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Ch. 4 - Applications of the Derivative
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 4 - Applications of the DerivativeProblem 107a
Chapter 4, Problem 107a

Interpreting the derivative The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


a. On what interval(s) is f increasing? Decreasing?

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To determine where the function f is increasing or decreasing, we need to analyze the sign of its derivative f'. A function f is increasing on intervals where f' is positive and decreasing where f' is negative.
Examine the graph of f' over the interval [-3, 2]. Identify the sections where the graph of f' is above the x-axis (positive) and where it is below the x-axis (negative).
List the intervals where f' is positive. These intervals correspond to where the function f is increasing.
List the intervals where f' is negative. These intervals correspond to where the function f is decreasing.
Ensure to consider any points where f' crosses the x-axis, as these are points where the behavior of f changes from increasing to decreasing or vice versa.

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Key Concepts

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

Derivative and Its Interpretation

The derivative of a function, denoted as f', represents the rate of change of the function f with respect to its variable. It provides critical information about the function's behavior, specifically whether it is increasing or decreasing. When f' is positive, the function f is increasing; when f' is negative, f is decreasing.
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Derivatives

Critical Points

Critical points occur where the derivative f' is zero or undefined. These points are essential for determining intervals of increase and decrease, as they can indicate potential local maxima or minima. Analyzing the sign of the derivative around these points helps in understanding the overall behavior of the function.
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Critical Points

Sign Chart

A sign chart is a visual tool used to determine the intervals where a function is increasing or decreasing based on the sign of its derivative. By plotting the critical points and testing intervals between them, one can easily see where the derivative is positive (indicating increase) or negative (indicating decrease), thus providing a clear picture of the function's behavior over the specified interval.
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Determining Where a Function is Increasing & Decreasing
Related Practice
Textbook Question

Consider the lim_x→∞ (√ ax + b) / √cx + d where a, b, c, and d are positive real numbers. Show that l’Hôpital’s Rule fails for this limit. Find the limit using another method.

Textbook Question

{Use of Tech} Graph carefully Graph the function f(x) = 60x⁵ - 901x³ + 27x in the window [-4,4] x [-10,000, 10,000]. How many extreme values do you see? Locate all the extreme values by analyzing f'. 

Textbook Question

The graph of f' on the interval [-3,2] is shown in the figure. <IMAGE>


c. At what point(s) does f have an inflection point?

Textbook Question

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x² ln x; x³

Textbook Question

Use limit methods to determine which of the two given functions grows faster, or state that they have comparable growth rates.

x²⁰ ; 1.0001ˣ

Textbook Question

Concavity of parabolas Consider the general parabola described by the function f(x) = ax² + bx + c. For what values of a, b, and c is f concave up? For what values of a, b, and c is f concave down?