Problem 112 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

Suppose \(f(x)\) is differentiable \(x=1\) and \(\lim _{h \rightarrow 0} \frac{1}{h} f(1+h)=5\), then \(f^{\prime}(1)\) equals \([2005]\) (A) 3 (B) 4 (C) 5 (D) 6

Step-by-Step Solution

Answer

The correct option is (C) 5.

Step 1: Understanding the Differentiation and the Limit

We need to evaluate the derivative of the function at the point where it's differentiable, particularly at \(x=1\). According to the definition of a derivative, \( f'(x) \) is given by \[ f'(1) = \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} \] We are given that \( \lim_{h \to 0} \frac{1}{h} f(1+h) = 5 \). This expression can be transformed to relate it to the derivative definition.

Step 2: Relating Given Limit to Derivative Definition

The given expression \( \lim_{h \to 0} \frac{1}{h} f(1+h) \) resembles the derivative definition but lacks the term \( -\frac{f(1)}{h} \). To make it fit the definition, consider breaking down the term: \[ \frac{1}{h} f(1+h) = \frac{f(1+h) - f(1) + f(1)}{h} \] Thus: \[ \frac{1}{h} f(1+h) = \frac{f(1+h) - f(1)}{h} + \frac{f(1)}{h} \] Taking the limit as \( h \to 0 \): \[ \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} + \lim_{h \to 0} \frac{f(1)}{h} = 5 \]

Step 3: Evaluate the Limits and Solve for Derivative

By the derivative definition, we know:\[ \lim_{h \to 0} \frac{f(1+h) - f(1)}{h} = f'(1) \] \( f(1) \) is a constant, making \( \lim_{h \to 0} \frac{f(1)}{h} = 0 \).Thus:\[ f'(1) + 0 = 5 \] Hence, \( f'(1) = 5 \).

Step 4: Determine the Correct Option

The calculated derivative \( f'(1) = 5 \) matches option (C) in the provided choices. Therefore, we conclude that the correct answer is \(5\).

Key Concepts

Understanding Differentiation

Differentiation is a fundamental concept in calculus, used to find the rate at which a function is changing at any given point. It is akin to finding the slope of a tangent line to a curve on a graph. This tells us how the function behaves around a specific value of its input.

• The process of differentiation applies to a function, resulting in a new function, known as the derivative.
• The derivative indicates the instantaneous rate of change and can reveal much about the function, such as its increasing or decreasing nature.

Differentiation is vital for solving practical problems, allowing us to determine maximum and minimum values of functions, which is essential in fields such as physics and economics.

Exploring Limits

Limits are central to calculus, enabling us to determine the value that a function approaches as the input approaches a given point. Understanding limits is essential for defining derivatives and integrals.

• In calculus, the limit of a function when approaching a point describes the behavior of the function nearby.
• The concept of limits allows us to handle values that the function cannot reach directly.

For a limit to exist, the value approached from the left and the right as you near the point must be the same. Evaluating limits involves examining this consistent approach from all directions.

Defining the Derivative

The derivative is defined as the limit of the average rate of change of the function as the interval approaches zero. Mathematically, for a function \( f(x) \) at point \( x = a \), it is defined as:
\[f'(a) = \lim_{h \to 0} \frac{f(a+h) - f(a)}{h}\]

• This definition represents the idea of calculating the slope of a tangent line at a given point on the function's graph.
• The derivative is the foundational tool for calculating things like velocity in physics, which is the rate of change of position over time.

The derivative's definition underscores its role in capturing the essence of changes as they occur instantaneously.

Introduction to Functional Analysis

Functional analysis is a branch of mathematical analysis dealing with function spaces and their properties. This broad area of study provides powerful tools for understanding phenomena described by differential equations and other operations on functions.

• It involves studying spaces of functions and transforming functions to explore algebraic structures and topology.
• Functional analysis bridges finite-dimensional spaces, like vectors, with infinite-dimensional space, offering insights into both theoretical and practical problems.

Through functional analysis, we can address complex systems more intuitively, which is crucial in advanced fields such as quantum mechanics and wave propagation.