Textbook Question
Finding Critical Points
In Exercises 41–50, determine all critical points and all domain endpoints for each function.
g(x) = √(2x − x²)
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 4.7.77
Initial Value Problems
Solve the initial value problems in Exercises 71–90.
ds/dt = 1 + cos t, s(0) = 4
Verified step by step guidance1
Identify the given differential equation and initial condition: \( \frac{ds}{dt} = 1 + \cos t \) with \( s(0) = 4 \).
Recognize that this is a first-order ordinary differential equation where \( s(t) \) is the unknown function and \( t \) is the independent variable.
Integrate both sides of the equation with respect to \( t \) to find \( s(t) \):
\[ s(t) = \int (1 + \cos t) \, dt + C \]
Compute the integral on the right-hand side by integrating each term separately:
\[ \int 1 \, dt = t \]
and
\[ \int \cos t \, dt = \sin t \]
Use the initial condition \( s(0) = 4 \) to solve for the constant of integration \( C \) by substituting \( t = 0 \) into the expression for \( s(t) \) and setting it equal to 4.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Differential Equations
A differential equation relates a function with its derivatives. Solving it means finding the original function that satisfies the given relationship between the function and its rate of change.
Recommended video:
07:39
Classifying Differential Equations
Initial Value Problem (IVP)
An IVP specifies a differential equation along with a condition that the solution must satisfy at a particular point, such as s(0) = 4. This condition helps determine the unique solution to the differential equation.
Recommended video:
05:03Initial Value Problems
Integration of Trigonometric Functions
Solving ds/dt = 1 + cos t requires integrating the right-hand side with respect to t. Knowing how to integrate basic trigonometric functions like cosine is essential to find the general solution.
Recommended video:
6:04Introduction to Trigonometric Functions
Related Practice
Textbook Question
119. Find the values of constants a, b, and c such that the graph of y = ax^3 + bx^2 + cx has a
local maximum at x = 3, local minimum at x =- 1, and inflection point at (1, 11).
Textbook Question
Sketch the graphs of the rational functions in Exercises 53–60.
𝓍⁴ ― 1
y = ------------------
𝓍²
Textbook Question
Theory and Examples
In Exercises 53 and 54, show that the function has neither an absolute minimum nor an absolute maximum on its natural domain.
y = 3x + tan x
Textbook Question
41. Among all triangles in the first quadrant formed by the x-axis, the y-axis, and tangent lines to the graph of y=3x-x^2, what is the smallest possible area?
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.
∫(8y − 2 / y¹ᐟ⁴) dy