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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.R.2e
Chapter 4, Problem 4.R.2e

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>
e. On what intervals (approximately) is f concave up?

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To determine where the function f is concave up, we need to analyze the second derivative, f''(x). A function is concave up on intervals where its second derivative is positive.
First, find the first derivative f'(x) of the function f. This will help us understand the slope of the tangent line at any point on the graph.
Next, find the second derivative f''(x) by differentiating f'(x). This will give us information about the curvature of the graph.
Examine the sign of f''(x) over the interval [-3, 3]. Identify the subintervals where f''(x) > 0, as these are the intervals where the function is concave up.
If the graph of f is provided, visually inspect the graph to confirm the intervals where the curvature is upwards, which corresponds to f''(x) being positive.

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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 graph opens upwards, resembling a cup. This occurs when the second derivative of the function is positive, indicating that the slope of the tangent line is increasing.
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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. If the second derivative, denoted as f''(x), is greater than zero on an interval, the function is concave up on that interval. Conversely, if f''(x) is less than zero, the function is concave down.
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The Second Derivative Test: Finding Local Extrema

Intervals of Concavity

Intervals of concavity are specific ranges on the x-axis where a function maintains a consistent concavity. To find these intervals, one typically analyzes the sign of the second derivative across the domain of the function, identifying where it remains positive or negative to determine where the function is concave up or down.
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Determining Concavity Given a Function
Related Practice
Textbook Question

{Use of Tech} A family of superexponential functions Let ƒ(x) = (a + x)ˣ , where a > 0.



c. Compute ƒ'. Then graphƒ and ƒ' for a = 0.5, 1, 2, and 3.

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

d. Give the approximate coordinates of the zero(s) of f.

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Textbook Question

60–81. Limits Evaluate the following limits. Use l’Hôpital’s Rule when needed. 

lim_t→0 (1 - cos 6t) / 2t

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

f. On what intervals (approximately) is f concave down?  

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Textbook Question

82–89. Comparing growth rates Determine which of the two functions grows faster, or state that they have comparable growth rates.


x¹⸍² and x¹⸍³

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Textbook Question

Locating extrema Consider the graph of a function ƒ on the interval [-3, 3]. <IMAGE>

c. Give the approximate coordinates of the inflection point(s) of f.

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