for those specific inputs.