Textbook Question
Use the graph of the greatest integer function y = ⌊x⌋, Figure 1.10 in Section 1.1, to help you find the limits in Exercises 21 and 22.
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b. limt→4−(t−⌊t⌋)
Ch. 2 - Limits and Continuity
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 2, Problem 2.1.4b
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(t)=2+cos t
b. [0,π]
Verified step by step guidance1
Identify the function g(t) = 2 + cos(t) and the interval [0, π].
Recall the formula for the average rate of change of a function g(t) over an interval [a, b], which is given by: \( \frac{g(b) - g(a)}{b - a} \).
Substitute the endpoints of the interval into the function: calculate g(0) and g(π).
Evaluate g(0) = 2 + cos(0) and g(π) = 2 + cos(π).
Substitute g(0) and g(π) into the average rate of change formula: \( \frac{g(\pi) - g(0)}{\pi - 0} \) and simplify the expression.
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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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Average Value of a Function
Function Evaluation
Function evaluation involves substituting a specific value into a function to determine its output. For the function g(t) = 2 + cos(t), evaluating it at the endpoints of the interval [0, π] means calculating g(0) and g(π). This step is crucial for finding the average rate of change, as it provides the necessary values to apply the average rate of change formula.
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4:26Evaluating Composed Functions
Trigonometric Functions
Trigonometric functions, such as cosine, are periodic functions that relate angles to ratios of sides in right triangles. The function g(t) = 2 + cos(t) combines a constant and a cosine function, which oscillates between -1 and 1. Understanding the behavior of cosine over the interval [0, π] is essential for accurately calculating the average rate of change, as it influences the function's values at the endpoints.
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6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
Theory and Examples
If limx→−2 f(x) / x² = 1, find
b. limx→−2 f(x) / x
Textbook Question
Limits and Continuity
On what intervals are the following functions continuous?
b. g(x) = x³/⁴
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let f(x) = (x² - 9) / (x + 3)
b. Support your conclusions in part (a) by graphing f near c = -3 and using Zoom and Trace to estimate y-values on the graph as x → −3.
Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let G(x)=(x + 6)/(x² + 4x − 12)
b. Support your conclusions in part (a) by graphing G and using Zoom and Trace to estimate y-values on the graph as x→−6.
Textbook Question
Finding Limits Graphically
Which of the following statements about the function y = f(x) graphed here are true, and which are false?
b. limx→2 f(x) does not exist