Ch. 4 - Applications of Derivatives

Hass15th EditionThomas' CalculusISBN: 9780137616077Non è quello che usi tu?Cambia libro di testo

Hass 15th Edition

Ch. 4 - Applications of Derivatives

Problema 4.3.1b

Capitolo 4, Problema 4.3.1b

Analyzing Functions from Derivatives

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

b. On what open intervals is f increasing or decreasing?

f′(x) = x(x − 1)

Guida verificata passo dopo passo

To determine where the function f is increasing or decreasing, we need to analyze the sign of its derivative f'(x). The function is increasing where f'(x) > 0 and decreasing where f'(x) < 0.

First, find the critical points by setting the derivative equal to zero: f'(x) = x(x - 1) = 0. Solve for x to find the critical points.

The solutions to the equation x(x - 1) = 0 are x = 0 and x = 1. These are the critical points where the derivative changes sign.

Next, determine the sign of f'(x) on the intervals defined by the critical points: (-∞, 0), (0, 1), and (1, ∞). Choose a test point from each interval and substitute it into f'(x) to determine the sign.

Based on the sign of f'(x) in each interval, conclude where the function f is increasing or decreasing. If f'(x) > 0 in an interval, f is increasing there; if f'(x) < 0, f is decreasing.

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Derivative and Critical Points

The derivative of a function, f'(x), provides information about the slope of the tangent line at any point x. Critical points occur where f'(x) = 0 or is undefined, indicating potential maxima, minima, or points of inflection. Identifying these points is crucial for determining intervals of increase or decrease in the function.

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Critical Points

Increasing and Decreasing Intervals

A function is increasing on an interval if its derivative is positive over that interval, meaning the function's slope is upward. Conversely, it is decreasing if the derivative is negative, indicating a downward slope. Analyzing the sign of f'(x) across different intervals helps identify where the function is increasing or decreasing.

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Determining Where a Function is Increasing & Decreasing

Sign Analysis of Derivatives

Sign analysis involves determining the sign of the derivative f'(x) over various intervals. For f'(x) = x(x - 1), factorization reveals critical points at x = 0 and x = 1. By testing values in intervals around these points, one can ascertain where f'(x) is positive or negative, thus identifying the function's behavior in terms of increasing or decreasing.

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Percorso guidato

06:51

Derivatives Applied To Acceleration Example 2

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Finding Antiderivatives

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Theory and Examples

Cubic functions Consider the cubic function f(x) = ax³ + bx² + cx + d.