Textbook Question
[Technology Exercise] Roots
Let ƒ(𝓍) = 𝓍³ ―𝓍― 1.
a. Use the Intermediate Value Theorem to show that ƒ has a zero between ―1 and 2 .
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 2.1.2a
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
g(x)=x²−2x
a. [1, 3]
Verified step by step guidance1
Identify the function given: \( g(x) = x^2 - 2x \).
Recall the formula for the average rate of change of a function \( g(x) \) over an interval \([a, b]\): \( \frac{g(b) - g(a)}{b - a} \).
Substitute the interval \([1, 3]\) into the formula, where \( a = 1 \) and \( b = 3 \).
Calculate \( g(1) \) by substituting \( x = 1 \) into the function: \( g(1) = 1^2 - 2 \times 1 \).
Calculate \( g(3) \) by substituting \( x = 3 \) into the function: \( g(3) = 3^2 - 2 \times 3 \).
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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 g(x) = x² - 2x, evaluating it at points 1 and 3 means calculating g(1) and g(3). This step is essential for finding the values needed to compute the average rate of change over the specified interval.
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4:26Evaluating Composed Functions
Quadratic Functions
Quadratic functions are polynomial functions of degree two, typically expressed in the form f(x) = ax² + bx + c. The function g(x) = x² - 2x is a quadratic function where a = 1, b = -2, and c = 0. Understanding the properties of quadratic functions, such as their parabolas' shape and vertex, is important for analyzing their behavior over intervals.
Recommended video:
6:04Introduction to Polynomial Functions
Related Practice
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⁺
Textbook Question
Limits and Continuity
On what intervals are the following functions continuous?
a. ƒ(x) = x¹/³
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
Suppose that limx→−2 p(x) = 4, limx→−2 r(x) = 0, and limx→−2 s(x) = −3. Find
a. limx→−2 (p(x) + r(x) + s(x))