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 75
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
Verified step by step guidance1
First, find the first derivative of the function g(t) = 3t⁵ - 30t⁴ + 80t³ + 100. This will help us understand the rate of change of the function.
Next, find the second derivative of the function. The second derivative will help us determine the concavity of the function. Use the first derivative to calculate this.
Set the second derivative equal to zero and solve for t. This will help identify potential inflection points where the concavity might change.
Analyze the sign of the second derivative on the intervals determined by the critical points found in the previous step. If the second derivative is positive, the function is concave up; if negative, concave down.
Identify any inflection points by checking where the second derivative changes sign. These points are where the function transitions from concave up to concave down or vice versa.
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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 curve bends. 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, a function is concave down if its second derivative is negative, meaning the slope of the tangent line is decreasing. Understanding concavity helps in analyzing the behavior of functions and identifying points of inflection.
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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 found by setting the second derivative equal to zero and solving for the variable. At an inflection point, the function transitions from concave up to concave down or vice versa. Identifying these points is crucial for understanding the overall shape and behavior of the function.
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04:50Critical Points
Related Practice
Textbook Question
Another pen problem A rancher is building a horse pen on the corner of her property using 1000 ft of fencing. Because of the unusual shape of her property, the pen must be built in the shape of a trapezoid (see figure). <IMAGE>
b. Suppose there is already a fence along the side of the property opposite the side of length y. Find the lengths of the sides that maximize the area of the pen, using 1000 ft of fencing.
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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.
f(x) = 2x⁴ + 8x³ + 12x² - x - 2
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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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