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 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 4.7.51
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.
∫(1 + tan²θ)dθ (Hint:1 + tan²θ = sec²θ)
Verified step by step guidance1
Recognize the integral given: \(\int (1 + \tan^{2}\theta) \, d\theta\).
Use the hint provided, which states that \(1 + \tan^{2}\theta = \sec^{2}\theta\). Substitute this into the integral to simplify it to \(\int \sec^{2}\theta \, d\theta\).
Recall the basic integral formula: \(\int \sec^{2}x \, dx = \tan x + C\), where \(C\) is the constant of integration.
Apply this formula to the integral, replacing \(x\) with \(\theta\), so the antiderivative is \(\tan \theta + C\).
Verify your result by differentiating \(\tan \theta + C\) to check if it returns the original integrand \(1 + \tan^{2}\theta\).
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45sKey Concepts
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Indefinite Integral
An indefinite integral represents the most general antiderivative of a function, including a constant of integration. It is written without limits and describes a family of functions whose derivative is the integrand.
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Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values in their domains. For example, the identity 1 + tan²θ = sec²θ simplifies integrals by rewriting expressions into more manageable forms.
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Verification by Differentiation
After finding an antiderivative, differentiating it should return the original integrand. This step confirms the correctness of the indefinite integral and ensures no errors were made during integration.
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Related Practice
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
22. A window is in the form of a rectangle surmounted by a semicircle. The rectangle is of clear glass, whereas the semicircle is of tinted glass that transmits only half as much light per unit area as clear glass does. The total perimeter is fixed. Find the proportions of the window that will admit the most light. Neglect the thickness of the frame.
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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) = x⁴ᐟ⁵, [0, 1]
Textbook Question
Theory and Examples
[Technology Exercise] Graph the functions in Exercises 63–66. Then find the extreme values of the function on the interval and say where they occur.
h(x) = |x + 2| − |x − 3|, −∞ < x < ∞