교과서 질문
Theory and Examples
Determine the values of constants a and b so that f(x) = ax² + bx has an absolute maximum at the point (1,2).
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Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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TOC.tableOfContents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- Ch. 11 - Parametric Equations and Polar Coordinates58
4장, 문제 4.7.102
Applications
Liftoff from Earth A rocket lifts off the surface of Earth with a constant acceleration of 20 m/sec². How fast will the rocket be going 1 min later?
검증된 단계별 안내1
Identify the known quantities: the acceleration \(a = 20 \ \text{m/s}^2\) and the time \(t = 1 \ \text{minute}\). Convert the time to seconds since acceleration is in meters per second squared. So, \(t = 60 \ \text{seconds}\).
Recall the kinematic equation that relates velocity, acceleration, and time when starting from rest: \(v = v_0 + a t\). Since the rocket starts from rest, the initial velocity \(v_0 = 0\).
Substitute the known values into the equation: \(v = 0 + (20)(60)\).
Simplify the expression to find the velocity after 60 seconds, which will give the speed of the rocket 1 minute after liftoff.
Interpret the result as the rocket's speed in meters per second at 1 minute after liftoff.
비슷한 문제에 대한 검증된 영상 답변:
이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m주요 개념
질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.
Constant Acceleration
Constant acceleration means the velocity of an object changes at a steady rate over time. In this problem, the rocket's acceleration is 20 m/s², which means its speed increases by 20 meters per second every second.
추천 영상:
가이드 코스
06:15
Derivatives Applied To Acceleration
Velocity-Time Relationship
When acceleration is constant, velocity changes linearly with time. The velocity after a certain time can be found using the formula v = at, where 'a' is acceleration and 't' is time elapsed.
추천 영상:
가이드 코스
06:29Derivatives Applied To Velocity
Unit Conversion and Time Calculation
Time must be expressed in consistent units when applying formulas. Since acceleration is in meters per second squared and time is given in minutes, converting 1 minute to 60 seconds is essential for accurate calculation.
추천 영상:
5:10Evaluate Composite Functions - Values on Unit Circle
관련 실천
교과서 질문
Theory and Examples
67. An inequality for positive integers Show that if a, b, c, and d are positive integers, then
[(a^2+1)(b^2+1)(c^2+1)(d^2+1)]/abcd ≥ 16
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교과서 질문
Identifying Extrema
In Exercises 19–40:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local extreme values, if any, saying where they occur.
f(x) = (x² − 3) / (x − 2), x ≠ 2
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교과서 질문
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = √(x(1 − x)), [0, 1]
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교과서 질문
93. The accompanying figure shows a portion of the graph of a twice-differentiable function y=f(x). At each of the five labeled points, classify y' and \(\y\)'' as positive, negative, or zero.
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교과서 질문
32. Answer Exercise 31 if one piece is bent into a square and the other into a circle.
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