Problem 9 - Calculus with Concepts in Calculus

Question

Find all critical numbers of the given function. $$ f(x)=\sin x $$

Step-by-Step Solution

Answer

Critical numbers are at \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer.

Step 1: Differentiate the Function

To find the critical numbers of the function, we must first find its derivative. The function given is \( f(x) = \sin x \). The derivative of \( \sin x \) is \( \cos x \). Thus, \( f'(x) = \cos x \).

Step 2: Set Derivative Equal to Zero

Critical numbers occur where the derivative is zero or undefined. Set \( f'(x) = \cos x \) equal to zero: \( \cos x = 0 \).

Step 3: Solve for x Where Derivative Equals Zero

\( \cos x = 0 \) is true for \( x = \frac{\pi}{2} + n\pi \), where \( n \) is any integer. This is because the cosine function is zero at these points on the unit circle.

Key Concepts

Critical numbers

Finding critical numbers in a function is crucial when trying to determine the behavior of the graph, such as identifying locations where local maxima, minima, or saddle points might occur. Critical numbers are the x-values in the domain of a function where the derivative is zero or undefined.

To find critical numbers:

• First, find the derivative of the function. This represents the slope of the tangent line to the graph at any point.
• Next, solve for where this derivative equals zero. This is important because the slope of the tangent line being zero can indicate potential peaks or valleys in the graph.
• Finally, determine where the derivative is undefined. These points also count as critical numbers.

For instance, in the function given, \( f(x) = \sin x \), we found that its derivative \( \cos x \) is zero at \( x = \frac{\pi}{2} + n\pi \), pinpointing the locations of its critical numbers across the unit circle.

Derivative

The derivative of a function represents the rate at which the function's value changes at any given point. It's a core component of calculus involving how quantities vary and change.

Key ideas about derivatives include:

• The derivative tells you if a function is increasing or decreasing at a particular point. When the derivative is positive, the function is increasing; when negative, it is decreasing.
• If the derivative equals zero, the function may be at a local maximum, minimum, or a point of inflection.
• To take a derivative, use differentiation rules like the power rule, product rule, quotient rule, and chain rule, depending on the complexity of the function.

In our exercise, differentiating \( f(x) = \sin x \) using basic differentiation rules, we find \( f'(x) = \cos x \). This derivative indicates how quickly the sine function is changing at each x-position.

Trigonometric functions

Trigonometric functions, like \( \sin x \), \( \cos x \), and \( \tan x \), originate from the relationships in right-angled triangles. They are pivotal in understanding periodic phenomena.

Some crucial properties of trigonometric functions include:

• These functions are periodic, meaning they repeat their values in regular intervals. For example, the sine and cosine functions have a period of \( 2\pi \).
• They play a significant role in modeling oscillatory behavior, such as waves and alternating currents.
• Their graphs are characterized by specific patterns, with sine waves rising and falling, while cosine waves do so but start at their maximum value.

For our exercise, knowing that the derivative of \( \sin x \) is \( \cos x \) helps locate critical numbers, leveraging the predictable repetition of zero values in the cosine function across one complete cycle.