Problem 117
Question
In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True and Reason(R) is True; Reason(R) is a correct explanation for Assertion(A) (B) Assertion(A) is True, Reason(R) is True; Reason(R) is not a correct explanation for Assertion(A) (C) Assertion(A) is True, Reason(R) is False (D) Assertion(A) is False, Reason(R) is True Assertion: Let \(f\) be a function such that \(f(x y)=f(x)\). \(f(y), \forall y \in R\) and \(f(1+x)=1+x(1+g(x))\), where \(\lim _{x \rightarrow 0} g(x)=0\), then $$ \begin{aligned} &\int_{1}^{2} \frac{f(x)}{f^{\prime}(x)} \cdot \frac{1}{1+x^{2}} d x=\frac{1}{2} \log \left(\frac{5}{2}\right) \\ &\text { Reason: } f^{\prime}(x)=\frac{f(x)}{x} \end{aligned} $$
Step-by-Step Solution
Verified Answer
(C) Assertion(A) is True, Reason(R) is False
1Step 1: Analyzing the Assertion
The assertion involves a function \( f \) having properties \( f(xy) = f(x)f(y) \) and \( f(1+x) = 1 + x(1 + g(x)) \), where \( \lim_{x \rightarrow 0} g(x) = 0 \). The goal is to evaluate the integral \[\int_{1}^{2} \frac{f(x)}{f'(x)} \cdot \frac{1}{1+x^{2}} \, dx = \frac{1}{2} \log \left(\frac{5}{2}\right)\] given these properties.By examining \( f(xy) = f(x)f(y) \), we infer that \( f \) might be an exponential function. Since \( f(1) = 1 \), and \( f(1+x) = 1 + x(1 + g(x)) \), as \( x \rightarrow 0 \), this suggests \( f \) is very close to linear around \( x = 0 \) and essentially, \( f'(x) \approx x \). This model could satisfy the assertion if chosen correctly.
2Step 2: Analyzing the Reason
The reason given states that \( f'(x) = \frac{f(x)}{x} \). Based on our analysis in Step 1, if \( f(x) = e^{x} \), then \( f'(x) = e^{x} \), which does not meet the reason. Instead, for \( f'(x) = \frac{f(x)}{x} \) to be true, it corresponds to function types such as logarithmic setups, but based on \( f(1+x) \), an exponential function is plausible around linear increments.
3Step 3: Evaluating the Consistency
To check the assertion: approximating an integral of the form that appears in the statement using our understanding of the derivative, if executed correctly, would potentially lead to the given result. However, if we strictly apply \( f'(x) = \frac{f(x)}{x} \), the function type doesn't align precisely with our earlier approximations concerning exponential behaviors or a polynomial analogous to the assertion's formulation.
4Step 4: Determination of Correct Answer
Given that \( f'(x) = \frac{f(x)}{x} \) doesn't align well with the properties outlined by both \( f(x) \) suggestions and our evaluations sketch a potential exponential, the reason's accountability in explaining or holding precisely to operation within the defined assertion's framework likely fails. Thus, it is valid to resolve with:\[ C: \textrm{Assertion(A) is True, Reason(R) is False} \] asserts.
Key Concepts
Integral EvaluationFunction PropertiesLimits and Continuity
Integral Evaluation
Integral evaluation is a fundamental aspect of calculus that involves finding the area under a curve or the accumulation of quantities over a certain interval.
In the given problem, the task is to evaluate the integral \[\int_{1}^{2} \frac{f(x)}{f^{\prime}(x)} \cdot \frac{1}{1+x^{2}} \, dx\] which essentially requires understanding the behavior of the function \(f(x)\) and its derivative \(f'(x)\).
One of the key steps in integral evaluation is to have a clear understanding of the function's form and properties. This helps in identifying potential simplifications or substitutions that can make the integral easier to solve.
By understanding these components and using function properties wisely, one can effectively solve the integral, yielding the result specified or questioning the reasoning as done in the exercise.
In the given problem, the task is to evaluate the integral \[\int_{1}^{2} \frac{f(x)}{f^{\prime}(x)} \cdot \frac{1}{1+x^{2}} \, dx\] which essentially requires understanding the behavior of the function \(f(x)\) and its derivative \(f'(x)\).
One of the key steps in integral evaluation is to have a clear understanding of the function's form and properties. This helps in identifying potential simplifications or substitutions that can make the integral easier to solve.
- Recognize the contribution of each part of the integrand: \(\frac{f(x)}{f'(x)}\) and \(\frac{1}{1+x^2}\).
- Analyze if the substitution or preference of properties aligns with the integral limits.
- Get familiar with common solutions and results that can be directly applied.
By understanding these components and using function properties wisely, one can effectively solve the integral, yielding the result specified or questioning the reasoning as done in the exercise.
Function Properties
Function properties are attributes or characteristics that define the behavior and features of a function. For the function \(f(x)\) discussed in this exercise, several properties were key to deciding its evaluation and subsequent integral computation.
The assertion in the problem gives us two properties:
These properties give insights into the nature of the function, particularly hinting towards an exponential form, especially for small increments. Knowing if the function is exponential, polynomial, or involving logarithmic behavior can influence not only the calculation of derivatives but also how integrals involving the function are approached.
In general, recognizing these properties helps in determining if one can apply standard calculus techniques or if specialized methods are necessary.
The assertion in the problem gives us two properties:
- \(f(xy) = f(x)f(y)\) which suggests a multiplicative property of the function, often associated with exponential functions.
- \(f(1+x) = 1 + x(1 + g(x))\) with \(\lim_{x \to 0} g(x) = 0\) indicating a behavior similar to linearity for small \(x\).
These properties give insights into the nature of the function, particularly hinting towards an exponential form, especially for small increments. Knowing if the function is exponential, polynomial, or involving logarithmic behavior can influence not only the calculation of derivatives but also how integrals involving the function are approached.
In general, recognizing these properties helps in determining if one can apply standard calculus techniques or if specialized methods are necessary.
Limits and Continuity
Limits and continuity are central concepts in calculus, critical for understanding the behavior of functions at specific points or under particular conditions.
In the problem, the limit is given as \(\lim_{x \to 0} g(x) = 0\), which is used to describe how the function \(f(x)\) behaves around \(x=0\). This means that near this point \(f(x)\) doesn't drastically change, allowing us to treat the function as linear or near-linear around these values.
Continuity is a major factor when dealing with integrals since it assures that a function does not have any breaks or jumps over the interval of integration.
To effectively analyze limits:
The coordination of limits and continuity allows mathematicians to comfortably extend function behavior from specific points, leading to reliable results in integration and evaluation contexts.
In the problem, the limit is given as \(\lim_{x \to 0} g(x) = 0\), which is used to describe how the function \(f(x)\) behaves around \(x=0\). This means that near this point \(f(x)\) doesn't drastically change, allowing us to treat the function as linear or near-linear around these values.
Continuity is a major factor when dealing with integrals since it assures that a function does not have any breaks or jumps over the interval of integration.
To effectively analyze limits:
- Understand \(g(x)\) behavior as \(x \to 0\), ensuring \(f(1+x)\) behaves predictably.
- Use the limit to derive approximations or assumptions for \(f(x)\) that simplify integration.
The coordination of limits and continuity allows mathematicians to comfortably extend function behavior from specific points, leading to reliable results in integration and evaluation contexts.
Other exercises in this chapter
Problem 115
The set of points of continuity of the function \(f(x)=\sqrt{\frac{1}{2}-\cos ^{2} x}\) is (A) \(\left\\{x: \frac{\pi}{4}+2 n \pi \leq x \leq \frac{3 \pi}{4}+2
View solution Problem 116
In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True
View solution Problem 118
In the following questions an Assertion \((A)\) is given followed by a Reason \((R) .\) Mark your responses from the following options: (A) Assertion(A) is True
View solution Problem 119
Let \(f(x)=\frac{1-\tan x}{4 x-\pi}, x \neq \frac{\pi}{4}, x \in\left[0, \frac{\pi}{2}\right] .\) If \(f(x)\) is continuous in \(\left[0, \frac{\pi}{2}\right]\)
View solution