Problem 22 - Calculus

Question

find \(k\) so that the function is continuous on any interval. $$h(x)=\left\\{\begin{array}{ll} k \cos x & 0 \leq x \leq \pi \\ 12-x & \pi \leq x \end{array}\right.$$

Step-by-Step Solution

Answer

The value of \( k \) is \( \pi - 12 \) for continuity.

Step 1: Evaluate continuity requirements

The function \( h(x) \) is a piecewise function that might have a discontinuity at the boundary \( x = \pi \) between \( k \cos x \) and \( 12 - x \). To ensure the function is continuous, both pieces must be equal at \( x = \pi \).

Step 2: Set expressions equal at boundary

At \( x = \pi \), set the two expressions equal to each other: \( k \cos(\pi) = 12 - \pi \).

Step 3: Solve for \( k \) using trigonometric identity

Since \( \cos(\pi) = -1 \), substitute into the equation: \( k(-1) = 12 - \pi \), simplifying to \(-k = 12 - \pi \).

Step 4: Solve for \( k \)

Rearrange the equation \(-k = 12 - \pi \) to find \( k \): \( k = \pi - 12 \).

Key Concepts

Piecewise Function

A piecewise function is a type of function that is defined by multiple sub-functions, each applied to a specific part of the function's domain. In the given problem, the function \( h(x) \) is defined in two parts:

• \( k \cos x \) for \( 0 \leq x \leq \pi \)
• \( 12 - x \) for \( \pi \leq x \)

This structure implies that depending on the value of \( x \), the function will behave differently:

• For values of \( x \) between \( 0 \) and \( \pi \), the function will follow the pattern of a cosine wave scaled by the constant \( k \).
• For \( x \) values greater than or equal to \( \pi \), the function becomes linear, decreasing with a slope of \(-1\).

Piecewise functions are common in many real-world applications where different conditions govern behavior, such as in physics or economics. Ensuring these functions transition seamlessly from one piece to another, especially at the boundaries, is crucial for continuity.

Continuity

Continuity in mathematics ensures that a function behaves predictably without any sudden jumps or breaks. For a function to be continuous:

• The function must be defined at the point in question.
• The limit of the function as it approaches the point from either side must exist.
• The value of the limit must equal the actual value of the function at that point.

In this exercise, the goal is to ensure continuity at the boundary \( x = \pi \). This requires:

• Checking if both pieces of the piecewise function meet at \( x = \pi \) and produce the same output.
• Setting \( k \cos(\pi) = 12 - \pi \) to match the function values at the boundary, thereby ensuring the function has no gaps.

By solving this equation, you adjust \( k \) to ensure the transition between the cosine and linear pieces is smooth, without a jump or discontinuity.

Trigonometric Identity

Trigonometric identities are mathematical equations that relate various angles and lengths in trigonometry. They are immensely useful in simplifying expressions and solving trigonometric equations.In this specific problem, the identity \( \cos(\pi) = -1 \) plays a crucial role. Because:

• The cosine function, \( \cos(\theta) \), is periodic with known values at critical points like \( \pi \), which is particularly helpful as cosine returns negative one at \( \pi \).
• This identity directly simplifies the equation \( k \cos(\pi) = 12 - \pi \) to \(-k = 12 - \pi \).

Understanding and applying these identities can transform otherwise complex problems into much simpler equations. Consequently, this allows you to directly solve for variables like \( k \) with greater ease, as demonstrated in the step-by-step solution by rearranging to find \( k = \pi - 12 \). Grasping these identities often gives insight into the behavior of trigonometric functions at key angles, which can be pivotal in both theoretical and applied mathematics.