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 4.3.5b

Capitolo 4, Problema 4.3.5b

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 − 1)(x + 2)(x − 3)

Guida verificata passo dopo passo

First, identify the critical points of the function f by setting the derivative f'(x) = (x - 1)(x + 2)(x - 3) equal to zero. Solve the equation (x - 1)(x + 2)(x - 3) = 0 to find the values of x where the derivative is zero.

The solutions to the equation are x = 1, x = -2, and x = 3. These are the critical points where the function f could change from increasing to decreasing or vice versa.

Next, determine the sign of f'(x) in the intervals defined by the critical points: (-∞, -2), (-2, 1), (1, 3), and (3, ∞). Choose test points from each interval and substitute them into f'(x) to check whether the derivative is positive or negative.

For each interval, if f'(x) is positive, the function f is increasing in that interval. If f'(x) is negative, the function f is decreasing in that interval.

Summarize the intervals where f is increasing and decreasing based on the sign of f'(x) in each interval. This will give you the open intervals where the function f is increasing or decreasing.

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Critical Points

Critical points of a function occur where its derivative is zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. In the given derivative f′(x) = (x − 1)(x + 2)(x − 3), the critical points are x = 1, x = -2, and x = 3, where the derivative equals zero.

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04:50

Critical Points

Increasing and Decreasing Intervals

A function is increasing on intervals where its derivative is positive and decreasing where its derivative is negative. By analyzing the sign of f′(x) = (x − 1)(x + 2)(x − 3) across different intervals determined by the critical points, we can determine where the function f is increasing or decreasing.

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Determining Where a Function is Increasing & Decreasing

Sign Chart Analysis

A sign chart helps visualize the sign changes of a derivative across different intervals. By testing points in each interval created by the critical points, we can determine the sign of the derivative in those intervals. This analysis helps identify where the function is increasing or decreasing, based on whether the derivative is positive or negative.

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Percorso guidato

06:51

Derivatives Applied To Acceleration Example 2

Pratica correlata

Domanda del libro di testo

[Technology Exercise] 16. Designing a box with a lid A piece of cardboard measures 10 in. by 15 in. Two equal squares are removed from the corners of a 10-in. side as shown in the figure. Two equal rectangles are removed from the other corners so that the tabs can be folded to form a rectangular box with lid.

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b. Find the domain of V for the problem situation and graph V over this domain.

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114. Parabolas

b. When is the parabola concave up? Concave down?

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Domanda del libro di testo

Identifying Extrema

In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).

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b. Either use the graph to determine which intervals f is positive on and which intervals f is negative on, or explain why this information cannot be determined from the graph.

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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.

-sec²(3x/2)

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Theory and Examples

In Exercises 51 and 52, give reasons for your answers.

Let f(x) = |x³ − 9x|.

b. Does f'(3) exist?