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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 4.2.52b
Chapter 4, Problem 4.2.52b

{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).
b. Sketch graphs of f and g to show that these functions do not differ by a constant.

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Identify the functions: Both f(x) and g(x) are given as f(x) = ln((x+1)/(x-1)) and g(x) = ln((x+1)/(x-1)). Notice that they are identical, which suggests they should be the same function.
Understand the concept: If two functions differ by a constant, their graphs will be vertically shifted versions of each other. This means that for all x in their domain, f(x) = g(x) + C, where C is a constant.
Determine the domain: The domain of both functions is x > 1 and x < -1, since the argument of the logarithm, (x+1)/(x-1), must be positive.
Sketch the graph: Plot the graph of f(x) = ln((x+1)/(x-1)) over its domain. Since f(x) and g(x) are identical, their graphs will overlap completely, indicating they do not differ by a constant.
Conclude from the graph: Since the graphs of f(x) and g(x) are identical and overlap completely, it confirms that they do not differ by a constant. If they did, one graph would be a vertical shift of the other.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Natural Logarithm

The natural logarithm, denoted as ln(x), is the logarithm to the base e, where e is approximately 2.71828. It is a fundamental function in calculus, particularly in relation to growth rates and areas under curves. Understanding the properties of the natural logarithm, such as its domain and range, is essential for analyzing functions like f(x) and g(x).
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Derivative of the Natural Logarithmic Function

Graphing Functions

Graphing functions involves plotting points on a coordinate system to visualize their behavior. For functions f(x) and g(x), sketching their graphs helps to identify key features such as intercepts, asymptotes, and overall shape. This visual representation is crucial for determining whether two functions differ by a constant, as it allows for direct comparison of their outputs across the same input values.
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Graph of Sine and Cosine Function

Difference of Functions

The difference of two functions, f(x) and g(x), is expressed as f(x) - g(x). If this difference is a constant for all x in the domain, it indicates that the two functions are parallel and differ only by that constant. In the context of the given functions, analyzing their difference will reveal whether they maintain a consistent vertical shift or if they diverge in behavior.
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Exponential Functions
Related Practice
Textbook Question

Pen problems


b. A rancher plans to make four identical and adjacent rectangular pens against a barn, each with an area of 100 m² (see figure). What are the dimensions of each pen that minimize the amount of fence that must be used? <IMAGE>

Textbook Question

{Use of Tech} Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45° to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x) = 32x² / v² + x + 8 (see figure). <IMAGE>



b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s = √ (x - 17.25)² + ( -(4x² / 81) + x - 2)² (Hint: The diameter of the basketball hoop is 18 inches.) 

Textbook Question

Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.


b. Does either g or h have a local extreme value at x = 2? Explain.

Textbook Question

{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>

b. Find the time and the displacement when the object reaches its lowest point.

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Textbook Question

Cylinder in a cone A right circular cylinder is placed inside a cone of radius R and height H so that the base of the cylinder lies on the base of the cone.


b. Find the dimensions of the cylinder with maximum lateral surface area (area of the curved surface).

Textbook Question

{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>


b. Find the critical point of T on (0, 50).

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