Textbook Question
Using the Sandwich Theorem
If √(5 −2x²) ≤ f(x) ≤ √(5−x²) for −1 ≤ x ≤ 1, find limx→0 f(x).
1. Limits and Continuity
Introduction to Limits
3:21 minutesProblem 2.1.3a
Textbook Question
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
h(t)=cot t
a. [π/4,3π/4]
Verified step by step guidance1
Identify the function and the interval: The function given is \( h(t) = \cot t \) and the interval is \([\frac{\pi}{4}, \frac{3\pi}{4}]\).
Recall the formula for the average rate of change of a function \( f(x) \) over an interval \([a, b]\): \( \frac{f(b) - f(a)}{b - a} \).
Calculate \( h(\frac{\pi}{4}) \): Since \( \cot t = \frac{1}{\tan t} \), find \( \tan(\frac{\pi}{4}) \) which is 1, so \( \cot(\frac{\pi}{4}) = 1 \).
Calculate \( h(\frac{3\pi}{4}) \): Similarly, find \( \tan(\frac{3\pi}{4}) \) which is -1, so \( \cot(\frac{3\pi}{4}) = -1 \).
Substitute the values into the average rate of change formula: \( \frac{h(\frac{3\pi}{4}) - h(\frac{\pi}{4})}{\frac{3\pi}{4} - \frac{\pi}{4}} = \frac{-1 - 1}{\frac{3\pi}{4} - \frac{\pi}{4}} \). Simplify the expression to find the average rate of change.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Average Rate of Change
The average rate of change of a function over an interval is defined as the change in the function's value divided by the change in the input value. Mathematically, it is expressed as (f(b) - f(a)) / (b - a), where [a, b] is the interval. This concept helps in understanding how a function behaves on average over a specified range.
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Cotangent Function
The cotangent function, denoted as cot(t), is the reciprocal of the tangent function, defined as cot(t) = cos(t)/sin(t). It is periodic with a period of π, meaning it repeats its values every π units. Understanding the properties of the cotangent function is essential for evaluating its behavior over specific intervals.
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Evaluating Functions at Specific Points
To find the average rate of change, one must evaluate the function at the endpoints of the given interval. This involves substituting the values of the interval into the function h(t) = cot(t) to find h(π/4) and h(3π/4). These evaluations are crucial for calculating the average rate of change accurately.
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