Problem 10 - Calculus 1 Course in Mathematics for the IIT JEE and Other Engineering Exams

Question

If \(y=f(x)\) is an even function such that \(f^{\prime}(0)\) exists, then \(f^{\prime}(0)=\) [IIT-1987] (a) 0 (b) \(-1\) (c) 1 (d) None of these

Step-by-Step Solution

Answer

The derivative \(f^{ ime}(0)\) is 0.

Step 1: Understanding Even Function

An even function is defined as a function where for all values of input, the relation \( f(x) = f(-x) \) holds true. This symmetry about the y-axis means that the function's graph is identical on both sides of the y-axis.

Step 2: Concept of Derivative at a Point

The derivative of a function at a certain point \(x = a\) is given by the limit \( f^{ ime}(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \). It represents the slope of the tangent to the graph of the function at that point.

Step 3: Apply Even Function Property

Given that \( f(x) = f(-x) \) for an even function, we know that \( f(0 + h) = f(0 - h) \), i.e., the function's value at positive and negative values of an interval around zero are the same.

Step 4: Calculate Derivative at Zero

To find the derivative at \( x = 0 \), use the limit definition: \[ f^{ ime}(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h}.\] Using the even function property, we can say \( f(h) = f(-h) \), which simplifies this limit to:\[ f^{ ime}(0) = \lim_{h \to 0} \frac{f(h) - f(0)}{h} = \lim_{h \to 0} \frac{f(-h) - f(0)}{-h}.\] Since both these limits lead to one another, the only possibility is that \( f^{ ime}(0) = 0 \).

Step 5: Conclude Derivative Value

Therefore, by the properties of even functions and considering the derivative limits from both positive and negative directions yielding the same result, we conclude that \( f^{ ime}(0) = 0 \).

Key Concepts

Derivative

In mathematics, the derivative provides the instantaneous rate of change of a function as the input changes. It's kind of like finding the slope of a line that's tangent to a point on the curve of the function. We use the derivative to understand how a function behaves at or around specific points.

• The derivative of a function at a certain point, say \(x = a\), is given by the expression \( f'(a) = \lim_{h \to 0} \frac{f(a + h) - f(a)}{h} \).
• This formula captures the idea of the tangent line slope at the point \((a, f(a))\).

Knowing the derivative allows us to predict the behavior of the function near that point. If you think of riding a roller coaster, the derivative gives you a clue whether you are about to rise steeply or just cruise along smoothly.

Symmetry

Symmetry in functions helps in understanding how the function's graph behaves. For even functions, symmetry is particularly interesting.

• An even function is defined mathematically as a function that satisfies \( f(x) = f(-x) \) for all \(x\).
• This means that the graph of the function looks identical on both sides of the \(y\)-axis.

Considering our example, the graph of an even function reflects itself across the \(y\)-axis. So, whatever happens on the positive side of the \(x\)-axis also happens in reverse on the negative side. This property plays a crucial role when we're finding derivatives, as it affects how we limit approach from both sides.

Limit Definition

The limit is a fundamental property used in calculus to define both derivatives and integrals. Using limits is essential to understand the derivative at a point.

• When finding a derivative, the limit definition is expressed as \( f'(x) = \lim_{h \to 0} \frac{f(x + h) - f(x)}{h} \).
• As \(h\) becomes incredibly small, this ratio approaches the slope of the tangent line at point \((x, f(x))\).

The limit helps us bridge the gap from the finite difference calculation to the exact slope at a single point. This becomes especially useful in cases involving even functions, as the function value approaches from both directions \(h\) and \(-h\), and these limits coincide, confirming the existence of the derivative.

Tangent Line

The tangent line is an essential concept that closely ties into the derivative. It provides a linear approximation of a function at a given point.

• The slope of this tangent line at a point \(a\) on the function \(f(x)\) is exactly the derivative, \( f'(a) \).
• The equation of the tangent line can be written as \( y = f'(a)(x - a) + f(a) \).

In the context of even functions, the tangent line at any point on the y-axis, say \((0, f(0))\), is horizontal if the derivative at zero is zero. For even functions, since \( f'(0) = 0 \), it means we have a horizontal tangent line at \(x = 0\). This symmetry is key to understanding why the tangent line behaves this way.