Domanda del libro di testo
Suppose S = x + 2y is an objective function subject to the constraint xy = 50, for x > 0 and y > 0.
b. Find the absolute minimum value of S subject to the given constraint.
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Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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TOC.tableOfContents
- Ch. 1 - Functions210
- Ch. 2 - Limits371
- Ch. 3 - Derivatives633
- Ch. 4 - Applications of the Derivative412
- Ch. 5 - Integration328
- Ch. 6 - Applications of Integration331
- Ch. 7 - Logarithmic, Exponential Functions, and Hyperbolic Functions177
- Ch. 8 - Integration Techniques394
- Ch. 9 - Differential Equations179
- Ch. 10 - Sequences and Infinite Series339
- Ch. 11 - Power Series218
- Ch.12 - Parametric and Polar Curves215
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Non è quello che usi tu?Cambia libro di testo
Capitolo 4, Problema 4.8.37a
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
a. Find the time at which the object first passes the rest position, y = 0.
Guida verificata passo dopo passo1
Identify the function representing the displacement: y(t) = 2.5e^(-t) * cos(2t).
Set the displacement function equal to zero to find when the object first passes the rest position: 2.5e^(-t) * cos(2t) = 0.
Since 2.5e^(-t) is never zero for any real value of t, focus on solving cos(2t) = 0.
Recall that cos(θ) = 0 at odd multiples of π/2, i.e., θ = (2n+1)π/2 for n being an integer.
Set 2t = (2n+1)π/2 and solve for t to find the first positive time when the object passes the rest position. Start with n = 0 to find the smallest positive t.
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Damped Oscillator
A damped oscillator is a system in which the amplitude of oscillation decreases over time due to energy loss, often modeled by an exponential decay function. In the given equation, the term '2.5e⁻ᵗ' represents this damping effect, indicating that the displacement diminishes as time progresses.
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03:07
Cases Where Limits Do Not Exist
Trigonometric Functions
The cosine function, represented as 'cos 2t' in the equation, is a periodic function that describes oscillatory motion. It oscillates between -1 and 1, and its argument '2t' indicates the frequency of oscillation, which affects how quickly the object moves up and down.
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Percorso guidato
6:04Introduction to Trigonometric Functions
Finding Roots of Equations
To find when the object first passes the rest position (y = 0), we need to solve the equation '2.5e⁻ᵗ cos 2t = 0'. This involves determining the values of 't' for which the cosine function equals zero, as the exponential term is never zero. The roots of the cosine function occur at specific intervals, which can be calculated to find the desired time.
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5:02Solving Logarithmic Equations
Pratica correlata
Domanda del libro di testo
Explain why or why not Determine whether the following statements are true and give an explanation or counterexample.
a. The function f(x) = √x has a local maximum on the interval [0,∞).
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Domanda del libro di testo
{Use of Tech} A damped oscillator The displacement of an object as it bounces vertically up and down on a spring is given by y(t) = 2.5e⁻ᵗ cos 2t, where the initial displacement is y(0) = 2.5 and y = 0 corresponds to the rest position (see figure). <IMAGE>
b. Find the time and the displacement when the object reaches its lowest point.
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Domanda del libro di testo
{Use of Tech} Elliptic curves The equation y² = x³ - ax + 3, where a is a parameter, defines a well-known family of elliptic curves.
a. Plot a graph of the curve when a = 3.
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Domanda del libro di testo
Two poles of heights m and n are separated by a horizontal distance d. A rope is stretched from the top of one pole to the ground and then to the top of the other pole. Show that the configuration that requires the least amount of rope occurs when Θ₁ = Θ₂ (see figure). <IMAGE>
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Domanda del libro di testo
{Use of Tech} Every second counts You must get from a point P on the straight shore of a lake to a stranded swimmer who is 50 from a point Q on the shore that is 50 m from you (see figure). Assuming that you can swim at a speed of 2 m/s and run at a speed of 4 m/s, the goal of this exercise is to determine the point along the shore, x meters from Q, where you should stop running and start swimming to reach the swimmer in the minimum time. <IMAGE>
a. Find the function T that gives the travel time as a function of x, where 0 ≤ x ≤ 50.
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