Problem 4 - Thomas Calculus

Question

Find the average rate of change of the function over the given interval or intervals. \(g(t)=2+\cos t\) a. \([0, \pi] \quad\) b. \([-\pi, \pi]\)

Step-by-Step Solution

Answer

a. \(-\frac{2}{\pi}\), b. 0

Step 1: Understand the Problem

We need to find the average rate of change of the function \(g(t) = 2 + \cos t\) on the given intervals \([0, \pi]\) and \([-\pi, \pi]\). The average rate of change formula for a function \(f(x)\) over an interval \([a, b]\) is given by \(\frac{f(b) - f(a)}{b - a}\).

Step 2: Calculate Values of the Function for Interval [0, π]

First, compute \(g(0)\) and \(g(\pi)\). \(g(0) = 2 + \cos(0) = 2 + 1 = 3\) \(g(\pi) = 2 + \cos(\pi) = 2 - 1 = 1\)

Step 3: Average Rate of Change for Interval [0, π]

Apply the average rate of change formula for the interval \([0, \pi]\):\[\text{AROC} = \frac{g(\pi) - g(0)}{\pi - 0} = \frac{1 - 3}{\pi} = \frac{-2}{\pi}\]

Step 4: Calculate Values of the Function for Interval [-π, π]

Compute \(g(-\pi)\) and re-use \(g(\pi)\) from Step 2: \(g(-\pi) = 2 + \cos(-\pi) = 2 - 1 = 1\)\(g(\pi) = 1\) as previously calculated.

Step 5: Average Rate of Change for Interval [-π, π]

Apply the average rate of change formula for the interval \([-\pi, \pi]\):\[\text{AROC} = \frac{g(\pi) - g(-\pi)}{\pi - (-\pi)} = \frac{1 - 1}{2\pi} = 0\]

Key Concepts

Trigonometric Functions

Trigonometric functions are a fundamental part of mathematics, often used in calculus to describe circular motion and oscillations. Functions like sine, cosine, and tangent are based on the ratios of the sides of a right triangle to its angles. The cosine function, in particular, is an even function and is defined as the x-coordinate of a point on a unit circle.

These functions are periodic, meaning they repeat values in regular intervals (every \(2\pi\) for cosine and sine). This property is vitally important when calculating the behavior of waves, or in this case, solving problems involving changes over specific intervals. In our problem, we use the cosine function's characteristics to explore how the function behaves over different parts of its cycle.

Interval Notation

Interval notation is a way of representing a range of values over which a function is examined. This is critical in calculus and helps specify the start and end points of our interest. These points are presented within brackets or parentheses:

• Closed Interval [a, b]: Includes both endpoints, denoting \( a \leq x \leq b \).
• Open Interval (a, b): Excludes both endpoints, indicating \( a < x < b \).
• Half-Open Interval [a, b): Includes one endpoint, specifically \( a \leq x < b \).

In our exercise, the intervals [0, \(\pi\)] and [-\(\pi\), \(\pi\)] are closed, allowing both end values to be considered when calculating averages.

Cosine Function

The cosine function, denoted as \(\cos(x)\), is a key trigonometric function that describes the horizontal coordinate of a point on a unit circle. It reflects the even and periodic nature of the circle, repeating every \(2\pi\). Here, the expression \( \cos(t) \) in our function \( g(t) = 2 + \cos(t) \) adds a trigonometric character to the function.

By examining the cosine function over given intervals, we investigate how its values fluctuate between 1 and -1. This fluctuation directly influences the shape and slope of our target function. For the intervals [0, \(\pi\)] and [-\(\pi\), \(\pi\)], cosine provides changing values at specific points, crucial to calculating the average rate of change.

Calculus Concepts

Calculus introduces us to the concept of understanding change within mathematical functions, primarily using tools such as derivatives and integrals. One essential idea is the "average rate of change," essentially a function's total change over a specified interval. It gives a 'smoothed out' view of how much the function increases or decreases over that period.

In finding the average rate of change for \( g(t) = 2 + \cos(t) \), we apply the formula \[ \frac{f(b) - f(a)}{b-a} \]. Here, \( f(a) \) and \( f(b) \) are the function's values at the start and end of intervals, providing a straightforward way of measuring change. By focusing on simple differences across the interval, we glean insights without diving into complex derivatives, yet still reaping valuable understandings of the function's behavior.