Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook

Briggs 3rd Edition

Ch. 4 - Applications of the Derivative

Problem 4.8.37a

Chapter 4, Problem 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.

Verified step by step guidance

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.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

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.

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.

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.

Recommended video:

5:02

Solving Logarithmic Equations

Related Practice

Textbook Question

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.

283

views

Textbook Question

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,∞).

234

views

Textbook Question

{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.

235

views

Textbook Question

{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.

997

views

Textbook Question

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>

210

views

Textbook Question

{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.

202

views