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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 2.1.2a
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]

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1
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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Evaluating 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.
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Related Practice
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⌋)

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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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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

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

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

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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))

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