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Ch. 2 - Limits and Continuity
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 2 - Limits and ContinuityProblem 2.1.3a
Chapter 2, Problem 2.1.3a

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 guidance
1
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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Introduction to Cotangent Graph

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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Related Practice
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On what intervals are the following functions continuous?


a. ƒ(x) = x¹/³

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Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.


lim ( 1 / x²/³ + 2 / (x − 1)²/³ ) as


a. x → 0⁺

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[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.


Let g(x) = (x² − 2) / (x − √2)


a. Make a table of the values of g at the points x=1.4,1.41,1.414, and so on through successive decimal approximations of √2. Estimate limx→√2 g(x).

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Textbook Question

Exercises 5–10 refer to the function

f(x) = { x² − 1, −1 ≤ x < 0

2x, 0 < x < 1

1, x = 1

−2x + 4, 1 < x < 2

0, 2 < x < 3

graphed in the accompanying figure.

<IMAGE>

a. Does f (1) exist?

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Textbook Question

Finding Limits Graphically


Let f(x) = {3 - x , x < 2

2, x = 2

x/2, x > 2


<IMAGE>


a. Find limx→2+ f(x), limx→2− f(x), and f(2).

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