Ch. 4 - Applications of the Derivative

Briggs - Calculus: Early Transcendentals 3rd Edition

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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.

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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.

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Textbook Question

Rectangles beneath a line

a. A rectangle is constructed with one side on the positive x-axis, one side on the positive y-axis, and the vertex opposite the origin on the line y = 10 - 2x. What dimensions maximize the area of the rectangle? What is the maximum area?

Textbook Question

Do dogs know calculus? A mathematician stands on a beach with his dog at point A. He throws a tennis ball so that it hits the water at point B. The dog, wanting to get to the tennis ball as quickly as possible, runs along the straight beach line to point D and then swims from point D to point B to retrieve his ball. Assume C is the point on the edge of the beach closest to the tennis ball (see figure). <IMAGE>

a. Assume the dog runs at speed r and swims at speed s, where r > s and both are measured in meters per second. Also assume the lengths of BC, CD, and AC are x, y, and z, respectively. Find a function T(y) representing the total time it takes for the dog to get to the ball.

Textbook Question

Sketch a graph of a function f with the following properties.

f' < 0 and f" < 0, for x < -1

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.

Textbook Question

107–110. {Use of Tech} Motion with gravity Consider the following descriptions of the vertical motion of an object subject only to the acceleration due to gravity. Begin with the acceleration equation a(t) = v' (t) = -g , where g = 9.8 m/s² .

a. Find the velocity of the object for all relevant times.

A payload is released at an elevation of 400 m from a hot-air balloon that is rising at a rate of 10 m/s.

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Textbook Question

{Use of Tech} Fixed points of quadratics and quartics Let f(x) = ax(1 -x), where a is a real number and 0 ≤ a ≤ 1. Recall that the fixed point of a function is a value of x such that f(x) = x (Exercises 48–51).

a. Without using a calculator, find the values of a, with 0 ≤ a ≤ 4, such that f has a fixed point. Give the fixed point in terms of a.