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,∞).
Ch. 4 - Applications of the Derivative
Briggs3rd EditionCalculus: Early TranscendentalsISBN: 9780136847243
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Table of contents
- 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: 9780136847243Not the one you use?Change textbook
Chapter 4, Problem 4.8.37b
{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.
Verified step by step guidance1
To find the time when the object reaches its lowest point, we need to find the critical points of the function y(t) = 2.5e^(-t) cos(2t). This involves taking the derivative of y(t) with respect to t.
Apply the product rule to differentiate y(t) = 2.5e^(-t) cos(2t). The product rule states that if you have a function h(t) = u(t)v(t), then h'(t) = u'(t)v(t) + u(t)v'(t). Here, u(t) = 2.5e^(-t) and v(t) = cos(2t).
Differentiate u(t) = 2.5e^(-t) to get u'(t) = -2.5e^(-t). Differentiate v(t) = cos(2t) to get v'(t) = -2sin(2t) using the chain rule.
Substitute the derivatives into the product rule: y'(t) = (-2.5e^(-t))cos(2t) + (2.5e^(-t))(-2sin(2t)). Simplify this expression to find y'(t).
Set y'(t) = 0 to find the critical points. Solve the resulting equation for t to find the times when the object reaches its lowest point. Evaluate y(t) at these times to find the corresponding displacements.
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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. The displacement function typically includes an exponential decay factor, which represents this loss of energy. In the given equation, the term '2.5e⁻ᵗ' indicates that the oscillation's amplitude diminishes as time progresses.
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Cases Where Limits Do Not Exist
Cosine Function in Oscillations
The cosine function is fundamental in describing periodic motion, such as oscillations. In the equation y(t) = 2.5e⁻ᵗ cos 2t, the 'cos 2t' part represents the oscillatory behavior of the system, where '2t' indicates the frequency of oscillation. The cosine function oscillates between -1 and 1, determining the position of the object at any given time.
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Finding Extrema
To find the lowest point of the oscillation, we need to determine the extrema of the displacement function. This involves taking the derivative of y(t) with respect to time, setting it to zero to find critical points, and then evaluating these points to identify the minimum displacement. The lowest point corresponds to the maximum negative value of the displacement function.
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05:58Finding Extrema Graphically
Related Practice
Textbook Question
{Use of Tech} Basketball shot A basketball is shot with an initial velocity of v ft/s at an angle of 45° to the floor. The center of the basketball is 8 ft above the floor at a horizontal distance of 18 feet from the center of the basketball hoop when it is released. The height h (in feet) of the center of the basketball after it has traveled a horizontal distance of x feet is modeled by the function h(x) = 32x² / v² + x + 8 (see figure). <IMAGE>
b. During the flight of the basketball, show that the distance s from the center of the basketball to the front of the hoop is s = √ (x - 17.25)² + ( -(4x² / 81) + x - 2)² (Hint: The diameter of the basketball hoop is 18 inches.)
Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
a. Evaluate g(2), h(2), g'(2), and h'(2).
Textbook Question
Values of related functions Suppose f is differentiable on (-∞,∞) and assume it has a local extreme value at the point x = 2, where f(2) = 0. Let g(x) = xf(x) + 1 and let h(x) = xf(x) + x +1, for all values of x.
b. Does either g or h have a local extreme value at x = 2? Explain.
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.
Textbook Question
{Use of Tech} Let f(x) = ln((x+1)/(x-1)) and g(x) = ln ((x+1)/(x-1)).
b. Sketch graphs of f and g to show that these functions do not differ by a constant.