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
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 Functions490
- Ch. 8 - Techniques of Integration399
- Ch. 9 - First-Order Differential Equations66
- Ch. 10 - Infinite Sequences and Series132
- 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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6mKey Concepts
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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.
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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.
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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.
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Related Practice
Textbook Question
113. If b, c, and d are constants, for what value of b will the curve y = x^3 + bx^2 + cx + d have a
point of inflection at x = 1? Give reasons for your answer.
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.
Textbook Question
Checking the Mean Value Theorem
Find the value or values of c that satisfy the equation (f(b) − f(a)) / (b − a) = f′(c) in the conclusion of the Mean Value Theorem for the functions and intervals in Exercises 1–6.
g(x) = {x³, −2 ≤ x ≤ 0
x², 0 < x ≤ 2
Textbook Question
Finding Indefinite Integrals
In Exercises 17–56, find the most general antiderivative or indefinite integral. You may need to try a solution and then adjust your guess. Check your answers by differentiation.
∫(√x + ³√x) dx
Textbook Question
37. What value of a makes f(x) = x^2 +(a/x) have
a. a local minimum at x = 2?
b. a point of inflection at x = 1?