교과서 질문
Average Rates of Change
In Exercises 1–6, find the average rate of change of the function over the given interval or intervals.
P(θ)=θ³ − 4θ² + 5θ; [1,2]
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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 2.2.59
Limits of Average Rates of Change
Because of their connection with secant lines, tangents, and instantaneous rates, limits of the form limh→0 (f(x+h) − f(x)) / h occur frequently in calculus. In Exercises 57–62, evaluate this limit for the given value of x and function f.
f(x) = 3x - 4, x = 2
검증된 단계별 안내1
Identify the function f(x) = 3x - 4 and the point x = 2 where we need to evaluate the limit.
Substitute f(x) and f(x+h) into the limit expression: lim(h→0) [(f(x+h) - f(x)) / h]. This becomes lim(h→0) [(3(x+h) - 4) - (3x - 4)] / h.
Simplify the expression inside the limit: (3(x+h) - 4) - (3x - 4) simplifies to 3x + 3h - 4 - 3x + 4, which further simplifies to 3h.
The limit expression now becomes lim(h→0) [3h / h]. Simplify this expression by canceling h in the numerator and denominator, resulting in lim(h→0) [3].
Since the expression is now a constant, the limit as h approaches 0 is simply the constant value, which is 3.
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Limits
In calculus, a limit is a fundamental concept that describes the behavior of a function as its input approaches a certain value. It helps in understanding how functions behave near specific points, which is crucial for defining derivatives and integrals. The notation lim h→0 f(x+h) indicates examining the function f as the increment h approaches zero, allowing us to analyze instantaneous rates of change.
추천 영상:
05:50
One-Sided Limits
Average Rate of Change
The average rate of change of a function over an interval is calculated 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) for the interval [a, b]. This concept is essential for understanding how a function behaves over an interval and serves as a foundation for the more precise instantaneous rate of change, which is explored through limits.
추천 영상:
가이드 코스
06:37Average Value of a Function
Derivative
The derivative of a function at a point represents the instantaneous rate of change of the function with respect to its variable. It is defined as the limit of the average rate of change as the interval approaches zero, formally expressed as f'(x) = lim h→0 (f(x+h) - f(x)) / h. Derivatives are crucial in calculus for analyzing the behavior of functions, finding slopes of tangent lines, and solving optimization problems.
추천 영상:
가이드 코스
05:44Derivatives
관련 실천
교과서 질문
Theory and Examples
Once you know limx→a+ f(x) and limx→a− f(x) at an interior point of the domain of f, do you then know limx→a f(x)? Give reasons for your answer.
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교과서 질문
Calculating Limits
Find the limits in Exercises 11–22.
limt→6 8(t−5)(t−7)
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교과서 질문
Use the Intermediate Value Theorem in Exercises 69–74 to prove that each equation has a solution. Then use a graphing calculator or computer grapher to solve the equations.
x³ − 3x − 1 = 0
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1
rank
교과서 질문
Slope of a Curve at a Point
In Exercises 7–18, use the method in Example 3 to find (a) the slope of the curve at the given point P, and (b) an equation of the tangent line at P.
y=√7−x, P(−2,3)
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교과서 질문
Using limθ→0 sin θ / θ = 1
Find the limits in Exercises 23–46.
limθ→0 sin θ / sin 2θ
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