Textbook Question
Finding Extreme Values
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
y = 𝓍³ ― 2𝓍 + 4
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- 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
Chapter 4, Problem 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?
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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.
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2mKey Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
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.
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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.
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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.
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Related Practice
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² + 2/x
Textbook Question
54. Fermat’s principle in optics Light from a source A is reflected by a plane mirror to a receiver at point B, as shown in the accompanying figure. Show that for the light to obey Fermat’s principle, the angle of incidence must equal the angle of reflection, both measured from the line normal to the reflecting surface. (This result can also be derived without calculus. There is a purely geometric argument, which you may prefer.)
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Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.