Problem 33 - Essential Calculus Early Transcendentals

Question

\(23-36=\) Find the critical numbers of the function. $$f(\theta)=2 \cos \theta+\sin ^{2} \theta$$

Step-by-Step Solution

Answer

The critical numbers are \( \theta = n\pi \), \( n \) an integer.

Step 1: Understanding the Problem

To find the critical numbers of the function \( f(\theta) = 2 \cos \theta + \sin^2 \theta \), we need to find where the derivative \( f'(\theta) \) is equal to zero or is undefined.

Step 2: Finding the Derivative

First, we need to differentiate the function. The derivative of \( 2 \cos \theta \) is \( -2 \sin \theta \), and using the chain rule, the derivative of \( \sin^2 \theta \) is \( 2 \sin \theta \cos \theta \) which is equivalent to \( \sin 2\theta \). Hence, the derivative \( f'(\theta) = -2\sin \theta + \sin 2\theta \).

Step 3: Setting the Derivative to Zero

Next, set the derivative equal to zero: \( -2\sin \theta + \sin 2\theta = 0 \). Using the identity \( \sin 2\theta = 2\sin\theta \cos\theta \), substitute it back: \( -2\sin \theta + 2\sin \theta \cos \theta = 0 \).

Step 4: Solving for \(\theta\)

Factor out \( 2\sin \theta \) to get \( 2\sin \theta (\cos \theta - 1) = 0 \). This gives us possible solutions: \( \sin \theta = 0 \) or \( \cos \theta = 1 \).

Step 5: Finding Critical Numbers

Solve \( \sin \theta = 0 \) which occurs at \( \theta = n\pi \) where \( n \) is an integer. Solve \( \cos \theta = 1 \) which occurs at \( \theta = 2m\pi \) where \( m \) is an integer. Combining these, the critical numbers are at integer multiples of \( \pi \).

Key Concepts

Derivatives and their Role in Finding Critical Numbers

When faced with the task of finding critical numbers for a function, understanding derivatives is essential. Derivatives indicate how a function changes at any point, providing insights into the function's slope.
To identify critical numbers, we specifically look for where the derivative equals zero or is undefined.

• A zero derivative signifies a horizontal tangent line, suggesting potential maxima, minima, or inflection points.
• An undefined derivative can occur when there are sharp turns or vertical tangent lines, also marking important points for further investigation.

In our specific exercise, we work with the function \[ f(\theta) = 2 \cos \theta + \sin^2 \theta \].
The task involves finding where its derivative, \( f'(\theta) \), drops to zero or fails to be defined. This step is crucial in pinpointing critical numbers and, subsequently, understanding the function's behavior.

Exploring Trigonometric Functions

Trigonometric functions, such as \( \sin \theta \) and \( \cos \theta \), are foundational in mathematics, especially when evaluating angles and oscillations.
These functions exhibit periodic behavior, which means they repeat values in regular intervals, typically \( 2\pi \).

• The cosine function, \( \cos \theta \), oscillates between -1 and 1, maximizing at multiples of \( 2\pi \).
• The sine function, \( \sin \theta \), also swings between -1 and 1, providing its distinctive wave pattern.

In the original problem, \( \cos \theta \) and \( \sin \theta \) contribute to the formula \( f(\theta) = 2 \cos \theta + \sin^2 \theta \).
Mastering these functions helps in calculating derivatives, factoring equations, and solving for critical numbers.

Understanding the Chain Rule in Calculating Derivatives

The chain rule is an essential technique in calculus, particularly when deriving composite functions. It allows us to differentiate a function that is nested within another.
Here's how the chain rule operates: if you have a composite function of the form \( h(g(\theta)) \), its derivative can be found by multiplying the derivative of \( h \) with respect to \( g \) by the derivative of \( g \) with respect to \( \theta \).

• This rule simplifies the process of dealing with functions of functions, crucial in engineering and physics applications.
• In our exercise, we encountered \( \sin^2 \theta \). Using the chain rule, its derivative becomes \( 2\sin \theta \cos \theta \), which is \( \sin 2\theta \).

By breaking it down, the chain rule becomes a powerful tool that enables us to obtain derivatives efficiently.
It ensures we're equipped to handle even complex mathematical tasks, like finding critical numbers across diverse function types.