Textbook Question
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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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.3.42
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
Verified step by step guidance1
First, find the derivative of the function f(x) = x⁴/4 - 8x³/3 + 15x²/2 + 8. The derivative, f'(x), will help us determine where the function is increasing or decreasing.
Calculate the derivative: f'(x) = d/dx (x⁴/4 - 8x³/3 + 15x²/2 + 8). Use the power rule for differentiation: d/dx (xⁿ) = n*xⁿ⁻¹.
Simplify the derivative: f'(x) = (4*x³/4) - (8*3*x²/3) + (15*2*x/2). This simplifies to f'(x) = x³ - 8x² + 15x.
Find the critical points by setting the derivative equal to zero: f'(x) = x³ - 8x² + 15x = 0. Factor the equation to find the values of x where the derivative is zero.
Determine the intervals of increase and decrease by testing values from the intervals created by the critical points in the derivative. If f'(x) > 0, the function is increasing; if f'(x) < 0, the function is decreasing.
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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 where the derivative of a function 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, you first compute the derivative of the function and then solve for x when the derivative equals zero.
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Critical Points
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, you can conclude whether the function is increasing or decreasing in the intervals around those points. 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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07:09The First Derivative Test: Finding Local Extrema
Intervals of Increase and Decrease
Intervals of increase and decrease refer to the ranges of x-values where a function is respectively rising or falling. A function is increasing on an interval if its derivative is positive throughout that interval, while it is decreasing if the derivative is negative. Identifying these intervals helps in understanding the overall behavior of the function and is crucial for graphing and optimization problems.
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07:32Determining Where a Function is Increasing & Decreasing
Related Practice
Textbook Question
Given a function f that is differentiable on its domain, write and explain the relationship between the differentials dx and dy.
Textbook Question
Avalanche forecasting Avalanche forecasters measure the temperature gradient dT/dh, which is the rate at which the temperature in a snowpack T changes with respect to its depth h. A large temperature gradient may lead to a weak layer in the snowpack. When these weak layers collapse, avalanches occur. Avalanche forecasters use the following rule of thumb: If dT/dh exceeds 10° C/m anywhere in the snowpack, conditions are favorable for weak-layer formation, and the risk of avalanche increases. Assume the temperature function is continuous and differentiable.
a. An avalanche forecaster digs a snow pit and takes two temperature measurements. At the surface (h = 0), the temperature is -16° C. At a depth of 1.1 m, the temperature is -2° C. Using the Mean Value Theorem, what can he conclude about the temperature gradient? Is the formation of a weak layer likely?
Textbook Question
Graphing functions Use the guidelines of this section to make a complete graph of f.
f(x) = 1/(e⁻ˣ - 1)
Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ 0⁺ (1 - ln x) / (1 + ln x)
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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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