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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 7.3.96c
7장, 문제 7.3.96c

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.


c. How long does it take for the BASE jumper to reach a speed of 45 m/s (roughly 100 mi/hr)?

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Recall that the velocity function \(v(t)\) is the derivative of the position function \(s(t)\) with respect to time \(t\). So, start by writing down the velocity function \(v(t) = s'(t)\).
From Exercise 95, the position function for a falling body with mass \(m\) and drag coefficient \(k\) is typically given by \(s(t) = \frac{mg}{k}t - \frac{m^2g}{k^2}(1 - e^{-\frac{k}{m}t})\). Differentiate this function to find the velocity function \(v(t)\).
After differentiating, you should get \(v(t) = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\). This formula shows how velocity changes over time considering air resistance.
Set the velocity equal to 45 m/s to find the time \(t\) when the BASE jumper reaches this speed: \(45 = \frac{mg}{k} (1 - e^{-\frac{k}{m}t})\).
Solve the equation for \(t\) by isolating the exponential term, then taking the natural logarithm: first, rearrange to get \(e^{-\frac{k}{m}t} = 1 - \frac{45k}{mg}\), then take \(\ln\) on both sides and solve for \(t\).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
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주요 개념

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

Position and Velocity Functions

The position function describes the location of a falling body over time, while the velocity function is its derivative, representing the rate of change of position. Understanding how to differentiate the position function is essential to find the velocity at any given time.
추천 영상:
가이드 코스
10:17
Using The Velocity Function

Solving for Time Given Velocity

To find the time when the velocity reaches a specific value, set the velocity function equal to that value and solve for time. This often involves algebraic manipulation or solving transcendental equations depending on the form of the velocity function.
추천 영상:
가이드 코스
10:17
Using The Velocity Function

Parameters in Motion Equations (Mass and Drag Coefficient)

Mass (m) and drag coefficient (k) affect the motion of the falling body by influencing acceleration and terminal velocity. These parameters appear in the differential equations governing motion and must be used correctly to model realistic velocity and position.
추천 영상:
04:11
Eliminate Parameter: Equations with Trig
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Properties of exp(x) Use the inverse relations between ln x and exp(x), and the properties of ln x, to prove the following properties:


c. (exp(x))ᵖ = exp(px), p rational

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Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.

c. ln(1 + √2) = −ln(−1 + √2)

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Oil consumption Starting in 2018 (t=0), the rate at which oil is consumed by a small country increases at a rate of 1.5%/yr, starting with an initial rate of 1.2 million barrels/yr.


c. How many years after 2018 will the amount of oil consumed since 2018 reach 10 million barrels?

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Evaluating hyperbolic functions Use a calculator to evaluate each expression or state that the value does not exist. Report answers accurate to four decimal places to the right of the decimal point.

c. csch⁻¹ 5

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Energy consumption On the first day of the year (t=0), a city uses electricity at a rate of 2000 MW. That rate is projected to increase at a rate of 1.3% per year.


c. Find a function that gives the total energy used (in MW-yr) between t=0 and any future time t>0.

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교과서 질문

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

c. Show that ln 2ⁿ > n/2 and ln 2^(−n) < −n/2.

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