Question

Is the function \(f(x)=\tan x . c o t x\) integrable on the closed intervals (i) \([\pi / 6, \pi / 4]\) and (ii) \([-1,1] ?\)

Short answer

Answer: Yes, the function \(f(x) = \tan x \cdot \cot x\) is integrable on both closed intervals (i) \([\pi / 6, \pi / 4]\) and (ii) \([-1,1]\).

Step-by-step solution

Simplify the function

First, let's simplify the given function. We know that \(\cot x = \frac{1}{\tan x}\), so we can substitute this into the function: $$f(x) = \tan x \cdot \frac{1}{\tan x}.$$ Now, we can simplify this expression: $$f(x) = 1.$$

Determine if the function is continuous

Since the function \(f(x) = 1\) is a simple constant function, it is continuous over any interval on the real number line, including our desired closed intervals: (i) \([\pi / 6, \pi / 4]\) and (ii) \([-1,1]\).

Determine if the function is bounded

Since \(f(x) = 1\) is a constant function, it is also bounded on our desired closed intervals. The function has the same constant value throughout the entire interval, so it is both upper and lower bounded by that value. For both intervals (i) and (ii), the function is bounded by 1.

Check the integrability of the function

A function is integrable on a closed interval if it is continuous and bounded on that interval. Since \(f(x) = 1\) is continuous and bounded on both closed intervals (i) \([\pi / 6, \pi / 4]\) and (ii) \([-1,1]\), the function is integrable on these intervals. In conclusion, the function \(f(x) = \tan x \cdot \cot x\) is integrable on the closed intervals (i) \([\pi / 6, \pi / 4]\) and (ii) \([-1,1]\).

Key concepts

Continuous Functions

Understanding continuous functions is crucial when studying integrability. A function is said to be continuous at a point when small changes in the input around that point result in small changes in the output. In other words, there aren't any abrupt jumps or breaks in the graph of the function. Imagine drawing the function without lifting your pencil – if this is possible, then the function is continuous.

Continuous functions have the property that they can be graphed as a single unbroken curve. They are important in calculus because the Fundamental Theorem of Calculus establishes that every continuous function on a closed interval is integrable. This is vital as integrals represent areas under curves, and we cannot have areas well-defined under curves that have breaks or jumps.

In our example, the function given is actually a constant function, which is inherently continuous over any interval. Therefore we can say with confidence that, for any closed interval, our function does not

Bounded Functions

When we talk about bounded functions, we refer to those with an output that doesn’t exceed certain limits. In more formal terms, a function is bounded if there exists a real number M such that for all x in the domain of the function, the absolute value of f(x) is less than or equal to M. This concept is seamlessly linked to integrability as one of the core ideas for a function to be integrable on a closed interval is that it must be bounded.

In the given exercise, the constant function keeps the output steady at 1, which means our M is simply 1. This is the easiest case for boundedness since the function will never take a value higher than 1 or lower than 1. Therefore, the function is bounded on any interval, indicating no obstacle to integrability from the standpoint of boundedness.

Constant Functions

A constant function is a special type of function where the output value is the same no matter what input value you choose. If we represent a constant function as f(x) = c, where c is a constant value, then for every value of x, f(x) will always be c. This makes constant functions both continuous and bounded, as they do not have any jumps or breaks (continuous) and the output does not go indefinitely high or low (bounded).

Thus, constant functions are always integrable over any closed interval. In the case of our exercise, the simplified function is a simple example of a constant function (\( f(x) = 1 \)), which can be easily integrated without running into any issues or special considerations.

Closed Intervals

The concept of closed intervals is very important in the study of integrability. A closed interval is a set of numbers that includes all the numbers between two endpoints, and includes the endpoints themselves. This is denoted as [a, b], where 'a' and 'b' represent the endpoint values, and every number x such that a ≤ x ≤ b belongs to the interval.

For a function to be integrable on a closed interval, it is essential that it is continuous and bounded within that closed interval. These properties ensure the existence of the integral on that interval, as the integral is essentially the accumulation of infinitely many infinitesimal areas under the curve from one endpoint to the other. The closed interval notation signifies that the integration includes the end points as well, which can have practical interpretations in various applications like calculating total distance traveled or the area of a region defined within those limits.