Problem 1 - Calculus Early Transcendentals: Pearson New International Edition

Question

In each of the Problems 1-21, a function is defined and a closed interval is given. Decide whether the Mean Value Theorem applies to the given function on the given interval. If it does, find all possible values of c; if not, state the reason. In each problem, sketch the graph of the given function on the given interval.$$ f(x)=|x| ;[1,2] $$

Step-by-Step Solution

Answer

Mean Value Theorem applies; all \( c \) in (1, 2) satisfy it.

Step 1: Check Continuity on the Interval

The function \( f(x) = |x| \) is defined for all real numbers and is continuous everywhere. Therefore, \( f(x) \) is continuous on the closed interval \([1, 2]\).

Step 2: Check Differentiability on the Interval

The function \( f(x) = |x| \) is differentiable everywhere except at \( x = 0 \). Since \( x = 0 \) is not within the interval \([1, 2]\), the function is differentiable on the open interval \((1, 2)\).

Step 3: Apply the Mean Value Theorem

The Mean Value Theorem states that if \( f(x) \) is continuous on \([a, b]\) and differentiable on \((a, b)\), there exists at least one \( c \) in \((a, b)\) such that \( f'(c) = \frac{f(b) - f(a)}{b - a} \). Here, \( a = 1 \) and \( b = 2 \).

Step 4: Calculate \( f'(x) \)

For \( x > 0 \), \( f(x) = x \), so \( f'(x) = 1 \).

Step 5: Calculate \( \frac{f(b) - f(a)}{b - a} \)

Calculate \( f(2) - f(1) \):\[ f(2) = |2| = 2 \]\[ f(1) = |1| = 1 \]Thus, \( \frac{f(2) - f(1)}{2 - 1} = \frac{2 - 1}{1} = 1 \).

Step 6: Solve \( f'(c) = 1 \) for \( c \)

Since \( f'(x) = 1 \) for all \( x \) in the interval \((1, 2)\), any \( c \) within this interval satisfies \( f'(c) = 1 \). Therefore, the Mean Value Theorem holds for \( c \) in \((1, 2)\).

Step 7: Conclude the Solution

The Mean Value Theorem applies to \( f(x) = |x| \) on \([1, 2]\), and any \( c \) in the open interval \((1, 2)\) satisfies the theorem.

Key Concepts

Continuity

Continuity is a key concept in calculus that describes a function's behavior at every point within a certain interval. In simple terms, a function is continuous if you can draw its graph without lifting your pencil from the paper. For a function to use the Mean Value Theorem, it must be continuous over the closed interval being considered. In this example with the function \( f(x) = |x| \) on the interval \([1, 2]\), the function is continuous throughout the interval.
Since absolute value functions are inherently continuous across their entire domain, they perfectly meet this requirement in our interval.

Differentiability

Differentiability refers to a function's ability to have a derivative at each point in an open interval. It indicates smoothness or the ability of the graph to have a tangent line at every point. For the Mean Value Theorem to apply, besides continuity on the closed interval, the function must be differentiable on the open interval. With \( f(x) = |x| \), differentiability is satisfied for \( x > 0 \), as it is differentiable everywhere except at \( x = 0 \).
Since 0 is not within our interval \( (1, 2) \), the function is differentiable for every \( x \) in the open interval, making it eligible for the theorem.

Absolute Value Function

The absolute value function characterizes any real number by its distance from zero, essentially removing any negative sign. As such, the function \( f(x) = |x| \) outputs the same value as its input if the input is positive, or the positive version of its input if it is negative.
It's defined as:

• \( f(x) = x \) if \( x \geq 0 \)
• \( f(x) = -x \) if \( x < 0 \)

Its properties of continuity over all real numbers make it suitable for calculus applications like the Mean Value Theorem, especially over strictly positive intervals, as in this case.

Calculus

Calculus provides a framework for understanding changes and areas under curves. It consists primarily of two main operations: differentiation and integration. Differentiation measures how a function changes at any given point, while integration concerns the accumulation of values.
The Mean Value Theorem in calculus combines these ideas, stating that if a function is continuous on a closed interval and differentiable on the open interval, there is at least one point within the interval where the function behaves as an average rate of change over that interval. This theorem was verified for our example function \( f(x) = |x| \) within the specified interval, showing that calculus allows us to derive important characteristics of functions and their graphs.