Textbook Question
17–83. Limits Evaluate the following limits. Use l’Hôpital’s Rule when it is convenient and applicable.
lim_x→ ∞ (e³ˣ ) / (3e³ˣ + 5)
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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 4.1.12
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>
Verified step by step guidance1
Begin by understanding the concept of absolute maximum and minimum values. An absolute maximum is the highest point over the entire interval, while an absolute minimum is the lowest point over the entire interval.
Examine the graph provided within the interval [a, b]. Look for the highest and lowest points on the graph within this interval. These points are candidates for absolute maximum and minimum values.
Identify any critical points within the interval [a, b]. Critical points occur where the derivative of the function is zero or undefined. These points can be potential locations for maximum or minimum values.
Check the endpoints of the interval [a, b]. Sometimes, the absolute maximum or minimum can occur at the endpoints of the interval, so it's important to evaluate the function at these points.
Compare the function values at the critical points and endpoints to determine which is the highest and which is the lowest. The highest value will be the absolute maximum, and the lowest value will be the absolute minimum.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Absolute Maximum and Minimum
An absolute maximum of a function on a given interval is the highest value that the function attains within that interval, while an absolute minimum is the lowest value. These extrema can occur at critical points, where the derivative is zero or undefined, or at the endpoints of the interval. Identifying these points is crucial for determining the overall behavior of the function.
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Finding Extrema Graphically Example 4
Critical Points
Critical points are values in the domain of a function where the derivative is either zero or does not exist. These points are significant because they are potential locations for local maxima and minima. To find absolute extrema, one must evaluate the function at these critical points as well as at the endpoints of the interval.
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04:50Critical Points
Evaluating Functions on an Interval
To find absolute extrema, one must evaluate the function at all critical points and at the endpoints of the specified interval. This involves substituting these values into the function to determine which yields the highest and lowest outputs. This process ensures that all potential candidates for absolute extrema are considered.
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Related Practice
Textbook Question
Sketching curves Sketch a graph of a function f that is continuous on (-∞,∞) and has the following properties.
f'(x) < 0 and f"(x) > 0 on (-∞,0); f'(x) > 0 and f"(x) < 0 on (0,∞)
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Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (√x(2x⁶ - 4³√)dx
59
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Textbook Question
Acceleration to position Given the following acceleration functions of an object moving along a line, find the position function with the given initial velocity and position.
a(t) = 2 + 3 sin t; v(0) = 1, s(0) = 10
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Textbook Question
Designer functions Sketch the graph of a function f that is continuous on (-∞,∞) and satisfies the following sets of conditions.
f"(x) > 0 on (-∞,-2); f"(-2) = 0; f'(1) = 0; f"(2) = 0; f'(3) = 0; f"(x) > 0 on (4,∞)
302
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Textbook Question
23–68. Indefinite integrals Determine the following indefinite integrals. Check your work by differentiation.
∫ (4/x√(x² - 1))dx
47
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