Textbook Question
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→0+ (1 − cos x) / |cos x − 1|
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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 2.1.1a
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
f(x)=x³+1
a. [2, 3]
Verified step by step guidance1
Identify the function given: \( f(x) = x^3 + 1 \).
Recall the formula for the average rate of change of a function \( f(x) \) over an interval \([a, b]\): \( \frac{f(b) - f(a)}{b - a} \).
Substitute the interval values into the formula: here \( a = 2 \) and \( b = 3 \).
Calculate \( f(2) \) by substituting \( x = 2 \) into the function: \( f(2) = 2^3 + 1 \).
Calculate \( f(3) \) by substituting \( x = 3 \) into the function: \( f(3) = 3^3 + 1 \).
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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 [a, b] 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). This concept is crucial for understanding how a function behaves over a specific range and is often used to analyze the function's growth or decline.
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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 f(x) = x³ + 1, evaluating it at points a and b (in this case, 2 and 3) allows us to find the corresponding function values f(2) and f(3). This step is essential for calculating the average rate of change.
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4:26Evaluating Composed Functions
Polynomial Functions
Polynomial functions are mathematical expressions that involve variables raised to whole number powers, combined using addition, subtraction, and multiplication. The function f(x) = x³ + 1 is a cubic polynomial, which means its graph can exhibit various behaviors, such as turning points and inflection points. Understanding the properties of polynomial functions helps in analyzing their rates of change and overall behavior.
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6:04Introduction to Polynomial Functions
Related Practice
Textbook Question
Theory and Examples
a. If limx→0 f(x) / x² = 1, find limx→0 f(x).
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Textbook Question
Find the limits in Exercises 59–62. Write ∞ or −∞ where appropriate.
lim ( 1 / x¹/³ − 1 / (x − 1)⁴/³ ) as
a. x → 0⁺
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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 One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→1+ (√2x (x − 1)) / |x − 1|
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Textbook Question
Estimating Limits
[Technology Exercise] You will find a graphing calculator useful for Exercises 67–74.
Let h(x)=(x² − 2x − 3)/(x² − 4x + 3)
b. Support your conclusions in part (a) by graphing h near c = 3 and using Zoom and Trace to estimate y-values on the graph as x→3.
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