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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 4.3.44
Chapter 4, Problem 4.3.44

Increasing and decreasing functions. Find the intervals on which f is increasing and the intervals on which it is decreasing.


f(x) = tan⁻¹ (x/(x²+2))

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First, understand that to determine where the function f(x) is increasing or decreasing, we need to find its derivative, f'(x). This will help us identify the critical points and analyze the behavior of the function.
Calculate the derivative f'(x) using the chain rule and quotient rule. The function f(x) = tan⁻¹(u) where u = x/(x²+2). The derivative of tan⁻¹(u) with respect to x is 1/(1+u²) * du/dx.
Apply the quotient rule to find du/dx for u = x/(x²+2). The quotient rule states that if u = v/w, then du/dx = (v'w - vw')/w². Here, v = x and w = x²+2.
Simplify the expression for f'(x) using the derivative found in the previous step. This involves substituting du/dx into the derivative of tan⁻¹(u) and simplifying the expression.
Determine the intervals where f'(x) > 0 (function is increasing) and f'(x) < 0 (function is decreasing). This involves solving inequalities and considering the critical points where f'(x) = 0 or is undefined.

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Key Concepts

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

Derivative

The derivative of a function measures the rate at which the function's value changes as its input changes. It is a fundamental tool in calculus used to determine the slope of the tangent line to the curve at any point. For a function to be increasing, its derivative must be positive, while a negative derivative indicates that the function is decreasing.
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Derivatives

Critical Points

Critical points occur where the derivative of a function is either zero or undefined. These points are essential for analyzing the behavior of the function, as they can indicate potential local maxima, minima, or points of inflection. To determine intervals of increase or decrease, one must evaluate the derivative at these critical points and test the sign of the derivative in the intervals created by these points.
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Critical Points

First Derivative Test

The First Derivative Test is a method used to determine whether a function is increasing or decreasing on specific intervals. By examining the sign of the derivative before and after each critical point, one can conclude whether the function transitions from increasing to decreasing or vice versa. This test provides a systematic approach to identifying the intervals of increase and decrease for a given function.
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The First Derivative Test: Finding Local Extrema
Related Practice
Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 6x² - x³

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Textbook Question

Differentials Consider the following functions and express the relationship between a small change in x and the corresponding change in y in the form dy = f'(x)dx.


f(x) = 2x + 1

Textbook Question

17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.


lim_x→π (cos x +1 ) / (x - π )²

Textbook Question

A car starting at rest accelerates at 16 ft/s² for 5 seconds on a straight road. How far does it travel during this time?

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Textbook Question

{Use of Tech} A pursuit curve A man stands 1 mi east of a crossroads. At noon, a dog starts walking north from the crossroads at 1 mi/hr (see figure). At the same instant, the man starts walking and at all times walks directly toward the dog at s > 1 mi/hr . The path in the xy-plane followed by the man as he pursues the dog is given by the function y = ƒ(x) = s/2 ((x(ˢ⁺¹)/ˢ) /(s+1) - (x(ˢ⁺¹)/ˢ / s-1)) + s/ s² - 1


Select various values of s > 1 and graph this pursuit curve. Comment on the changes in the curve as s increases. <IMAGE>


Textbook Question

Second Derivative Test Locate the critical points of the following functions. Then use the Second Derivative Test to determine (if possible) whether they correspond to local maxima or local minima.


f(x) = 2x² ln x - 11x²

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