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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
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Tutti i libri di testoHass 15th EditionCh. 4 - Applications of DerivativesProblema 4.3.3a
Capitolo 4, Problema 4.3.3a

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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To find the critical points of the function f, we need to determine where its derivative f'(x) is equal to zero or undefined. In this case, f'(x) = (x - 1)²(x + 2).
Set the derivative equal to zero: (x - 1)²(x + 2) = 0. This equation will help us find the values of x where the derivative is zero, indicating potential critical points.
Solve the equation (x - 1)²(x + 2) = 0 by setting each factor equal to zero: (x - 1)² = 0 and (x + 2) = 0.
For (x - 1)² = 0, solve for x to get x = 1. This is a critical point because the derivative is zero at this value.
For (x + 2) = 0, solve for x to get x = -2. This is another critical point because the derivative is zero at this value.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are significant because they can indicate local maxima, minima, or points of inflection. To find critical points, set the derivative equal to zero and solve for the variable, or identify where the derivative does not exist.
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Critical Points

Factoring Polynomials

Factoring polynomials involves expressing a polynomial as a product of its factors. This is crucial for solving equations where the polynomial is set to zero, as it allows us to find the roots or solutions. In the given derivative, f′(x) = (x − 1)²(x + 2), factoring helps identify the values of x that make the derivative zero.
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Introduction to Polynomial Functions

Derivative Analysis

Analyzing the derivative of a function provides insights into the function's behavior, such as increasing or decreasing intervals and concavity. The sign and value of the derivative indicate the slope of the tangent line at any point on the function. For f′(x) = (x − 1)²(x + 2), examining the derivative helps determine where the function changes direction or has potential extrema.
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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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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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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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Finding Antiderivatives

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

(3/2)√x

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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

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