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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.PE.92
Chapter 4, Problem 4.PE.92

Initial Value Problems
Solve the initial value problems in Exercises 89–92.
d^3 r/dt^3 = - cos t; r''(0) = r'(0) = 0 , r(0) = -1

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Identify the given differential equation and initial conditions: \(\frac{d^3 r}{dt^3} = -\cos t\), with \(r''(0) = 0\), \(r'(0) = 0\), and \(r(0) = -1\).
Integrate the third derivative \(\frac{d^3 r}{dt^3} = -\cos t\) once with respect to \(t\) to find the second derivative \(r''(t)\). Remember to add an integration constant \(C_1\).
Integrate \(r''(t)\) to find the first derivative \(r'(t)\), adding another integration constant \(C_2\).
Integrate \(r'(t)\) to find the original function \(r(t)\), adding a third integration constant \(C_3\).
Use the initial conditions \(r''(0) = 0\), \(r'(0) = 0\), and \(r(0) = -1\) to set up equations and solve for the constants \(C_1\), \(C_2\), and \(C_3\).

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Key Concepts

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

Higher-Order Differential Equations

Higher-order differential equations involve derivatives of order two or more. In this problem, the third derivative of r with respect to t is given, requiring integration multiple times to find the original function r(t). Understanding how to reduce the order by successive integration is essential.
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Higher Order Derivatives

Initial Conditions in Differential Equations

Initial conditions specify the values of a function and its derivatives at a particular point, allowing for the determination of integration constants. Here, values of r(0), r'(0), and r''(0) are given, which help uniquely solve for the constants after integrating the differential equation.
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Solutions to Basic Differential Equations

Integration of Trigonometric Functions

Solving the differential equation requires integrating the function -cos(t) multiple times. Familiarity with the integrals of sine and cosine functions, including their signs and constants of integration, is crucial to correctly find r(t) from its third derivative.
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Introduction to Trigonometric Functions
Related Practice
Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 1/( r + 5)²dr

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ 𝓍³ (1 + 𝓍⁴ )⁻¹/⁴ d𝓍

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ (𝓍³ + 5𝓍 ―7) d𝓍

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) = sin x − cos x,0 ≤ x ≤ 2π

Textbook Question

Identifying Extrema


In Exercises 41–52:


a. Identify the function’s local extreme values in the given domain, and say where they occur.


f(x) = (x + 1)², −∞ < x ≤ 0

Textbook Question

Finding Indefinite Integrals

Find the indefinite integrals (most general antiderivatives) in Exercises 73–88. You may need to try a solution and then adjust your guess. Check your answers by differentiation.

∫ sec θ/3 tan θ/3 dθ