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Ch. 3 - Derivatives
Briggs - Calculus: Early Transcendentals 3rd Edition
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243Not the one you use?Change textbook
All textbooksBriggs 3rd EditionCh. 3 - DerivativesProblem 3.7.101a
Chapter 3, Problem 3.7.101a

{Use of Tech} A damped oscillator The displacement of a mass on a spring suspended from the ceiling is given by y=10e−t2cos(πt8)y=10e^{-\(\frac{t}{2}\)}\(\cos\[\left\)(\(\frac{\pi t}{8}\]\right\)).
a. Graph the displacement function. 

Verified step by step guidance
1
Identify the components of the displacement function: The function is y = 10e^{-t/2} * cos(πt/8). It consists of an exponential decay term 10e^{-t/2} and a cosine oscillation term cos(πt/8).
Understand the effect of each component: The exponential term 10e^{-t/2} represents damping, which means the amplitude of the oscillation decreases over time. The cosine term cos(πt/8) represents the oscillatory motion of the mass on the spring.
Determine the behavior of the function over time: As time t increases, the exponential term decreases, causing the overall amplitude of the oscillation to decrease. The cosine term will continue to oscillate between -1 and 1, but with a decreasing amplitude due to the damping effect.
Set up a graphing tool or software: Use a graphing calculator or software capable of plotting functions to visualize the displacement function. Input the function y = 10e^{-t/2} * cos(πt/8) into the tool.
Plot the function: Observe the graph to see how the displacement changes over time. Notice the oscillations with decreasing amplitude, which is characteristic of a damped oscillator. The graph should show a wave-like pattern that diminishes as time progresses.

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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 from friction or resistance. In the context of the given function, the term 'e^{-t/2}' represents the damping effect, indicating that the displacement will gradually diminish as time progresses.
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Exponential Function

An exponential function is a mathematical function of the form f(t) = a * e^{kt}, where 'e' is the base of natural logarithms, 'a' is a constant, and 'k' determines the rate of growth or decay. In the displacement function, the exponential term '10e^{-t/2}' signifies how the displacement decreases exponentially over time due to damping.
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Cosine Function

The cosine function is a periodic function that describes oscillatory motion, characterized by its amplitude, frequency, and phase shift. In the displacement equation, 'cos(πt/8)' indicates the oscillation of the mass, with a specific frequency that affects how quickly the mass moves back and forth, contributing to the overall behavior of the damped oscillator.
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Related Practice
Textbook Question

7–14. Find the derivative the following ways:

a. Using the Product Rule (Exercises 7–10) or the Quotient Rule (Exercises 11–14). Simplify your result.

g(s) = 4s³ - 8s² +4s / 4s

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If two opposite sides of a rectangle increase in length, how must the other two opposite sides change if the area of the rectangle is to remain constant?

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Textbook Question

31–32. Velocity functions A projectile is fired vertically upward into the air, and its position (in feet) above the ground after t seconds is given by the function s(t).

a. For the following functions s(t), find the instantaneous velocity function v(t). (Recall that the velocity function v is the derivative of the position function s.)

s(t)= −16t²+100t

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Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = -7x; P(-1,7)

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Use definition (2) (p. 135) to find the slope of the line tangent to the graph of f at P.

f(x) = 1/x; P (1,1)

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Use definition (1) (p. 133) to find the slope of the line tangent to the graph of f at P.

f(x) = 2/√x; P(4,1)

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