Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
f(x) = x(4 − x)³
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 4.7.83
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
d²y/dx² = 2 − 6x; y′(0) = 4, y(0) = 1
Verified step by step guidance1
Identify the given differential equation and initial conditions: \( \frac{d^2 y}{dx^2} = 2 - 6x \), with \( y'(0) = 4 \) and \( y(0) = 1 \).
Integrate the second derivative \( \frac{d^2 y}{dx^2} \) once with respect to \( x \) to find the first derivative \( y'(x) \). Remember to add an integration constant \( C_1 \):
\[ y'(x) = \int (2 - 6x) \, dx = 2x - 3x^2 + C_1 \]
Use the initial condition \( y'(0) = 4 \) to solve for the constant \( C_1 \) by substituting \( x = 0 \) into the expression for \( y'(x) \).
Integrate \( y'(x) \) with respect to \( x \) to find \( y(x) \), adding another integration constant \( C_2 \):
\[ y(x) = \int (2x - 3x^2 + C_1) \, dx = x^2 - x^3 + C_1 x + C_2 \]
Use the initial condition \( y(0) = 1 \) to solve for \( C_2 \) by substituting \( x = 0 \) into the expression for \( y(x) \).
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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
A second-order differential equation involves the second derivative of a function. Solving it requires integrating twice, often leading to a general solution with two arbitrary constants. Understanding how to handle these equations is essential for modeling systems with acceleration or curvature.
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Classifying Differential Equations
Initial Conditions
Initial conditions specify the values of a function and its derivatives at a particular point, allowing us to find the unique solution to a differential equation. For example, y(0) = 1 and y′(0) = 4 provide specific values to determine the constants after integration.
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05:03Initial Value Problems
Integration Techniques
Solving the given differential equation requires integrating the right-hand side function step-by-step. Mastery of basic integration rules and applying them correctly to find the antiderivatives is crucial to progress from the differential equation to the explicit solution.
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06:18Integration by Parts for Definite Integrals
Related Practice
Textbook Question
Estimate the open intervals on which the function y = ƒ(𝓍) is
a. increasing.
b. decreasing.
c. Use the given graph of ƒ' to indicate where any local extreme
values of the function occur, and whether each extreme
is a relative maximum or minimum.
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Textbook Question
Roots (Zeros)
Show that the functions in Exercises 19–26 have exactly one zero in the given interval.
r(θ) = 2θ − cos²θ + √2, (−∞, ∞)
Textbook Question
Finding Extrema from Graphs
In Exercises 15–20, sketch the graph of each function and determine whether the function has any absolute extreme values on its domain. Explain how your answer is consistent with Theorem 1.
g(x) = {−x, 0 ≤ x < 1
x − 1, 1 ≤ x ≤ 2
Textbook Question
Identify the inflection points and local maxima and minima of the functions graphed in Exercises 1–8. Identify the open intervals on which the functions are differentiable and the graphs are concave up and concave down.
5. y=x+sin(2x), -2π/3≤x≤2π/3
Textbook Question
Checking the Mean Value Theorem
Which of the functions in Exercises 7–12 satisfy the hypotheses of the Mean Value Theorem on the given interval, and which do not? Give reasons for your answers.
f(x) = {2x − 3, 0 ≤ x ≤ 2
6x − x² − 7, 2 < x ≤ 3