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Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Briggs 3rd EditionCh. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions문제 7.3.97a
7장, 문제 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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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.
추천 영상:
06:11
Limits of Rational Functions: Denominator = 0

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.
추천 영상:
05:13
Slopes of Tangent Lines

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.
추천 영상:
06:29
Derivatives Applied To Velocity
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a. Find an exponential decay function V₁(t) that equals the total volume of the quiescent cells in the tumor t days after treatment.

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a. Are there numbers 0 < a < 1 such that ∫₁₋ₐ¹⁺ᵃ f(x) dx = 0?

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Velocity of falling body Refer to Exercise 95, which gives the position function for a falling body. Use m = 75 kg and k = 0.2.


a. Confirm that the BASE jumper’s velocity t seconds after jumping is v(t) = d'(t) = √(mg/k) tanh (√(kg/m) t).

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Shallow-water velocity equation

a. Confirm that the linear approximation to ƒ(x) = tanh x at a = 0 is L(x) = x.

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Evaluating hyperbolic functions Evaluate each expression without using a calculator or state that the value does not exist. Simplify answers as much as possible.


a. cosh 0

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Falling body When an object falling from rest encounters air resistance proportional to the square of its velocity, the distance it falls (in meters) after t seconds is given by d(t) = (m/k) ln (cosh (√(kg/m) t)), where m is the mass of the object in kilograms, g = 9.8 m/s² is the acceleration due to gravity, and k is a physical constant.


a. A BASE jumper (m = 75 kg) leaps from a tall cliff and performs a ten-second delay (she free-falls for 10 s and then opens her chute). How far does she fall in 10 s? Assume k = 0.2.

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