교과서 질문
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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TOC.tableOfContents
- 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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
2장, 문제 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]
검증된 단계별 안내1
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: \( f(3) = 3^3 + 1 \).
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
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.
추천 영상:
가이드 코스
06:37
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.
추천 영상:
가이드 코스
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.
추천 영상:
6:04Introduction to Polynomial Functions
관련 실천
교과서 질문
Theory and Examples
a. If limx→0 f(x) / x² = 1, find limx→0 f(x).
255
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교과서 질문
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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교과서 질문
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?
255
views
교과서 질문
Finding One-Sided Limits Algebraically
Find the limits in Exercises 11–20.
a. limx→1+ (√2x (x − 1)) / |x − 1|
235
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교과서 질문
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.
233
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