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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 4.3.43
Chapter 4, Problem 4.3.43

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


f(x) = xe⁻(ˣ²/₂)

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To determine where the function f(x) = x * e^(-x²/2) is increasing or decreasing, we first need to find its derivative, f'(x). Use the product rule for differentiation, which states that if you have a function h(x) = u(x) * v(x), then h'(x) = u'(x) * v(x) + u(x) * v'(x). Here, u(x) = x and v(x) = e^(-x²/2).
Differentiate u(x) = x to get u'(x) = 1. For v(x) = e^(-x²/2), use the chain rule. The derivative of e^u is e^u * u', where u = -x²/2. So, v'(x) = e^(-x²/2) * (-x).
Apply the product rule: f'(x) = u'(x) * v(x) + u(x) * v'(x) = 1 * e^(-x²/2) + x * (-x) * e^(-x²/2). Simplify this to get f'(x) = e^(-x²/2) - x² * e^(-x²/2).
Factor out e^(-x²/2) from f'(x): f'(x) = e^(-x²/2) * (1 - x²). The sign of f'(x) depends on the sign of (1 - x²) since e^(-x²/2) is always positive.
Solve the inequality 1 - x² > 0 to find where f'(x) > 0 (increasing) and 1 - x² < 0 for f'(x) < 0 (decreasing). The solution to 1 - x² > 0 is -1 < x < 1, so f(x) is increasing on (-1, 1). For 1 - x² < 0, x < -1 or x > 1, so f(x) is decreasing on (-∞, -1) and (1, ∞).

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Key Concepts

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

Critical Points

Critical points are values of x in a function where the derivative is either zero or undefined. These points are essential for determining where a function changes from increasing to decreasing or vice versa. To find critical points, we first compute the derivative of the function and set it equal to zero, solving for x.
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First Derivative Test

The First Derivative Test is a method used to determine the behavior of a function at its critical points. By analyzing the sign of the derivative before and after each critical point, we can conclude whether the function is increasing or decreasing in those intervals. If the derivative changes from positive to negative, the function is decreasing; if it changes from negative to positive, the function is increasing.
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The First Derivative Test: Finding Local Extrema

Exponential Functions

Exponential functions, such as f(x) = xe⁻(ˣ²/₂), involve a constant raised to a variable exponent. In this case, the function combines polynomial and exponential components, which can affect its growth and decay rates. Understanding the behavior of exponential functions is crucial for analyzing their derivatives and determining intervals of increase and decrease.
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Related Practice
Textbook Question

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


f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8

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Textbook Question

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Textbook Question

Use the following graphs to identify the points (if any) on the interval [a, b] at which the function has an absolute maximum or an absolute minimum value <IMAGE>

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Textbook Question

Solving initial value problems Find the solution of the following initial value problems.


y'(Θ) = ((√2 cos³ Θ + 1)/cos² Θ); y (π/4) = 3, -π/2 < Θ < π/2

Textbook Question

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Textbook Question

23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.


∫ sec Θ(tan Θ + sec Θ + cos Θ)dΘ