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Ch. 3 - Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Hass 15th EditionCh. 3 - Derivatives문제 3.4.10e
3장, 문제 3.4.10e

Lunar projectile motion A rock thrown vertically upward from the surface of the moon at a velocity of 24 m/sec (about 86 km/h) reaches a height of s = 24t − 0.8t² m in t sec.
e. How long is the rock aloft?

검증된 단계별 안내
1
To determine how long the rock is aloft, we need to find the time 't' when the rock returns to the surface of the moon. This occurs when the height 's' is zero.
Set the height equation to zero: \( s = 24t - 0.8t^2 = 0 \).
This is a quadratic equation in the form \( at^2 + bt + c = 0 \), where \( a = -0.8 \), \( b = 24 \), and \( c = 0 \).
Use the quadratic formula \( t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for 't'.
Calculate the discriminant \( b^2 - 4ac \) and then find the two possible values for 't'. The positive value will be the time the rock is aloft.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m
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주요 개념

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

Projectile Motion

Projectile motion refers to the motion of an object thrown or projected into the air, subject to only the acceleration of gravity. In this problem, the rock's motion is described by a quadratic equation, which models its vertical displacement over time on the moon, where gravity is weaker than on Earth.
추천 영상:
가이드 코스
06:51
Derivatives Applied To Acceleration Example 2

Quadratic Equations

Quadratic equations are polynomial equations of the form ax² + bx + c = 0. The height function s = 24t − 0.8t² is a quadratic equation representing the rock's height over time. Solving this equation for when s = 0 will determine the time when the rock returns to the moon's surface, indicating how long it is aloft.
추천 영상:
5:02
Solving Logarithmic Equations

Roots of Equations

Finding the roots of an equation involves determining the values of the variable that make the equation true, typically where the function equals zero. For the height equation s = 24t − 0.8t², the roots represent the times when the rock is at ground level. Solving for these roots will reveal the duration the rock remains in the air.
추천 영상:
5:02
Solving Logarithmic Equations
관련 실천
교과서 질문

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


e. When is it moving fastest (highest speed)? Slowest?


s = 4 - 7t + 6t² - t³, 0 ≤ t ≤ 4

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교과서 질문

Suppose that functions ƒ(x) and g(x) and their first derivatives have the following values at x = 0 and x = 1.


x ƒ(x) g(x) ƒ'(x) g'(x)

0 1 1 -3 1/2

1 3 5 1/2 -4


Find the first derivatives of the following combinations at the given value of x.


d. ƒ(g(x)), x = 0

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교과서 질문

Suppose that functions f and g and their derivatives with respect to x have the following values at x = 2 and x = 3.


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Find the derivatives with respect to x of the following combinations at the given value of x.


f. √f(x), x = 2

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교과서 질문

Suppose that the functions f and g and their derivatives with respect to x have the following values at x = 0 and x = 1.


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Find the derivatives with respect to x of the following combinations at the given value of x.


f. (x¹¹ + f(x))⁻², x = 1

221
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교과서 질문

Analyzing Motion Using Graphs


[Technology Exercise] Exercises 31–34 give the position function s = f(t) of an object moving along the s-axis as a function of time t. Graph f together with the velocity function v(t) = ds/dt = f'(t) and the acceleration function a(t) = d²s/dt² = f''(t). Comment on the object’s behavior in relation to the signs and values of v and a. Include in your commentary such topics as the following:


f. When is it farthest from the axis origin?


s = t³ - 6t² + 7t, 0 ≤ t ≤ 4

223
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교과서 질문

Average single-family home prices P (in thousands of dollars) in Sacramento, California, are shown in the accompanying figure from the beginning of 2006 through the end of 2015.



d. During what year did home prices drop most rapidly and what is an estimate of this rate?

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