Problem 19 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

Determine whether or not each function is periodic. If a function is periodic, determine its period. (a) \(f(x)=\cos |x|\) (b) \(g(x)=\sin |x|\) (c) \(h(x)=|\cos x|\) (d) \(j(x)=|\sin x|\) (c) \(k(x)=\sin \left(x^{2}\right)\) (f) \(l(x)=\sin ^{2} x\)

Step-by-Step Solution

Answer

(a) Not periodic, (b) Not periodic, (c) Periodic with period π, (d) Periodic with period π, (e) Not periodic, (f) Periodic with period π.

Step 1: Solution for (a)

The function \(f(x) = \cos |x|\) is not a periodic function. The reason being that the function is non-repetitive over any given interval of \('x'\). This is because absolute function |x| is non-periodic and we are applying cosine function on the non-periodic function.

Step 2: Solution for (b)

Similarly, \(g(x) = \sin |x|\) is not a periodic function. The reason being that the function is non-repetitive over any interval of \('x'\). It has the same reason as in (a), the absolute function |x| is non-periodic and we are applying sine function on it.

Step 3: Solution for (c)

However, \(h(x) = |\cos x|\) is a periodic function. The absolute value of cosine function repeats its values in the interval of [0, π]. So, the period of this function, \(h(x)\) is π.

Step 4: Solution for (d)

Like \(h(x)\), \(j(x) = |\sin x|\) is also a periodic function. The absolute value of sine function also repeats its values in the interval of [0, π]. So, the period of this function, \(j(x)\) is π.

Step 5: Solution for (e)

The function \(k(x) = \sin (x^2)\) is not a periodic function, as squaring the variable \('x'\) within the sine function has an effect of 'stretching' the normal period of sinusoidal function i.e. [0, 2π]. This makes the intuition of repetition of values over a certain interval impossible.

Step 6: Solution for (f)

Finally, \(l(x)=\sin^{2}x\) is a periodic function. Squaring function does not affect periodicity. As the base function is sine, which is periodic, squaring this function will also yield a periodic function. The base function sine has a period of 2π, but squaring it halves the period. Thus, period of function \(l(x)\) is π.

Key Concepts

Function Characteristics

Functions are mathematical expressions that describe the relationship between two quantities. They have certain features and behaviors that help classify their properties. One such property is periodicity, where a function repeats its values in regular intervals known as the period.
For a function to be periodic, it must show repetition over intervals of its input variable, usually represented by 'x'. Mathematically, a function is periodic with period 'T' if for all values of 'x', the equation \( f(x + T) = f(x) \) holds true. Periodicity is common in trigonometric functions like sine and cosine which naturally have periods due to their wave-like patterns.
Understanding whether a function is periodic helps determine how it behaves over time or space. This can be crucial for applications in engineering, physics, and signal processing among others.

Trigonometric Functions

Trigonometric functions are fundamental in mathematics for describing relationships between angles and distances. The primary trigonometric functions include sine, cosine, and tangent. They stem from the unit circle and are essential for the study of periodic functions.
These functions also have natural periodic characteristics. For instance, sine and cosine have a period of \(2\pi\), meaning they repeat every \(2\pi\) units. Their periodic nature makes them ideal for modeling cyclical phenomena such as sound waves, light waves, or even tides.
In addition, trigonometric functions can be combined with other mathematical constructs like absolute values to explore more complex behaviors and characteristics like the ones we see in this exercise.

Absolute Value in Functions

The absolute value function, denoted as \(|x|\), takes any real number and returns its non-negative value. In function analysis, this can affect the periodicity or symmetry of the function when applied to other functions like sine and cosine.
For example, when you apply an absolute value to cosine as in \(|\cos x|\), it affects the periodic characteristics by reflecting any negative parts of the cosine wave to positive, thereby potentially altering the observed period and symmetry. Such transformations create functions that mirror over a certain line, typically the x-axis, making them always non-negative.
Absolute values in trigonometric functions lead to unique behaviors, influencing whether the resulting function remains periodic and what the new period might be, as seen with functions \(|\cos x|\) and \(|\sin x|\).

Sine and Cosine Functions

Sine and cosine functions are quintessential examples of periodic functions, defined by the unit circle. They are oscillating functions that represent a smooth, repetitive wave.
The sine function gives the vertical coordinate or the y-value of a point on the unit circle, written as \( \sin x \). It starts at 0, rises to 1, drops to -1, and returns to 0 in a span of \(2\pi\). Its periodicity is immediately apparent, repeating every \(2\pi\) units.
Similarly, the cosine function, \( \cos x \), gives the horizontal coordinate or the x-value of a point on the unit circle. It starts at 1, drops to 0, reaches -1, and returns to 1 over the same \(2\pi\) span. Like sine, cosine is periodic, indicative of many natural patterns. The exploration of absolute values of these functions, like in \(|\cos x|\) or \(|\sin x|\), modifies the wave but keeps a reduced period due to symmetrical reflection, leading to their specific periodic patterns.