Problem 3 - Calculus Early Transcendentals: Pearson New International Edition

Question

, show that \(f\) has an inverse by showing that it is strictly monotonic. $$ f(\theta)=\cos \theta, 0 \leq \theta \leq \pi $$

Step-by-Step Solution

Answer

The function \(f(\theta) = \cos \theta\) is strictly decreasing on \([0, \pi]\) and has an inverse.

Step 1: Understand the Function

The given function is \( f(\theta) = \cos \theta \). We need to determine if this function is strictly monotonic on the interval \([0, \pi]\). A function is considered strictly monotonic if it is either strictly increasing or strictly decreasing over the specified interval.

Step 2: Consider the Derivative

To determine monotonicity, consider the derivative of the function. The derivative of \( f(\theta) = \cos \theta \) is \( f'(\theta) = -\sin \theta \). Analyzing this derivative will help us determine if the function is strictly monotonic.

Step 3: Analyze the Derivative

On the interval \( 0 \leq \theta \leq \pi \), \( \sin \theta \) is non-negative and strictly positive except at the endpoints (i.e., it ranges from 0 to 1 and back to 0). Therefore, \( f'(\theta) = -\sin \theta \) is non-positive and strictly negative in the open interval \( (0, \pi) \).

Step 4: Conclude Monotonicity

Since \( f'(\theta) = -\sin \theta \) is strictly negative for \(0 < \theta < \pi\), this indicates that \( f(\theta) = \cos \theta \) is strictly decreasing on \(0 < \theta < \pi\). As the derivative does not change sign (it remains non-positive and strictly negative), the function is strictly monotonic over the interval \([0, \pi]\).

Step 5: Determine Invertibility

Since \( f(\theta) = \cos \theta \) is strictly monotonic (specifically, strictly decreasing) on \([0, \pi]\), it is therefore invertible on this interval. This means \( f(\theta) = \cos \theta \) has an inverse on \([0, \pi]\).

Key Concepts

Strictly Monotonic Functions

In mathematics, a function is described as strictly monotonic if it consistently moves in one direction, either increasing or decreasing, over its entire domain. Understanding monotonicity helps us to determine certain properties of functions, particularly invertibility. For a function to be classified as strictly monotonic, it must either

• never fall as we move along the domain (strictly increasing), or
• never rise as we move along the domain (strictly decreasing).

For the function \( f(\theta) = \cos \theta \) over the interval \([0, \pi]\), monotonicity is determined by examining its behavior on this specified range. Since cosine changes from 1 to -1 as \(\theta\) moves from 0 to \(\pi\), it continuously decreases and exhibits strictly decreasing behavior over the interval when examined via its derivative.

The Role of Derivatives in Monotonicity

Derivatives are a fundamental concept in calculus used to understand the behavior of functions. When we talk about the derivative of a function, we refer to its rate of change or its gradient.
The derivative tells us whether a function is increasing, decreasing, or remaining constant over a certain interval. Importantly,

• If the derivative is positive over an interval, the function is increasing.
• If it is negative, the function is decreasing.
• A zero derivative may indicate a horizontal tangent, possibly a local minimum or maximum.

For \( f(\theta) = \cos \theta \), the derivative is \( f'(\theta) = -\sin \theta \). Within the interval \(0 \leq \theta \leq \pi\), \(-\sin \theta\) remains non-positive, meaning \( f(\theta) \) is strictly decreasing on this interval. This negativity is crucial for establishing the strictly monotonic nature of the function.

Understanding Trigonometric Functions

Trigonometric functions, such as sine and cosine, are functions of an angle and are fundamental in understanding various mathematical concepts. They are periodic and have specific patterns of increase and decrease within their cycles.

Cosine, denoted as \( \cos \theta \), measures the adjacent side in a right-angled triangle compared to the hypotenuse for an angle \( \theta \). Within the context of our function \( f(\theta) = \cos \theta \):

• It's important to note the cycle progresses from maximum value 1 at \( \theta=0 \) to minimum -1 at \( \theta=\pi \).
• Monotonic behavior can be influenced by the restricted intervals due to its periodic nature.

This helps in understanding how it behaves over specified intervals such as \([0, \pi]\), where it strictly decreases — moving from 1 to -1, indicating a clear pattern suitable for determining invertibility.

Invertibility and Its Connection to Monotonicity

The concept of invertibility is key when dealing with functions that have an inverse. For a function to be invertible over an interval, it should be both one-to-one (injective) and onto (surjective) over that interval.
A strictly monotonic function automatically satisfies these criteria because

• It is one-to-one, with each input mapping to a unique output, since no two values in its domain can yield the same value without breaking monotonicity.
• As a result, its inverse can be defined clearly and without ambiguity over its range.

For \( f(\theta) = \cos \theta \) over \([0, \pi]\), we observe strict monotonic decreasing behavior, meaning the function maintains a consistent decrease across the interval. This quality makes it invertible, affirming that a unique inverse exists in this context.