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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic FunctionsProblem 7.3.97a
Chapter 7, Problem 7.3.97a

Terminal velocity Refer to Exercises 95 and 96.


a. Compute a jumper’s terminal velocity, which is defined as lim t → ∞ v(t) = lim t → ∞ √(mg/k) tanh (√(kg/m) t).

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Identify the given velocity function: \(v(t) = \sqrt{\frac{mg}{k}} \tanh \left( \sqrt{\frac{kg}{m}} t \right)\), where \(m\) is mass, \(g\) is acceleration due to gravity, and \(k\) is a constant related to air resistance.
Recall the definition of terminal velocity as the limit of \(v(t)\) as time \(t\) approaches infinity: \(\lim_{t \to \infty} v(t)\).
Focus on the behavior of the hyperbolic tangent function \(\tanh(x)\) as \(x \to \infty\). Remember that \(\tanh(x)\) approaches 1 when \(x\) becomes very large.
Apply this limit to the velocity function: replace \(\tanh \left( \sqrt{\frac{kg}{m}} t \right)\) with 1 as \(t \to \infty\).
Conclude that the terminal velocity is \(\lim_{t \to \infty} v(t) = \sqrt{\frac{mg}{k}} \times 1 = \sqrt{\frac{mg}{k}}\).

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Key Concepts

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

Limit of a Function as t Approaches Infinity

The limit of a function as t approaches infinity describes the behavior of the function as the input grows without bound. In this problem, evaluating lim t → ∞ v(t) helps find the steady-state velocity, known as terminal velocity, where acceleration ceases and velocity stabilizes.
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Hyperbolic Tangent Function (tanh)

The hyperbolic tangent function, tanh(x), is a smooth, continuous function that approaches 1 as x approaches infinity and -1 as x approaches negative infinity. Understanding tanh's limiting behavior is crucial to simplifying the velocity expression for large time values.
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Terminal Velocity in Physics

Terminal velocity is the constant speed an object reaches when the force of gravity is balanced by the drag force, resulting in zero net acceleration. Mathematically, it is the limit of the velocity function as time goes to infinity, representing the maximum velocity attainable during free fall.
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Related Practice
Textbook Question

61–62. Points of intersection and area

a. Sketch the graphs of the functions f and g and find the x-coordinate of the points at which they intersect.


f(x) = sech x, g(x) = tanh x; the region bounded by the graphs of f, g, and the y-axis

Textbook Question

Chemotherapy In an experimental study at Dartmouth College, mice with tumors were treated with the chemotherapeutic drug Cisplatin. Before treatment, the tumors consisted entirely of clonogenic cells that divide rapidly, causing the tumors to double in size every 2.9 days. Immediately after treatment, 99% of the cells in the tumor became quiescent cells which do not divide and lose 50% of their volume every 5.7 days. For a particular mouse, assume the tumor size is 0.5 cm³ at the time of treatment.

a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.

Textbook Question

Depreciation of equipment A large die-casting machine used to make automobile engine blocks is purchased for \$2.5 million. For tax purposes, the value of the machine can be depreciated by 6.8% of its current value each year.


a. What is the value of the machine after 10 years?

Textbook Question

ln x is unbounded Use the following argument to show that lim (x → ∞) ln x = ∞ and lim (x → 0⁺) ln x = −∞.

a. Make a sketch of the function f(x) = 1/x on the interval [1, 2]. Explain why the area of the region bounded by y = f(x) and the x-axis on [1, 2] is ln 2.

Textbook Question

Projection sensitivity

According to the 2014 national population projections published by the U.S. Census Bureau, the U.S. population is projected to be 334.4 million in 2020 with an estimated growth rate of 0.79%/yr.

a. Based on these figures, find the doubling time and the projected population in 2050. Assume the growth rate remains constant.

Textbook Question

"Integral formula Carry out the following steps to derive the formula ∫ csch x dx = ln |tanh(x / 2)| + C (Theorem 7.6).


b. Use the identity for sinh(2u) to show that 2 / sinh(2u) = sech² u / tanh u."