Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x − 3x²ᐟ³
Ch. 4 - Applications of Derivatives
Hass15th EditionThomas' CalculusISBN: 9780137616077
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Table of contents
- Ch. 1 - Functions155
- Ch. 2 - Limits and Continuity240
- Ch. 3 - Derivatives337
- Ch. 4 - Applications of Derivatives293
- Ch. 5 - Integrals41
- Ch. 6 - Applications of Definite Integrals25
- Ch. 7 - Transcendental Functions489
- Ch. 8 - Techniques of Integration398
- Ch. 9 - First-Order Differential Equations60
- Ch. 10 - Infinite Sequences and Series119
- Ch. 11 - Parametric Equations and Polar Coordinates58
Chapter 4, Problem 4.7.85
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²r/dt² = 2/t³; dr/dt|ₜ ₌ ₁ =1, r(1) = 1
Verified step by step guidance1
Identify the given differential equation and initial conditions: \( \frac{d^{2}r}{dt^{2}} = \frac{2}{t^{3}} \), with \( \frac{dr}{dt}\bigg|_{t=1} = 1 \) and \( r(1) = 1 \).
Integrate the second derivative \( \frac{d^{2}r}{dt^{2}} \) with respect to \( t \) to find the first derivative \( \frac{dr}{dt} \). This means computing \( \int \frac{2}{t^{3}} \, dt \). Remember to add an integration constant \( C_1 \).
Use the initial condition \( \frac{dr}{dt}\big|_{t=1} = 1 \) to solve for the constant \( C_1 \) after finding the general form of \( \frac{dr}{dt} \).
Integrate the expression for \( \frac{dr}{dt} \) with respect to \( t \) to find \( r(t) \). Again, include a second integration constant \( C_2 \).
Apply the initial condition \( r(1) = 1 \) to solve for \( C_2 \), completing the solution for \( r(t) \).
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Second-Order Differential Equations
These are differential equations involving the second derivative of a function. Solving them requires integrating twice and applying initial conditions to find the specific solution that fits the problem.
Recommended video:
07:39
Classifying Differential Equations
Initial Value Problems (IVP)
An IVP specifies the value of the function and its derivatives at a particular point. This information is used to determine the constants of integration after solving the differential equation.
Recommended video:
05:03Initial Value Problems
Integration Techniques for Variable Coefficients
When the differential equation has terms like 2/t³, integration involves handling variable coefficients carefully, often requiring substitution or direct integration of power functions.
Recommended video:
07:15Separation of Variables
Related Practice
Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
f(x) = x³ + 4x² + 7, (−∞, 0)
Textbook Question
In Exercises 9–66, graph the function using appropriate methods from the graphing procedures presented just before Example 9, identifying the coordinates of any local extreme points and inflection points. Then find coordinates of absolute extreme points, if any.
53. y = x * √(8 - x²)
Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
y = x² + 2/x
Textbook Question
In Exercises 1–10, find the extreme values (absolute and local) of the function over its natural domain, and where they occur.
______
y = √𝓍² ― 1
Textbook Question
Identifying Extrema
In Exercises 15–18:
a. Find the open intervals on which the function is increasing and those on which it is decreasing.
b. Identify the function’s local and absolute extreme values, if any, saying where they occur.