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Ch. 4 - Applications of Derivatives
Hass - Thomas' Calculus 15th Edition
Hass15th EditionThomas' CalculusISBN: 9780137616077Not the one you use?Change textbook
All textbooksHass 15th EditionCh. 4 - Applications of DerivativesProblem 4.5.52a
Chapter 4, Problem 4.5.52a

52. Two masses hanging side by side from springs have positions s_1 = 2 sin t and s_2 = sin 2t,
respectively.

a. At what times in the interval 0 < t do the masses pass each other? (Hint: sin 2t = 2 sint cost.)

Verified step by step guidance
1
To find the times when the masses pass each other, we need to set their positions equal: s_1 = s_2. Given s_1 = 2 sin t and s_2 = sin 2t, we equate them: 2 sin t = sin 2t.
Use the hint provided: sin 2t = 2 sin t cos t. Substitute this into the equation: 2 sin t = 2 sin t cos t.
Simplify the equation by dividing both sides by 2 sin t, assuming sin t is not zero: 1 = cos t.
Solve the equation 1 = cos t for t in the interval 0 < t. The cosine function equals 1 at t = 0, but we need to find other solutions within the interval.
Since cos t = 1 at t = 0 and t = 2π, within the interval 0 < t, the solution is t = 2π. However, check for other possible solutions within the given interval.

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Key Concepts

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

Trigonometric Functions

Trigonometric functions, such as sine and cosine, are fundamental in calculus and physics, representing periodic phenomena. In this problem, the positions of the masses are given by sine functions, which oscillate between -1 and 1. Understanding how these functions behave over time is crucial for determining when the two masses pass each other.
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Equating Positions

To find when the two masses pass each other, we need to set their position equations equal to each other: s_1 = s_2. This involves solving the equation 2 sin(t) = sin(2t). This step is essential as it translates the physical scenario into a mathematical problem that can be solved using algebraic techniques.
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Double Angle Identity

The double angle identity for sine states that sin(2t) = 2 sin(t) cos(t). This identity simplifies the equation we need to solve, allowing us to express sin(2t) in terms of sin(t) and cos(t). Utilizing this identity is a key step in solving the problem, as it helps to reduce the complexity of the equation.
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Related Practice
Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫√(2x + 1) dx = √(x² + x +C)

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Textbook Question

Identifying Extrema


In Exercises 53–60:


a. Find the local extrema of each function on the given interval, and say where they occur.


f(x) = x / 2 − 2sin (x/2), 0 ≤ x ≤ 2π

Textbook Question

Theory and Examples


Sketch the graph of a differentiable function y = f(x) that has a local minimum at (1, 1) and a local maximum at (3, 3).

Textbook Question

Checking Antiderivative Formulas


Right, or wrong? Say which for each formula and give a brief reason for each answer.


∫tanθ sec²θ dθ = sec³θ / 3 + C

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Textbook Question

Identifying Extrema


In Exercises 63 and 64, the graph of f' is given. Assume that f has domain (-2, 2).


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a. Either use the graph to determine which intervals f is increasing on and which intervals f is decreasing on, or explain why this information cannot be determined from the graph.

Textbook Question

Finding Antiderivatives

In Exercises 1–16, find an antiderivative for each function. Do as many as you can mentally. Check your answers by differentiation.

1 / x²

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