Textbook Question
Turning a corner with a pole
What is the length of the longest pole that can be carried horizontally around a corner at which a corridor that is a ft wide and a corridor that is b ft wide meet at right angles?
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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 76
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
f(x) = 2x⁴ + 8x³ + 12x² - x - 2
Verified step by step guidance1
To determine concavity, we need to find the second derivative of the function f(x). Start by finding the first derivative f'(x) of the function f(x) = 2x⁴ + 8x³ + 12x² - x - 2.
Calculate the first derivative: f'(x) = d/dx [2x⁴ + 8x³ + 12x² - x - 2]. Use the power rule for differentiation: f'(x) = 8x³ + 24x² + 24x - 1.
Next, find the second derivative f''(x) by differentiating f'(x): f''(x) = d/dx [8x³ + 24x² + 24x - 1]. Again, apply the power rule: f''(x) = 24x² + 48x + 24.
To find intervals of concavity, solve f''(x) = 0 to find critical points. Set 24x² + 48x + 24 = 0 and solve for x. This will give potential inflection points where concavity changes.
Use the critical points to test intervals on the number line. Choose test points in each interval and evaluate the sign of f''(x) at those points. If f''(x) > 0, the function is concave up; if f''(x) < 0, it is concave down. Identify any inflection points where the concavity changes.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Concavity
Concavity refers to the direction in which a function curves. A function is concave up on an interval if its second derivative is positive, indicating that the slope of the tangent line is increasing. Conversely, it is concave down if the second derivative is negative, meaning the slope is decreasing. Understanding concavity helps in analyzing the behavior of functions and identifying intervals of increase or decrease.
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Determining Concavity Given a Function
Second Derivative Test
The second derivative test is a method used to determine the concavity of a function and locate inflection points. By calculating the second derivative of a function, we can assess where it changes sign. If the second derivative is positive, the function is concave up; if negative, it is concave down. Inflection points occur where the second derivative equals zero or is undefined, indicating a change in concavity.
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06:02The Second Derivative Test: Finding Local Extrema
Inflection Points
Inflection points are specific points on a curve where the concavity changes. These points are significant because they indicate a transition in the behavior of the function, which can affect its graph and the nature of its extrema. To find inflection points, one must solve for values of x where the second derivative is zero or undefined, and then verify that the concavity changes at those points.
Recommended video:
04:50Critical Points
Related Practice
Textbook Question
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
g(t) = 3t⁵ - 30t⁴ + 80t³ + 100
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Textbook Question
Two methods Evaluate the following limits in two different ways: Use the methods of Chapter 2 and use l’Hôpital’s Rule.
lim_x→0 (e²ˣ + 4eˣ - 5) / (e²ˣ - 1)
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Textbook Question
The arbelos An arbelos is the region enclosed by three mutually tangent semicircles; it is the region inside the larger semicircle and outside the two smaller semicircles (see figure). <IMAGE>
b. Show that the area of the arbelos is the area of a circle whose diameter is the distance BD in the figure.
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Textbook Question
Concavity Determine the intervals on which the following functions are concave up or concave down. Identify any inflection points.
h(t) = 2 + cos 2t on [0,π]
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Textbook Question
{Use of Tech} Optimal boxes Imagine a lidless box with height h and a square base whose sides have length x. The box must have a volume of 125 ft³.
b. Based on your graph in part (a), estimate the value of x that produces the box with a minimum surface area.
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