Problem 20 - Calculus

Question

find \(k\) so that the function is continuous on any interval. $$f(x)=\left\\{\begin{array}{ll} k x & 0 \leq x<2 \\ 3 x^{2} & 2 \leq x \end{array}\right.$$

Step-by-Step Solution

Answer

The value of \(k\) is 6 for continuity on any interval.

Step 1: Understanding the Problem

We are given a piecewise function \(f(x)\) with two parts: \(kx\) when \(0 \leq x < 2\), and \(3x^2\) when \(x \geq 2\). We need to find \(k\) such that \(f(x)\) is continuous at \(x = 2\), as continuity at other points is ensured by the definition for each segment.

Step 2: Definition of Continuity

For the function \(f(x)\) to be continuous at \(x = 2\), the left-hand limit, right-hand limit, and the function value at \(x = 2\) must all be equal. Thus, we need to compute \(\lim_{x \to 2^-} f(x)\), \(\lim_{x \to 2^+} f(x)\), and \(f(2)\).

Step 3: Compute Left-Hand Limit

The left-hand limit as \(x\) approaches 2 from the left (\(x \to 2^-\)) using \(f(x) = kx\) is \(\lim_{x \to 2^-} kx = k \cdot 2 = 2k\).

Step 4: Compute Right-Hand Limit

The right-hand limit as \(x\) approaches 2 from the right (\(x \to 2^+\)) using \(f(x) = 3x^2\) is \(\lim_{x \to 2^+} 3x^2 = 3 \cdot 2^2 = 12\).

Step 5: Check Function Value at x=2

Since the function definition for \(x \geq 2\) is \(3x^2\), the value at \(x = 2\) is \(f(2) = 3 \cdot 2^2 = 12\).

Step 6: Equalize Limits and Function Value

For continuity at \(x=2\), set the left-hand limit equal to the right-hand limit and the function value: \(2k = 12\).Solve for \(k\) to find: \(k = 6\).

Step 7: Conclusion

For the function to be continuous on any interval, \(k\) must be 6.

Key Concepts

Piecewise Functions

A piecewise function is a function defined by different expressions over different parts of its domain. This allows for flexibility in modeling real-world scenarios, where a one-size-fits-all equation might not be sufficient.
For example, a piecewise function might have one rule for small values of a variable and a different rule for larger values.

• The notation for piecewise functions involves specifying the domain of each piece, using braces to denote the different cases.
• A piecewise function can also describe a situation where the physical characteristics of a system change at a certain threshold, such as different pricing tiers in a service plan.

Understanding how to work with piecewise functions is essential for analyzing situations where behavior changes at certain intervals. When dealing with such functions, it's crucial to evaluate each section according to its specified rule. Recognizing how each part fits into the whole is important for ensuring that the function behaves as expected across its entire domain.

Limits

Limits are a fundamental concept in calculus that describe the behavior of a function as the input approaches a specified value. Limits help us understand what a function is approaching, even if the function is not explicitly defined at that point.
They are especially useful when dealing with piecewise functions, where transitions occur between different segments.

• The left-hand limit (\( \lim_{x \to c^-} f(x) \)) considers the value that the function approaches as the variable comes from values less then \(c\).
• The right-hand limit (\( \lim_{x \to c^+} f(x) \)) examines the approach from values greater than \(c\).

To determine the overall limit at a junction point, both the left and right-hand limits must be equivalent, ensuring that the function approaches a single value from both sides. This process is crucial in analyzing piecewise functions, as it helps verify that the transition between different segments is seamless.

Calculus Continuity

In calculus, continuity is a property of a function that indicates it has no breaks, jumps, or holes at any given point within its domain. For a function to be continuous at a specific point \(x = c\), three conditions must be satisfied:

• The function \(f(x)\) must be defined at \(x = c\).
• The limit of \(f(x)\) as \(x\) approaches \(c\) must exist.
• The limit and the actual value of the function at that point must be equal: \(\lim_{x \to c} f(x) = f(c)\).

For piecewise functions, ensuring continuity at the boundaries between pieces is vital. This often involves equating the left and right-hand limits of the function at those points where the function's definition changes. Successfully matching these limits ensures there is seamless flow from one section of the function to the next, making the transition appear smooth and continuous.