Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problem 4.2.15ai

Chapter 4, Problem 4.2.15ai

Roots (Zeros)

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

i. y = x² − 4

Verified step by step guidance

First, identify the zeros of the polynomial y = x² − 4. To find the zeros, set the equation equal to zero: x² − 4 = 0.

Solve the equation x² − 4 = 0 by factoring it as (x - 2)(x + 2) = 0. This gives the zeros x = 2 and x = -2.

Next, find the first derivative of the polynomial y = x² − 4. The derivative, using basic differentiation rules, is y' = 2x.

Set the first derivative equal to zero to find its zeros: 2x = 0. Solving this gives x = 0.

Plot the zeros of the original polynomial (x = 2 and x = -2) and the zero of its first derivative (x = 0) on a number line. This visual representation helps in understanding the behavior of the function and its slope at these points.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Roots 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 = x² − 4, the roots can be found by setting the equation to zero and solving for x, resulting in x = ±2. These roots represent the points where the graph of the polynomial intersects the x-axis.

First Derivative of a Polynomial

The first derivative of a polynomial provides the rate of change of the function and is used to find critical points, such as maxima, minima, and points of inflection. For y = x² − 4, the first derivative is y' = 2x. Setting the derivative to zero helps identify the x-values where the slope of the tangent is zero, indicating potential extrema.

Plotting on a Number Line

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. For y = x² − 4, plot the roots x = ±2 and the zero of the derivative x = 0. This visualization aids in understanding how the function behaves around these critical points.

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Related Practice

Textbook Question

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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Textbook Question

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)

260

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Textbook Question

Identifying Extrema

In Exercises 19–40:

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

f(θ) = 3θ² − 4θ³

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Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

csc x cot x

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Textbook Question

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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Textbook Question

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

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Roots (Zeros)a. Plot the zeros of each polynomial on a line together with the zeros of its first derivative.i. y = x² − 4

Verified video answer for a similar problem:

Key Concepts

Roots of a Polynomial

First Derivative of a Polynomial

Plotting on a Number Line

Roots (Zeros)

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

i. y = x² − 4