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Magazine Chapter 12: Problem 95
If the function \(f(x)\), defined as \(f(x)= \begin{cases}\frac{a(1-x \sin x)+b \cos x+5}{x^{2}}, x<0 \\ 3 & , x=0 \\ \left\\{1+\left(\frac{c x+d x^{3}}{x^{2}}\right)\right\\}^{1 / x} & , x>0\end{cases}\) is continuous at \(x=0\), then (A) \(a=-1\) (B) \(b=-4\) (C) \(c=0\) (D) \(\log _{e}{ }^{3}\)
Short Answer
Expert verified
(A) \(a = -1\), (B) \(b = -4\), (C) \(c = 0\) ensure continuity; \(f(0) = 3\).
Step by step solution
01
Understanding Continuity
The function is continuous at a point if the left-hand limit, right-hand limit, and the function's value at that point are all equal. We need to evaluate the limits of \( f(x) \) as \( x \to 0^- \) and \( x \to 0^+ \), and ensure they are equal to \( f(0) = 3 \).
02
Evaluate the Left-Hand Limit
For the left-hand limit as \( x \to 0^- \), the function is \( f(x) = \frac{a(1-x \sin x)+b \cos x+5}{x^2} \). Simplifying, we note that \( \sin x \approx x \) and \( \cos x \approx 1 \) when \( x \to 0 \), giving us:\( \lim_{{x \to 0^-}} \frac{a(1-x^2)+b + 5}{x^2} = \lim_{{x \to 0^-}} \frac{a - ax^2 + b + 5}{x^2} \).To make the function continuous, the denominator's degree must match the numerator's, so solving \((a + b + 5) = 0\).
03
Solve for the Coefficients
From the left-hand limit equation, setting the terms:\[ a + b + 5 = 0 \]So, \( b = -a - 5 \).
04
Evaluate the Right-Hand Limit
For \( x > 0 \), \( f(x) = \left(1 + \left(\frac{cx + dx^3}{x^2}\right)\right)^{1/x} = \left(1 + \frac{c}{x} + dx \right)^{1/x} \).As \( x \to 0^+ \), \( \frac{c}{x} \) becomes infinite unless \( c = 0 \). Therefore, set \( c = 0 \).
05
Function Value at Zero
\( f(0) = 3 \), and it should match the limits found at \( x \to 0^- \) and \( x \to 0^+ \). Using the calculations \( b = -a - 5 \) and substituting \( c = 0 \), we ensure the left-hand limit equals 3, solving for:\[ a + (-a - 5) + 5 = 0 \right] \rightarrow 3a = -15 \rightarrow a = -1 \].
06
Verify
Substituting \( a = -1 \), \( b = -4 \), and \( c = 0 \), both limits and the value at \( x = 0 \) are consistent, confirming that both \( a = -1 \) and \( b = -4 \) ensure the function's continuity.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Limits
Limits are a fundamental concept in calculus and play a crucial role in analyzing the behavior of functions near a particular point. A limit helps us understand the value that a function approaches as the input (usually denoted as \( x \)) gets infinitely close to a specific value. This concept is the backbone of determining function continuity, differentiability, and integrality.
When examining the continuity of a function at a certain point, we often talk about the left-hand limit and the right-hand limit:
In our exercise, evaluating limits on both sides of \( x = 0 \) is essential for determining the function's continuity. Left and right-hand limits are calculated separately for \( x < 0 \) and \( x > 0 \) to ensure they both point towards the same value at \( x = 0 \).
When examining the continuity of a function at a certain point, we often talk about the left-hand limit and the right-hand limit:
- **Left-Hand Limit**: This is the value approached by the function as the variable approaches the point from the left side (i.e., \( x \to c^- \)).
- **Right-Hand Limit**: Conversely, this is the value approached by the function as the variable approaches the point from the right side (i.e., \( x \to c^+ \)).
In our exercise, evaluating limits on both sides of \( x = 0 \) is essential for determining the function's continuity. Left and right-hand limits are calculated separately for \( x < 0 \) and \( x > 0 \) to ensure they both point towards the same value at \( x = 0 \).
Unpacking Piecewise Functions
A piecewise function is a type of mathematical function defined by multiple sub-functions, each applicable to a certain interval of the main function's domain. These functions change their behavior based on the input value, making them very flexible.
For example, the exercise function is piecewise, with different formulas used depending on whether \( x \) is less than, greater than, or equal to zero. Such structuring allows complex behaviors to be modeled in a single mathematical construct. The challenge with piecewise functions is often ensuring the transitions between different sub-functions are smooth, particularly at the boundaries where the function segmentation occurs.
In this exercise, the piecewise function required an evaluation to ensure continuity at \( x = 0 \). This means we need to ascertain that no matter from which side the \( x \) approaches zero, the function's value remains consistent (i.e., there is no jump or break at \( x = 0 \)). By analyzing each piece and solving for coefficients, we found values that smoothed out the transitions.
For example, the exercise function is piecewise, with different formulas used depending on whether \( x \) is less than, greater than, or equal to zero. Such structuring allows complex behaviors to be modeled in a single mathematical construct. The challenge with piecewise functions is often ensuring the transitions between different sub-functions are smooth, particularly at the boundaries where the function segmentation occurs.
In this exercise, the piecewise function required an evaluation to ensure continuity at \( x = 0 \). This means we need to ascertain that no matter from which side the \( x \) approaches zero, the function's value remains consistent (i.e., there is no jump or break at \( x = 0 \)). By analyzing each piece and solving for coefficients, we found values that smoothed out the transitions.
Exploring Mathematical Function Properties
Mathematical functions have intrinsic properties that determine their behavior and guide their applications across various fields. Some key concepts include continuity, differentiability, and integrability.
- **Continuity**: Refers to a function that has no breaks, jumps, or holes when graphed. For a function to be continuous at a point, the left-hand limit, right-hand limit, and the actual function value at that point must all agree.- **Differentiability**: Differentiability means that the function has a derivative at all points in its domain. A function must be continuous before being differentiable.- **Integrability**: This property refers to if and how a function can be integrated over an interval. Continuous functions over a closed interval are typically integrable.
In our problem, the focus is on ensuring continuity of the piecewise function at \( x = 0 \). By solving for the coefficients \( a \), \( b \), and \( c \), it was confirmed that with specific values (\( a = -1 \), \( b = -4 \), \( c = 0 \)), the function remains continuous over its domain, thereby satisfying one of the key attributes of a well-behaved mathematical function.
- **Continuity**: Refers to a function that has no breaks, jumps, or holes when graphed. For a function to be continuous at a point, the left-hand limit, right-hand limit, and the actual function value at that point must all agree.- **Differentiability**: Differentiability means that the function has a derivative at all points in its domain. A function must be continuous before being differentiable.- **Integrability**: This property refers to if and how a function can be integrated over an interval. Continuous functions over a closed interval are typically integrable.
In our problem, the focus is on ensuring continuity of the piecewise function at \( x = 0 \). By solving for the coefficients \( a \), \( b \), and \( c \), it was confirmed that with specific values (\( a = -1 \), \( b = -4 \), \( c = 0 \)), the function remains continuous over its domain, thereby satisfying one of the key attributes of a well-behaved mathematical function.
