Domanda del libro di testo
62–65. {Use of Tech} Graphing f and f'
c. Verify that the zeros of f' correspond to points at which f has a horizontal tangent line.
f(x)=(x²−1)sin^−1 x on [−1,1]
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Ch. 3 - Derivatives
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 3, Problema 3.6.55c
{Use of Tech} Spring oscillations A spring hangs from the ceiling at equilibrium with a mass attached to its end. Suppose you pull downward on the mass and release it 10 inches below its equilibrium position with an upward push. The distance x (in inches) of the mass from its equilibrium position after t seconds is given by the function x(t) = 10sin t−10cos t, where x is positive when the mass is above the equilibrium position. <IMAGE>
c. At what times is the velocity of the mass zero?
Guida verificata passo dopo passo1
To find when the velocity of the mass is zero, we first need to determine the velocity function. The velocity is the derivative of the position function x(t) with respect to time t.
Given the position function x(t) = 10sin(t) - 10cos(t), we differentiate it to find the velocity function v(t). The derivative of sin(t) is cos(t), and the derivative of cos(t) is -sin(t).
Applying these derivatives, the velocity function v(t) is: v(t) = d/dt [10sin(t) - 10cos(t)] = 10cos(t) + 10sin(t).
To find when the velocity is zero, we set the velocity function equal to zero: 10cos(t) + 10sin(t) = 0.
Solve the equation 10cos(t) + 10sin(t) = 0 for t. This involves finding the values of t where the sum of the cosine and sine terms equals zero, which can be done by using trigonometric identities or algebraic manipulation.
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Velocity and Its Calculation
Velocity is the rate of change of position with respect to time. In calculus, it is found by taking the derivative of the position function. For the given function x(t) = 10sin(t) - 10cos(t), the velocity function v(t) can be determined by differentiating x(t) with respect to t, which will provide the times when the mass is momentarily at rest.
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Percorso guidato
06:29
Derivatives Applied To Velocity
Finding Critical Points
Critical points occur where the derivative of a function is zero or undefined. In the context of the velocity function, setting v(t) = 0 will help identify the times when the mass's velocity is zero, indicating moments when the mass changes direction. This is essential for understanding the oscillatory motion of the mass attached to the spring.
Video consigliato:
04:50Critical Points
Trigonometric Functions and Their Properties
The position function x(t) involves sine and cosine functions, which are periodic and have specific properties regarding their maxima, minima, and zeros. Understanding the behavior of these trigonometric functions is crucial for analyzing the motion of the mass, particularly in determining the intervals of time when the velocity is zero, as these functions oscillate between -1 and 1.
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Percorso guidato
06:21Properties of Functions
Pratica correlata
Domanda del libro di testo
Suppose a stone is thrown vertically upward from the edge of a cliff on Earth with an initial velocity of 64 ft/s from a height of 32 ft above the ground. The height (in feet) of the stone above the ground t seconds after it is thrown is s(t) = -16t²+64t+32.
c. What is the height of the stone at the highest point?
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Domanda del libro di testo
Derivatives from a graph If possible, evaluate the following derivatives using the graphs of f and f'. <IMAGE>
c. (f^-1)'(f(2))
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Domanda del libro di testo
A rectangular swimming pool 10 ft wide by 20 ft long and of uniform depth is being filled with water.
c. At what rate is the water level rising if the pool is filled at a rate of 10ft³/min?
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Domanda del libro di testo
Finding derivatives from a table Find the values of the following derivatives using the table. <IMAGE>
c. d/dx ((f(x)g(x)) |x=3
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Domanda del libro di testo
Witch of Agnesi Let y(x²+4)=8 (see figure). <IMAGE>
c. Solve the equation y(x²+4)=8 for y to find an explicit expression for y and then calculate dy/dx.
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