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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 4.2.15aiii
4장, 문제 4.2.15aiii

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iii. y = x³ − 3x² + 4 = (x + 1)(x − 2)²

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First, identify the zeros of the polynomial y = x³ − 3x² + 4. To do this, set y equal to zero and solve for x. The polynomial is factored as (x + 1)(x − 2)², so the zeros are x = -1 and x = 2.
Next, find the first derivative of the polynomial y = x³ − 3x² + 4. Use the power rule to differentiate each term: the derivative of x³ is 3x², the derivative of -3x² is -6x, and the derivative of 4 is 0. Therefore, the first derivative is y' = 3x² - 6x.
Set the first derivative equal to zero to find its zeros: 3x² - 6x = 0. Factor out the common term, which is 3x, resulting in 3x(x - 2) = 0. The zeros of the derivative are x = 0 and x = 2.
Plot the zeros of the original polynomial and its derivative on a number line. The zeros of the polynomial are x = -1 and x = 2, and the zeros of the derivative are x = 0 and x = 2. Note that x = 2 is a common zero.
Consider the behavior of the function and its derivative around these zeros. The zero at x = 2 is a repeated zero in the polynomial, indicating a change in concavity or a local extremum. The zero at x = 0 in the derivative suggests a potential inflection point or change in slope.

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Roots (Zeros) of a Polynomial

The roots or zeros of a polynomial are the values of x for which the polynomial equals zero. For the polynomial y = (x + 1)(x − 2)², the zeros are x = -1 and x = 2. These points are where the graph of the polynomial intersects the x-axis.
추천 영상:
6:04
Introduction to Polynomial Functions

First Derivative and Critical Points

The first derivative of a function, denoted as y', represents the rate of change or slope of the function. Finding the zeros of the first derivative helps identify critical points, which are potential locations for local maxima, minima, or points of inflection. For y = x³ − 3x² + 4, the derivative is y' = 3x² - 6x, and its zeros are found by solving 3x(x - 2) = 0, giving x = 0 and x = 2.
추천 영상:
04:50
Critical Points

Graphical Representation of Zeros

Plotting the zeros of a polynomial and its derivative on a number line helps visualize the relationship between the function and its rate of change. The zeros of the polynomial indicate where the graph crosses the x-axis, while the zeros of the derivative indicate where the slope is zero, marking potential turning points or changes in concavity.
추천 영상:
6:37
Zero and Negative Rules
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교과서 질문

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


iv. y = x³ − 33x² + 216x = x(x - 9)(x − 24)

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교과서 질문

Identifying Extrema


In Exercises 19–40:


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


f(θ) = 3θ² − 4θ³

139
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교과서 질문

Identifying Extrema


In Exercises 19–40:


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


f(r) = 3r³ + 16r

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교과서 질문

Analyzing Functions from Derivatives


Answer the following questions about the functions whose derivatives are given in Exercises 1–14:


a. What are the critical points of f?


f′(x) = (x − 1)²(x + 2)

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교과서 질문

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


g(x) = (x − 2) / (x²−1), 0 ≤ x < 1

208
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교과서 질문

Roots (Zeros)


a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.


i. y = x² − 4

231
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