Problem 6 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

The left-hand derivative of \(f(x)=[x] \sin (\pi x)\) at \(x=k\), \(k\) an integer and \([x]=\) greatest integer \(\leq x\), is (A) \((-1)^{k}(k-1) \pi\) (B) \((-1)^{k-1} \cdot(k-1) \pi\) (C) \((-1)^{k} \cdot k \pi\) (D) \((-1)^{k-1} \cdot k \pi\).

Step-by-Step Solution

Answer

The left-hand derivative is (A) \((-1)^{k}(k-1) \pi\).

Step 1: Understand the Function

The function is defined by two parts: the greatest integer function \([x]\) and the sine function. At integer points, the left-hand limit is important because the greatest integer function changes value.

Step 2: Left-Hand Derivative Definition

The left-hand derivative of a function \( f \) at a point \( x=c \) is defined as \( \lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h} \).

Step 3: Evaluate \( f(x) \) at \( x=k \)

For integer \( k \), \( [k] = k \) and \( f(k) = k \sin(\pi k) = 0 \), because \( \sin(n \pi) = 0 \) for integer \( n \).

Step 4: Evaluate \( f(x) \) for Slightly Less than \( k \)

For \( x = k-h \) with \( h > 0 \), \( [x] = [k-h] = k-1 \). Thus, \( f(k-h) = (k-1) \sin(\pi (k-h)) = (k-1) \sin(\pi k - \pi h) \).

Step 5: Simplify Using Trigonometric Identity

Using the identity \( \sin(a-b) = \sin a \cos b - \cos a \sin b \), we have \( \sin(\pi k - \pi h) = \sin(\pi k) \cos(\pi h) - \cos(\pi k) \sin(\pi h) \).

Step 6: Simplify Further

Since \( \sin(\pi k) = 0 \) and \( \cos(\pi k) = (-1)^k \), it simplifies to \(-(-1)^k (k-1) \sin(\pi h)\). For \( \sin(\pi h) \approx \pi h \) as \( h \) approaches 0, \( f(k-h) \approx -(-1)^k(k-1) \pi h \).

Step 7: Substitute and Simplify the Derivative Expression

By substituting into the left-hand derivative formula: \( \lim_{h \to 0^-} \frac{(-1)^k (k-1) \pi h}{h} \), it simplifies to \( (-1)^k (k-1) \pi \).

Step 8: Conclusion

The left-hand derivative matches option (A), \((-1)^{k}(k-1) \pi\).

Key Concepts

Left-Hand Derivative

The concept of the left-hand derivative is pivotal in understanding points where functions are not smooth or linear. It essentially describes the rate of change of a function as you approach a point from the left side.

• The left-hand derivative of a function \( f(x) \) at a point \( x = c \) is expressed as \( \lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h} \).
• It gives a picture of how the function behaves just before reaching \( x = c \).

For the given function \( f(x) = [x] \sin(\pi x) \), the left-hand derivative is calculated by considering integer values of \( x \). At these integers, the behavior on the left can be different from the right because the greatest integer function can jump in value.

Greatest Integer Function

The greatest integer function, denoted by \( [x] \), is a special type of step function. It takes any real number \( x \) and rounds it down to the nearest integer less than or equal to \( x \). This can lead to jumps at integer points:

• For any real number \( x = 3.7 \), \([x] = 3\).
• At exactly integer values, such as \( x = 4 \), \([x] = 4\).

When incorporated into functions, especially when paired with trigonometric functions like \( \sin(\pi x) \), it can produce interesting behaviors, particularly at integer boundaries. For instance, around \( x = k \), where \( k \) is an integer, \([x] \) faces a change just before reaching \( k \), which defines the need for calculating things like left-hand derivatives carefully.

Trigonometric Identities

Trigonometric identities are essential tools for simplifying expressions in calculus. These identities help in making expressions easier to work with and to identify limits or derivatives. A particularly useful identity in this context is:

• \( \sin(a-b) = \sin a \cos b - \cos a \sin b \)

This identity helps in breaking down complex trigonometric expressions into simpler parts:- For instance, considering \( \sin(\pi k - \pi h) \), where \( k \) is an integer, you can use the identity to write this as \( \sin(\pi k)\cos(\pi h) - \cos(\pi k)\sin(\pi h) \).- Since \( \sin(\pi k) \) equals 0 (because \( k \) is an integer), the expression simplifies significantly, especially when evaluating limits.

Limit Evaluation

Limit evaluation examines how a function behaves as it approaches a particular point, whether from the left, the right, or from both sides. It's critical for defining derivatives at points where functions may not be smooth. For evaluating limits in the given function:

• You use the formula for left-hand derivative: \( \lim_{h \to 0^-} \frac{f(c+h) - f(c)}{h} \). This helps determine how the function's slope behaves just before hitting an integer \( x = k \).
• In the exercise, by substituting \( f(k-h) \approx (-1)^k(k-1)\pi h \) into this formula, you find the behavior near \( x = k \).
• The simplification of this limit provided the result, aligning with option (A): \((-1)^{k}(k-1) \pi\).

This process mandates careful attention to limit rules and identities, allowing precise calculation of derivatives even for more complex functions involving integer steps.