Skip to main content
Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 4 - Applications of Derivatives문제 4.2.6
4장, 문제 4.2.6

Checking the Mean Value Theorem


Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.


g(x) = {x³, −2 ≤ x ≤ 0
x², 0 < x ≤ 2

검증된 단계별 안내
1
First, understand the Mean Value Theorem (MVT). It states that for a function g(x) that is continuous on the closed interval [a, b] and differentiable on the open interval (a, b), there exists at least one c in (a, b) such that (g(b) - g(a)) / (b - a) = g'(c).
Identify the intervals and the function pieces: g(x) = x³ for -2 ≤ x ≤ 0 and g(x) = x² for 0 < x ≤ 2. The function is continuous on [-2, 2] but we need to check differentiability at x = 0.
Check differentiability at x = 0 by finding the derivatives of each piece: For x³, the derivative is g'(x) = 3x², and for x², the derivative is g'(x) = 2x. Evaluate the left-hand and right-hand derivatives at x = 0 to ensure they are equal.
Calculate g(-2) and g(2) to apply the MVT: g(-2) = (-2)³ = -8 and g(2) = (2)² = 4. Use these to find the average rate of change: (g(2) - g(-2)) / (2 - (-2)) = (4 - (-8)) / 4.
Set the average rate of change equal to the derivative g'(c) and solve for c in each interval: For -2 < c < 0, use g'(c) = 3c², and for 0 < c < 2, use g'(c) = 2c. Solve these equations to find the value(s) of c that satisfy the MVT.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
6m
도움이 되었나요?

주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

Mean Value Theorem

The Mean Value Theorem (MVT) states that for a continuous function on a closed interval [a, b] that is differentiable on the open interval (a, b), there exists at least one point c in (a, b) such that the derivative at c equals the average rate of change over [a, b]. This is expressed as f'(c) = (f(b) - f(a)) / (b - a).
추천 영상:
가이드 코스
06:11
Fundamental Theorem of Calculus Part 1

Piecewise Functions

A piecewise function is defined by different expressions over different intervals. In this problem, g(x) is defined as x³ for -2 ≤ x ≤ 0 and x² for 0 < x ≤ 2. Understanding how to evaluate and differentiate each piece within its respective interval is crucial for applying the Mean Value Theorem.
추천 영상:
가이드 코스
05:36
Piecewise Functions

Differentiability and Continuity

For the Mean Value Theorem to apply, the function must be continuous on the closed interval and differentiable on the open interval. This requires checking that g(x) is continuous at x = 0, where the pieces meet, and ensuring differentiability across the entire interval, except possibly at the endpoints.
추천 영상:
05:34
Intro to Continuity
관련 실천
교과서 질문

117. Suppose that the second derivative of the function y = f(x) isy" =(x+1)(x-2).

For what x-values does the graph of f have an inflection point?

222
views
교과서 질문

Initial Value Problems


Solve the initial value problems in Exercises 71–90.


dv/dt = (1/2)sec t tan t, v(0) = 1

13
views
교과서 질문

Identifying Extrema


In Exercises 19–40:


a. Find the open intervals on which the function is increasing and those on which it is decreasing.


b. Identify the function’s local extreme values, if any, saying where they occur.


f(x) = x³ / (3x² + 1)

159
views
교과서 질문

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² + 2/x

159
views
교과서 질문

"Roots (Zeros) Show that the functions in Exercises 19–26 have exactly one zero

206
views
교과서 질문

Finding Critical Points


In Exercises 41–50, determine all critical points and all domain endpoints for each function.


y = x² − 32√x

200
views